---
title: "Nonnormality in Control Systems, Transient Amplification, Covariance Propagation, and Their Role in Reinforcement Learning Control and Safe RL: A Comprehensive Academic Research Report"
date: "2026-05-23"
author: "Systematic Literature Review"
---

# Nonnormality in Control Systems, Transient Amplification, Covariance Propagation, and Their Role in Reinforcement Learning Control and Safe RL

## A Comprehensive Academic Research Report

**Date:** May 23, 2026

**Based on:** PRISMA-compliant systematic literature review of 1,122 papers with 20 included studies

---

## Table of Contents

1. [Executive Summary](#1-executive-summary)
2. [Concept Map](#2-concept-map)
3. [Literature Review by Theme](#3-literature-review-by-theme)
   - 3.1 Pseudospectra, Nonnormality, and Transient Growth
   - 3.2 Nonnormality in Feedback Control and Robust Control
   - 3.3 Stochastic Forcing, Covariance Amplification, and Covariance Control
   - 3.4 Koopman / EDMD / Data-Driven Surrogate and Stability Certificates
   - 3.5 Reinforcement Learning: Action Variance, Smooth Control, Residual RL, and Safe RL
4. [DOI Table](#4-doi-table)
5. [Research Gap Analysis](#5-research-gap-analysis)
6. [How to Position a New Paper](#6-how-to-position-a-new-paper)
7. [Final Summary](#7-final-summary)
   - Part A: Mature Knowledge
   - Part B: Unresolved Problems
   - Part C: Possible New Paper Contributions
8. [References](#references)

---


## 1. Executive Summary

Nonnormal matrices and operators arise throughout control systems when the eigenvectors of the linearized dynamics are non-orthogonal. A matrix $A$ is normal if $A A^* = A^* A$, meaning its eigenvectors form an orthonormal basis; when this condition fails, the system is nonnormal. The distinction matters because eigenvalue-based stability analysis—which guarantees asymptotic decay when all eigenvalues lie in the stable region—can severely underestimate transient behavior in nonnormal systems [5] [13]. Even when the spectral radius is less than one (discrete time) or all eigenvalues have negative real parts (continuous time), non-orthogonal eigenvectors enable temporary amplification of perturbations before eventual decay [16]. This transient growth is quantified by pseudospectra, level sets of the resolvent norm $\|(zI - A)^{-1}\|$ that reveal how perturbations at frequency $z$ are amplified [5] [13]. In fluid dynamics, pseudospectral analysis explains why linearly stable shear flows exhibit amplification factors exceeding $10^5$ and transition to turbulence despite stable eigenvalues [16]. The Kreiss constant, which bounds the maximum transient growth over all time, provides a scalar measure of nonnormality [1] [7].

The gap between asymptotic stability and transient behavior becomes critical under stochastic forcing. When a nonnormal system is driven by white noise or persistent random perturbations, the steady-state variance can vastly exceed the variance predicted by modal decay rates in an equivalent normal system with identical eigenvalues [9] [11]. Farrell and Ioannou demonstrated that stochastic excitation of the linearized Navier–Stokes equations produces high variance levels sustained by energy transfer from the mean flow to perturbations, despite exponential stability of all normal modes [11]. This amplification arises because the dominant forcing functions—solutions of a Lyapunov equation—are distinct from the system's eigenmodes and project energy onto response functions that exploit non-orthogonality [11] [15]. Bamieh and Dahleh quantified this effect analytically, showing that three-dimensional streamwise-constant disturbances in channel flow achieve $O(R^3)$ energy amplification, where $R$ is the Reynolds number [8]. The mechanism couples wall-normal velocity and vorticity disturbances through mean shear and spanwise variation, a consequence of operator nonnormality rather than near-resonance or algebraic modal growth [8]. Ioannou proved that variance amplification under unbiased stochastic forcing is a general property of nonnormal evolution operators, implying that atmospheric response to forcing is systematically underestimated when analysis is restricted to normal-mode decay [9].

This theoretical foundation has direct implications for safe reinforcement learning when learned policies are deployed on physical systems. Reinforcement learning produces policies with residual action variance from multiple sources: exploration noise during training, stochastic policy parameterizations, and learned residual corrections atop nominal controllers in residual RL architectures. When the closed-loop system—comprising plant dynamics and learned feedback—is nonnormal, this residual action variance acts as stochastic forcing that can be amplified into unsafe state excursions even when the nominal closed-loop eigenvalues guarantee asymptotic stability. The causal chain runs from learned action covariance (source) through the input channel into the nonnormal closed-loop dynamics (amplifier), producing amplified state covariance that may violate safety constraints defined by Lyapunov functions, control barrier functions, or probabilistic reach-avoid specifications. Grigoriev and Handel showed that localized control in spatially extended systems fails under extremely small noise due to nonnormality-induced sensitivity [2], and Higham and Mao demonstrated that systems stable without noise can become exponentially unstable in mean-square under vanishingly small noise perturbations as nonnormality increases [12]. These results suggest that RL policies nominally certified as stable may still produce unsafe behavior when residual action variance interacts with closed-loop nonnormality.

A systematic review of 1,122 papers yielded 20 included studies addressing components of this problem, but revealed that the complete causal chain from residual action covariance through nonnormal amplification to safety risk remains unaddressed. The RL literature treats safe RL, action smoothing, and residual RL as separate concerns. Safe RL methods enforce safety constraints via control barrier functions, constrained policy optimization, or Lyapunov-based certificates, but typically assume deterministic policies or do not explicitly model how action variance propagates through nonnormal dynamics [14]. Action smoothing techniques penalize temporal variation in control inputs to improve performance, but do not connect smoothing objectives to transient amplification or covariance propagation. Residual RL architectures learn corrective actions atop nominal controllers to improve performance while preserving stability guarantees, yet do not model the covariance structure of learned residuals or its interaction with closed-loop nonnormality. Data-driven surrogate methods such as Koopman operator theory and extended dynamic mode decomposition (EDMD) construct linear or bilinear approximations of nonlinear dynamics for model predictive control and certificate synthesis, but existing Koopman-based safe control frameworks focus on deterministic trajectories and do not incorporate stochastic forcing or covariance amplification analysis.

The control theory literature extensively studies nonnormality, transient growth, and covariance amplification, but primarily in the context of fluid dynamics, atmospheric science, and classical feedback design rather than learned control policies. Pseudospectral methods quantify transient growth and have been extended to structured perturbations [3], delay differential equations [6], and iterative learning control [4], but these tools are not integrated into RL training or deployment pipelines. Covariance control and $H_2$ optimal control minimize the trace of the steady-state covariance under stochastic forcing by solving Lyapunov equations, providing a natural framework for suppressing variance amplification, yet this approach has not been adapted to penalize learned residual action covariance in directions that excite nonnormal transients. Blanchard et al. developed optimally time-dependent (OTD) modes to control transient instabilities in high-dimensional systems, demonstrating superiority over eigenvalue-based modal control in fluid flows [14].

Only one paper in the systematic review partially addresses the intersection of these concerns. Yue et al. (details not provided in references but cited in the brief) consider input-side variance suppression under nonnormal transient amplification, but do not establish the complete framework linking learned residual action covariance as source, local closed-loop surrogate nonnormality as amplifier, and certificate-weighted state covariance or violation probability as risk metric. No existing work explicitly models the scenario where: (1) a learned RL policy produces residual actions with empirically measured or theoretically bounded covariance; (2) the closed-loop dynamics are approximated by a local linear or Koopman surrogate exhibiting nonnormality; (3) the residual action covariance is propagated through the nonnormal surrogate via Lyapunov equations to compute state covariance; and (4) a directional penalty of the form $\Phi_u^T P \Phi_u$, where $P$ is a certificate matrix (Lyapunov or barrier function Hessian) and $\Phi_u$ captures input-to-state covariance directions, is used to suppress variance in directions that threaten safety.

The research gap is therefore real and specific: existing literature separately studies action smoothing (temporal regularization without covariance propagation), safe RL (constraint enforcement without stochastic forcing analysis), residual RL (performance improvement without variance modeling), Koopman control (surrogate construction without covariance amplification), Lyapunov certificates (deterministic stability without transient stochastic growth), nonnormal transient amplification (classical theory without learned control), and covariance control (optimal $H_2$ design without RL integration). The missing synthesis is a unified framework that treats learned residual action covariance as a first-class design variable, explicitly accounts for nonnormal amplification in local closed-loop surrogates, and employs certificate-weighted covariance metrics to guide policy training toward safe operation under realistic stochastic conditions. This gap is particularly salient given recent results showing that nonnormal eigenvector amplification can produce heavy-tailed statistics and critical-like behavior in spectrally stable systems [10], suggesting that RL policies certified under deterministic or weakly stochastic assumptions may exhibit rare but severe safety violations when deployed.

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## 2. Concept Map

The causal chain linking residual action covariance to safety violations through nonnormal amplification comprises five interconnected nodes. Each node represents a distinct theoretical construct studied extensively within its own domain, yet the complete chain—from learned policy variance through nonnormal dynamics to probabilistic safety risk—remains unaddressed in existing literature. This section traces the theoretical foundations, representative literature, and inter-node connections that define this gap.

## Node 1: Residual Action Covariance

Residual action covariance quantifies the variance structure of control actions produced by learned policies. In reinforcement learning, this covariance arises from three primary sources. First, exploration noise injected during training—typically Gaussian perturbations added to deterministic policies or entropy regularization terms that encourage stochastic exploration—persists as residual variance even after convergence. Second, stochastic policy parameterizations such as Gaussian policies $\pi(a|s) = \mathcal{N}(\mu_\theta(s), \Sigma_\theta(s))$ explicitly model action distributions, where the covariance matrix $\Sigma_\theta(s)$ captures inherent policy uncertainty. Third, residual RL architectures learn corrective actions $\Delta u$ atop nominal controllers $u_{\text{nom}}$, producing total actions $u = u_{\text{nom}} + \Delta u$ where the learned residual $\Delta u$ exhibits covariance due to function approximation error, training stochasticity, and deliberate smoothing objectives.

The theoretical foundation rests on the representation of policies as probability distributions over action spaces. For continuous control, Gaussian policies dominate due to their tractability and compatibility with gradient-based optimization. The covariance structure $\Sigma_u$ of the action distribution determines how control effort is distributed across actuators and how temporal correlations in actions evolve. In residual RL, the nominal controller provides baseline stability while the learned residual improves performance, but the residual's covariance—shaped by neural network training dynamics, replay buffer sampling, and policy gradient estimators—becomes a stochastic forcing term when the policy is deployed.

Existing literature treats action variance primarily as a performance concern rather than a safety-critical design variable. Action smoothing techniques penalize temporal variation in control inputs to reduce wear, improve tracking, and stabilize learning, but do not model how action covariance propagates through system dynamics. Safe RL methods enforce constraints via control barrier functions or Lyapunov certificates, yet typically assume deterministic policies or treat stochasticity as a secondary perturbation rather than a first-class source of risk. Residual RL frameworks focus on preserving nominal stability guarantees while enhancing performance, but do not characterize the covariance of learned residuals or its interaction with closed-loop dynamics. The gap is that residual action covariance is not explicitly modeled as a source term in a variance propagation analysis that accounts for downstream amplification.

## Node 2: Input Channel

The input channel maps control actions to system inputs, transforming the action space into forces, torques, or other physical quantities that drive state evolution. In linear control theory, this mapping is represented by the input matrix $B$ in the state-space model $\dot{x} = Ax + Bu$, where $u \in \mathbb{R}^m$ is the control input and $B \in \mathbb{R}^{n \times m}$ defines how each actuator influences each state variable. The structure of $B$ encodes actuator placement, control authority, and the directional coupling between inputs and states. For example, in underactuated systems, $B$ has fewer columns than rows, concentrating control effort into a lower-dimensional subspace of the state space.

The theoretical foundation is rooted in controllability and input-output analysis. The controllability Gramian $W_c = \int_0^\infty e^{At} B B^T e^{A^T t} dt$ quantifies the reachable set from the origin under bounded control energy, with its eigenvectors revealing the directions in state space most easily influenced by inputs. The input matrix $B$ also determines how stochastic forcing enters the system: when actions are drawn from a distribution with covariance $\Sigma_u$, the effective forcing covariance in state space becomes $B \Sigma_u B^T$. This projection amplifies variance in directions aligned with the columns of $B$ and suppresses variance in orthogonal directions.

Existing literature extensively studies input design for controllability, optimal actuator placement, and robust control under input uncertainty. Gramian-based methods identify optimal sensor and actuator locations by maximizing controllability or observability measures [16]. However, these approaches focus on deterministic control objectives or worst-case disturbance rejection rather than on how learned action covariance propagates through $B$ into state covariance. The connection between residual action covariance (Node 1) and the input channel (Node 2) is that $B$ transforms the action covariance $\Sigma_u$ into a forcing covariance $B \Sigma_u B^T$ that drives state variance. This transformation is directional: if $\Sigma_u$ has large variance in directions poorly aligned with the columns of $B$, the resulting state forcing may be weak, whereas variance aligned with high-gain input directions produces strong state excitation.

## Node 3: Nonnormal Closed-Loop Amplifier

Nonnormal closed-loop dynamics arise when the closed-loop system matrix $A_{\text{cl}} = A + BK$, formed by combining plant dynamics $A$ and feedback gain $K$, has non-orthogonal eigenvectors. A matrix is normal if $A_{\text{cl}} A_{\text{cl}}^* = A_{\text{cl}}^* A_{\text{cl}}$, meaning its eigenvectors form an orthonormal basis. When this condition fails, the system is nonnormal, and eigenvalue-based stability analysis—which guarantees asymptotic decay when all eigenvalues have negative real parts—can severely underestimate transient behavior. Even when $A_{\text{cl}}$ is spectrally stable, non-orthogonal eigenvectors enable temporary amplification of perturbations before eventual decay.

The theoretical foundation rests on pseudospectral analysis and transient growth bounds. Pseudospectra, defined by level sets of the resolvent norm $\|(zI - A_{\text{cl}})^{-1}\|$, reveal how perturbations at frequency $z$ are amplified [1] [2]. The Kreiss constant $K(A_{\text{cl}}) = \sup_{t \geq 0} \|e^{A_{\text{cl}} t}\|$ bounds the maximum transient growth over all time [3]. For nonnormal systems, $K(A_{\text{cl}})$ can vastly exceed unity even when all eigenvalues are stable, indicating that initial perturbations or stochastic forcing can produce large transient excursions. Trefethen and Embree demonstrated that fluid flows with stable eigenvalues can exhibit large amplification factors due to nonnormality [2]. Grigoriev and Handel showed that localized control in spatially extended systems fails under extremely small noise due to nonnormality-induced sensitivity, with transient amplification and spectral focusing of disturbances playing crucial roles [4]. Grigoriev further examined these mechanisms in spatially extended systems [19].

Existing literature extensively studies nonnormality in fluid dynamics, atmospheric science, and classical feedback control. Pseudospectral methods quantify transient growth and have been extended to structured perturbations, delay differential equations, and iterative learning control [8]. Feedback design methods minimize transient energy growth by optimizing the Kreiss constant or solving linear matrix inequalities to bound $\|e^{A_{\text{cl}} t}\|$ [10] [12] [20]. Apkarian and Noll showed how feedback can reduce the Kreiss constant in closed loop, thereby reducing transients [3]. However, these methods focus on deterministic initial conditions or worst-case perturbations rather than on stochastic forcing with known covariance structure. The connection between the input channel (Node 2) and nonnormal amplification (Node 3) is that the forcing covariance $B \Sigma_u B^T$ enters the nonnormal dynamics $\dot{x} = A_{\text{cl}} x + B u$, where $u$ is the stochastic residual action. The nonnormality of $A_{\text{cl}}$ determines how this forcing covariance is amplified into state covariance.

## Node 4: State Covariance Propagation

State covariance propagation quantifies how input covariance is transformed into state covariance through linear dynamics. For a linear system $\dot{x} = A_{\text{cl}} x + B u$ driven by white noise $u$ with covariance $\Sigma_u$, the steady-state state covariance $\Sigma_x$ satisfies the continuous-time Lyapunov equation $A_{\text{cl}} \Sigma_x + \Sigma_x A_{\text{cl}}^T + B \Sigma_u B^T = 0$. The solution $\Sigma_x$ encodes the variance and cross-correlations of state variables under persistent stochastic forcing. The trace of $\Sigma_x$ measures total variance, while the eigenvalues and eigenvectors of $\Sigma_x$ reveal the dominant directions of state uncertainty.

The theoretical foundation is rooted in stochastic control and $H_2$ optimal control. The $H_2$ norm of the transfer function from input to output, $\|G\|_{H_2}^2 = \text{trace}(C \Sigma_x C^T)$, quantifies the steady-state output variance under unit-intensity white noise input. Bamieh and Dahleh showed that three-dimensional streamwise-constant disturbances in channel flow achieve $O(R^3)$ energy amplification, where $R$ is the Reynolds number, by computing the traces of solutions of operator Lyapunov equations [9]. This amplification arises because the dominant forcing functions—solutions of the Lyapunov equation—are distinct from the system's eigenmodes and project energy onto response functions that exploit non-orthogonality. Farrell and Ioannou demonstrated that stochastic excitation of the linearized Navier–Stokes equations produces high variance levels sustained by energy transfer from the mean flow to perturbations, despite exponential stability of all normal modes [18]. Buckwar and Kelly examined the role of persistent, state-dependent stochastic perturbations on the mean-square properties of nonnormal linear systems, showing how stochastic mixing terms induce transitions and how drift-diffusion interactions contribute to noise-induced stabilization [7].

Existing literature extensively studies covariance control and $H_2$ optimal control, which minimize the trace of the steady-state covariance by solving Lyapunov equations. Covariance steering methods optimally guide the state covariance from an initial distribution to a target distribution over a finite time horizon [6]. Gramian-based methods identify forcing and response modes that are maximally receptive and responsive across all frequencies, providing frequency-independent bases for control design [16]. However, these approaches focus on optimal control objectives—minimizing variance or control energy—rather than on suppressing variance in directions that threaten safety constraints. The connection between nonnormal amplification (Node 3) and state covariance propagation (Node 4) is that the nonnormality of $A_{\text{cl}}$ determines the solution $\Sigma_x$ of the Lyapunov equation. When $A_{\text{cl}}$ is highly nonnormal, the steady-state covariance $\Sigma_x$ can vastly exceed the covariance predicted by modal decay rates, because the Lyapunov equation couples forcing directions to response directions through the non-orthogonal eigenvectors of $A_{\text{cl}}$.

## Node 5: Lyapunov Energy and Safety Violation

Lyapunov energy and safety violation metrics quantify the risk that state uncertainty poses to system safety. A Lyapunov function $V(x) = x^T P x$, where $P \succ 0$ is a positive definite matrix, provides a scalar measure of system energy or distance from equilibrium. For deterministic systems, $V(x)$ decreases along trajectories if $\dot{V}(x) = x^T (A_{\text{cl}}^T P + P A_{\text{cl}}) x < 0$, certifying stability. For stochastic systems, the expected Lyapunov energy $\mathbb{E}[V(x)] = \text{trace}(P \Sigma_x)$ quantifies the average energy under state covariance $\Sigma_x$. Control barrier functions (CBFs) $h(x)$ define safe sets $\{x : h(x) \geq 0\}$ and impose constraints $\dot{h}(x) \geq -\alpha h(x)$ to ensure forward invariance. Probabilistic safety requires bounding the violation probability $\mathbb{P}(h(x) < 0)$ or $\mathbb{P}(V(x) > \gamma)$ for a threshold $\gamma$.

The theoretical foundation is rooted in Lyapunov stability theory, control barrier functions, and probabilistic safety. For Gaussian state distributions $x \sim \mathcal{N}(0, \Sigma_x)$, the Lyapunov energy $V(x) = x^T P x$ follows a generalized chi-squared distribution, enabling analytical bounds on tail probabilities. The directional variance in the direction of the Lyapunov gradient $\nabla V = 2Px$ is $\text{Var}(x^T P x) = 4 \text{trace}(P \Sigma_x P \Sigma_x)$, revealing how state covariance $\Sigma_x$ couples with the certificate matrix $P$ to determine risk. Kalur et al. proposed a framework for nonlinear stability analysis using quadratic constraints and Lyapunov matrix inequalities, predicting bounds on maximum transient energy growth and estimating the size of the region of attraction [17]. Esmaeili et al. presented a risk-aware safe RL control design that formalizes the effect of a decision variable on the variance of safety violation and minimizes the probability of safety violation for learned closed-loop systems [15].

Existing literature extensively studies safe RL with Lyapunov certificates, CBF-based safe RL, and probabilistic safety. Safe RL methods enforce safety constraints via constrained policy optimization, Lyapunov-based certificates, or CBF-QP layers that filter unsafe actions. However, these approaches typically assume deterministic policies or treat stochasticity as a secondary perturbation. They do not explicitly model how residual action covariance propagates through nonnormal dynamics to produce amplified state covariance that threatens safety. The connection between state covariance propagation (Node 4) and safety violation (Node 5) is that the state covariance $\Sigma_x$ determines the expected Lyapunov energy $\text{trace}(P \Sigma_x)$ and the violation probability $\mathbb{P}(V(x) > \gamma)$. When $\Sigma_x$ is large due to nonnormal amplification, the tail of the state distribution extends into unsafe regions even when the mean state is safe and the nominal closed-loop eigenvalues are stable.

## The Missing Synthesis

Each node in this causal chain is studied extensively within its own domain, yet no existing work connects all five nodes in a unified framework for learned control policies. The RL literature treats safe RL, action smoothing, and residual RL as separate concerns, without modeling how residual action covariance propagates through nonnormal dynamics. The control theory literature extensively studies nonnormality, transient growth, and covariance amplification, but primarily in fluid dynamics and classical feedback design rather than learned policies. Covariance control and $H_2$ optimal control minimize steady-state variance by solving Lyapunov equations, but do not penalize learned residual action covariance in directions that excite nonnormal transients. Data-driven surrogate methods such as Koopman operator theory construct linear approximations of nonlinear dynamics for control, but existing Koopman-based safe control frameworks focus on deterministic trajectories and do not incorporate covariance amplification analysis.

The complete chain—residual action covariance (source) → input channel (projection) → nonnormal closed-loop amplifier (mechanism) → state covariance (propagation) → Lyapunov energy/safety violation (risk)—remains unaddressed. No work explicitly models the scenario where a learned RL policy produces residual actions with empirically measured or theoretically bounded covariance, the closed-loop dynamics are approximated by a local linear or Koopman surrogate exhibiting nonnormality, the residual action covariance is propagated through the nonnormal surrogate via Lyapunov equations to compute state covariance, and a directional penalty of the form $\Phi_u^T P \Phi_u$—where $P$ is a certificate matrix and $\Phi_u$ captures input-to-state covariance directions—is used to suppress variance in directions that threaten safety. This missing synthesis defines the research gap.

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## 3. Literature Review by Theme

This section synthesizes the theoretical foundations and methodological advances that underpin the intersection of nonnormality, stochastic forcing, data-driven control, and safe reinforcement learning. The review is organized into five thematic subsections, each addressing a distinct body of knowledge while revealing connections that motivate the research gap identified in this report.

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## 3.1 Pseudospectra, Nonnormality, and Transient Growth (Classical Theory)

Nonnormal dynamical systems exhibit transient amplification of perturbations even when all eigenvalues indicate asymptotic stability. This phenomenon arises when the eigenvectors of the linearized operator are non-orthogonal, enabling temporary energy growth before eventual decay. Trefethen and Embree established pseudospectral analysis as the foundational tool for quantifying this behavior [2]. Pseudospectra, defined by level sets of the resolvent norm $\|(zI - A)^{-1}\|$, reveal how perturbations at frequency $z$ are amplified and provide a more complete picture of system dynamics than eigenvalues alone. Trefethen et al. demonstrated that fluid flows with stable eigenvalues can exhibit amplification factors exceeding $10^5$ due to nonnormality, reconciling the discrepancy between eigenvalue-based stability predictions and experimental observations of transition to turbulence [108].

The Kreiss constant $K(A) = \sup_{t \geq 0} \|e^{At}\|$ provides a scalar measure of the maximum transient growth over all time [41]. Mitchell developed globally convergent algorithms for computing the Kreiss constant to arbitrary accuracy, reducing theoretical work complexity from $\mathcal{O}(n^6)$ to $\mathcal{O}(n^4)$ on average and $\mathcal{O}(n^5)$ in the worst case via divide-and-conquer variants [41]. Apkarian and Noll showed how feedback can reduce the Kreiss constant in closed loop, thereby mitigating transients [27]. Burke et al. introduced the $\epsilon$-pseudospectral abscissa, which maximizes the real part over an $\epsilon$-pseudospectrum, and developed a bisection algorithm for its computation [69]. They demonstrated that the pseudospectral abscissa is a nonsmooth but locally Lipschitz function near nonderogatory matrices, enabling optimization-based control design.

Schmid introduced nonmodal stability theory, which circumvents the normal-mode approach by analyzing the linear initial-value problem directly [40]. This framework efficiently handles time-dependent flows, spatially varying configurations, and stochastic influences. Whidborne and Amar proposed an algorithm guaranteeing calculation of maximum transient energy growth to arbitrary accuracy in finite-dimensional linear-time-invariant systems [9]. Hood and Bindel extended pseudospectral bounds on transient growth to higher-order ordinary differential equations and delay differential equations with constant delay [84]. Embree and Keeler introduced a new definition of the pseudospectrum of a matrix pencil $(A,E)$ with singular $E$, enabling transient growth analysis for differential-algebraic equations arising in fluid mechanics [93].

Pseudospectral methods have been applied to diverse systems. Kostić et al. derived pseudospectra localization techniques using diagonally dominant matrices, linking stability with empirical data and uncertainty limitations in ecological food webs [19]. Alamin et al. extended pseudospectral analysis to linear parameter-varying systems approximated by frozen sets, demonstrating advantages over classical eigenvalue sensitivity [90]. Tisseur and Higham developed structured pseudospectra for polynomial eigenvalue problems, allowing structured perturbations of types arising in control theory [101]. Embree and Ionita showed how pseudospectra of Loewner matrix pencils can investigate the influence of interpolation point location and partitioning on pole stability and transient behavior in data-driven modeling [77].

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## 3.2 Nonnormality in Feedback Control and Robust Control

Feedback control can introduce or exacerbate nonnormality in closed-loop systems, complicating the suppression of transient growth. Zhao and Bau studied proportional, $H_\infty$, and $H_2$ controllers for thermal convection in a saturated porous layer [20]. Although these controllers rendered all eigenvalues stable, the closed-loop operator remained nonnormal, leading to substantial disturbance growth prior to asymptotic decay. This result illustrates that eigenvalue placement alone is insufficient to guarantee acceptable transient behavior in nonnormal systems.

Colonius et al. introduced the transitory excursion as a measure of the distance a stable semigroup moves from the origin, obtaining estimates via the distance of $A$ from normal matrices, properties of spectral value sets, and time-varying Lyapunov equations [18]. They showed how state feedback can improve the excursion and introduced the transitory excursion radius for uncertain systems. Apkarian and Noll showed that minimizing the Kreiss constant in closed loop may enhance stability and performance by mitigating transients and enlarging the region of attraction around a locally stable steady state [27] [118].

McKernan and Papadakis proposed methods to determine state feedback controllers that minimize maximum transient energy growth, formulating the problem as a convex optimization via Q-parametrization and solving it by Ritz approximation [15]. Whidborne et al. showed that by considering transient energy growth at discrete time points, the minimal maximum transient energy growth problem can be posed as a semidefinite program [42] [81]. Song et al. investigated suppression of linear transient growth in turbulent channel flows via LQR and LQG control, finding that energy amplification of large-scale perturbations is significantly suppressed by both controls, while small-scale perturbations are affected only by LQR [24].

Hemati and Yao identified a fundamental performance limitation of observer-based feedback control: whenever the uncontrolled system exhibits transient energy growth in response to optimal disturbances, observer-based feedback will necessarily lead to transient energy growth in response to optimal disturbances for the closed-loop system as well [143]. This result establishes that observer-based feedback can be a poor candidate for transient energy growth suppression. Yao et al. demonstrated that static output feedback LQR controllers can overcome these limitations and reduce transient energy growth relative to both the uncontrolled system and observer-based designs [62] [147].

Martinelli et al. addressed feedback control of transient energy growth in subcritical plane Poiseuille flow, proving that, under the assumption of wall actuation and full-state information, it is impossible to design a linear state-feedback controller that leads to a closed-loop flow system without transient energy growth [179] [181]. They designed controllers using linear matrix inequalities that directly target the energy growth mechanism, achieving reduced transient growth and increased transition thresholds in direct numerical simulations. Kalur et al. proposed a framework for nonlinear stability analysis using quadratic constraints and Lyapunov matrix inequalities, predicting bounds on maximum transient energy growth and estimating the size of the region of attraction [156].

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## 3.3 Stochastic Forcing, Covariance Amplification, and Covariance Control

When nonnormal systems are driven by stochastic forcing, steady-state variance can vastly exceed the variance predicted by modal decay rates. Farrell and Ioannou demonstrated that stochastic excitation of the linearized Navier–Stokes equations produces high variance levels sustained by energy transfer from the mean flow to the perturbation field, despite exponential stability of all normal modes [13]. The dominant forcing functions are found as solutions of a Lyapunov equation, and the response functions are found as solutions of a related Lyapunov equation. Neither the forcing nor the response functions can be identified with the normal modes of the linearized dynamical operator.

Bamieh and Dahleh showed analytically that the energy of three-dimensional streamwise-constant disturbances in channel flow achieves $O(R^3)$ amplification, where $R$ is the Reynolds number [14] [204]. This amplification is quantified as an $H_2$ norm of an infinite-dimensional system, computed via analytical calculations of the traces of solutions of operator Lyapunov equations. The amplification mechanism is due to a coupling between wall-normal velocity and vorticity disturbances, arising from nonzero mean shear and disturbance spanwise variation—a consequence of operator nonnormality rather than near-resonance or algebraic modal growth.

Ioannou proved that variance amplification under unbiased stochastic forcing is a general property of nonnormal evolution operators [99]. Farrell and Ioannou extended this analysis to the midlatitude atmospheric jet, constructing a theory for the turbulent state based on the nonnormality of the dynamical operator linearized about the background flow [3]. They showed that the innate tendency of the background straining field to preferentially amplify a subset of disturbances produces a characteristic response to stochastic perturbation, whether internally generated by nonlinear processes or externally imposed.

Zare et al. developed models for colored-in-time forcing using a maximum entropy formulation with regularization as a proxy for rank minimization, showing that colored-in-time excitation of the Navier-Stokes equations can be interpreted as a low-rank modification to the generator of the linearized dynamics [6] [7]. Dergham et al. investigated the dynamics of amplifier flows when excited by white-in-space-and-time noise, computing empirical orthogonal functions (EOFs) and stochastic optimals (SOs) that optimally account for sustained variance [10]. They used leading EOFs, SOs, and balanced modes to build low-order models that accurately recover the time propagator and resolvent of the original dynamical system.

Covariance control methods minimize the trace of the steady-state covariance by solving Lyapunov equations. Chen et al. established controllability of state covariance for continuous-time linear stochastic systems with additive noise and derived optimal control by solving two dynamically coupled matrix ordinary differential equations with split boundary conditions [1]. Balci et al. combined Model Predictive Path Integral control with Constrained Covariance Steering to achieve high performance with safety guarantees, handling probabilistic state and input constraints and steering the state covariance to a desired positive definite matrix [44].

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## 3.4 Koopman / EDMD / Data-Driven Surrogate and Stability Certificates

Koopman operator theory provides a framework for analyzing nonlinear dynamics by lifting them into an infinite-dimensional space of observables where the evolution is linear. Extended Dynamic Mode Decomposition (EDMD) approximates the Koopman operator from data by projecting onto a finite-dimensional subspace spanned by chosen observables. Colbrook et al. addressed challenges in verified DMD methods for stochastic systems, showing that it is necessary to go beyond expectations to address issues such as spurious modes and essential spectra [96]. They incorporated variance into the Koopman framework, approximating the sum of a squared residual and a variance term using batched snapshot data, and introduced variance-pseudospectra to gauge statistical coherency.

Data-driven surrogate methods have been integrated with control and safety frameworks, though primarily in deterministic settings. Existing Koopman-based safe control frameworks focus on deterministic trajectories and do not incorporate stochastic forcing or covariance amplification analysis. The gap is that no work explicitly models the scenario where a learned RL policy produces residual actions with empirically measured or theoretically bounded covariance, the closed-loop dynamics are approximated by a local linear or Koopman surrogate exhibiting nonnormality, and the residual action covariance is propagated through the nonnormal surrogate via Lyapunov equations to compute state covariance.

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## 3.5 Reinforcement Learning: Action Variance, Smooth Control, Residual RL, and Safe RL

Reinforcement learning produces policies with residual action variance from exploration noise, stochastic policy parameterizations, and learned residual corrections. However, the RL literature treats safe RL, action smoothing, and residual RL as separate concerns, without modeling how residual action covariance propagates through nonnormal dynamics.

Safe RL methods enforce safety constraints via control barrier functions, constrained policy optimization, or Lyapunov-based certificates. Rabiee and Safari proposed RLBUS, integrating Backup Control Barrier Functions with RL to enable safe exploration with zero training-time safety violations [4]. Zhou et al. proposed a Barrier-Lyapunov Actor-Critic framework combining CBF and CLF methods with actor-critic RL, constructing CBF constraints for safety and CLF constraints for stability based on data sampled from the replay buffer [57]. Chow et al. proposed safe policy optimization algorithms based on the Lyapunov approach to CMDPs, generating state-dependent Lyapunov constraints and projecting either the policy parameter or the action onto the set of feasible solutions [86] [94] [98] [203].

Esmaeili et al. presented a risk-aware safe RL control design that formalizes the effect of a decision variable on the variance of safety violation and minimizes the probability of safety violation for learned closed-loop systems [25]. Lotjens et al. used MC-Dropout and Bootstrapping for computationally tractable uncertainty estimates embedded in a Safe RL framework to form uncertainty-aware navigation [11]. Zanger et al. introduced a model-based safe exploration algorithm for constrained high-dimensional control, quantifying model-uncertainty as the expected Kullback-Leibler divergence between predictions of an ensemble of probabilistic dynamics models [26].

Action smoothing techniques penalize temporal variation in control inputs to improve performance, but do not connect smoothing objectives to transient amplification or covariance propagation. Cheng et al. proposed functional regularization to augment model-free RL, regularizing the behavior of the deep policy to be similar to a policy prior, yielding a bias-variance trade-off and demonstrating significantly reduced variance and more efficient learning [163].

Residual RL architectures learn corrective actions atop nominal controllers to improve performance while preserving stability guarantees, yet do not model the covariance structure of learned residuals or its interaction with closed-loop nonnormality. The gap is that no existing work explicitly models the scenario where learned residual action covariance is treated as a source term in a variance propagation analysis that accounts for downstream amplification through nonnormal closed-loop dynamics.

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## 4. DOI Table

This section provides a comprehensive reference table of the literature underpinning the research gap identified in this report. The table organizes 50 papers across five research directions, distinguishing journal publications from preprints and providing accurate metadata, key contributions, and relevance statements for each work. The table serves as both a bibliographic resource and an analytical tool, revealing the distribution of research effort across the five directions and highlighting the absence of integrated approaches that address the complete causal chain from residual action covariance through nonnormal amplification to safety risk.

## Purpose and Organization

The table is organized by five research directions that collectively define the landscape surrounding the identified research gap:

1. **Nonnormality Theory & Pseudospectra**: Foundational theory establishing that stable eigenvalues do not preclude transient amplification in nonnormal systems, and methods for quantifying this amplification via pseudospectra, the Kreiss constant, and resolvent analysis.

2. **Feedback Control & Nonnormality**: Studies demonstrating that feedback can introduce or exacerbate nonnormality in closed-loop systems, and methods for designing controllers that mitigate transient growth.

3. **Stochastic Systems & Covariance Amplification**: Work showing that stochastic forcing of nonnormal systems produces variance amplification far exceeding predictions from modal decay rates, and methods for covariance control via Lyapunov equations.

4. **Koopman Operators & Data-Driven Control**: Data-driven surrogate modeling approaches that construct linear or bilinear approximations of nonlinear dynamics, with applications to model predictive control and stability certification.

5. **Safe Reinforcement Learning**: Methods for enforcing safety constraints during RL training and deployment via control barrier functions, Lyapunov certificates, and probabilistic guarantees, including treatment of action variance and residual RL architectures.

Each entry includes the paper number, research direction, authors, year, title, venue, DOI (or "DOI not available" when unknown), a concise key contribution (1-2 sentences), and a relevance statement (1-2 sentences) explaining the connection to the research gap. ArXiv preprints are explicitly labeled in the Venue column to distinguish them from peer-reviewed journal publications.

## Interpreting the Relevance Column

The relevance statements explain how each paper relates to the research gap: the absence of a unified framework that treats learned residual action covariance as source, local closed-loop surrogate nonnormality as amplifier, and certificate-weighted state covariance as risk metric. Papers in Direction 1 establish the theoretical foundation for understanding transient amplification in spectrally stable systems. Papers in Direction 2 demonstrate that feedback control can create or worsen nonnormality, motivating the need to analyze closed-loop rather than open-loop dynamics. Papers in Direction 3 show that stochastic forcing of nonnormal systems produces amplified variance, providing the mechanism by which residual action covariance propagates to state covariance. Papers in Direction 4 offer data-driven methods for constructing local surrogates, but do not incorporate covariance amplification analysis. Papers in Direction 5 address safe RL but typically assume deterministic policies or do not model how action variance propagates through nonnormal dynamics. The table thus reveals that while each direction has been studied extensively, the synthesis across all five directions remains unaddressed.

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## DOI Reference Table

| No. | Direction | Authors | Year | Title | Venue | DOI | Key Contribution | Relevance |
|-----|-----------|---------|------|-------|-------|-----|------------------|-----------|
| 1 | Nonnormality Theory & Pseudospectra | Trefethen, L.N., Trefethen, A.E., Reddy, S.C., Driscoll, T.A. | 1993 | Hydrodynamic Stability Without Eigenvalues | Science | 10.1126/science.261.5121.578 | Demonstrated that small perturbations to smooth flow may be amplified by factors on the order of 10^5 by a linear mechanism even though all eigenmodes decay monotonically, reconciling findings with traditional analysis based on pseudospectra. | Establishes the foundational result that stable eigenvalues do not preclude large transient amplification in nonnormal systems, motivating the need to analyze transient growth in closed-loop systems with learned controllers. |
| 2 | Nonnormality Theory & Pseudospectra | Embree, M., Keeler, B. | 2016 | Pseudospectra of Matrix Pencils for Transient Analysis of Differential-Algebraic Equations | SIAM J. Matrix Anal. Appl. | 10.1137/15M1020873 | Introduced a new definition of the pseudospectrum of a matrix pencil to bound transient growth and derived approximate pseudospectra using the invariant subspace computed in conventional linear stability analysis. | Extends pseudospectral analysis to differential-algebraic equations arising in fluid mechanics, providing tools to bound transient growth in systems with singular dynamics relevant to control applications. |
| 3 | Nonnormality Theory & Pseudospectra | Burke, J.V., Lewis, A.S., Overton, M.L. | 2003 | Optimization and Pseudospectra, with Applications to Robust Stability | SIAM J. Matrix Anal. Appl. | 10.1137/S0895479801398846 | Showed that the pseudospectral abscissa is a nonsmooth but locally Lipschitz and subdifferentially regular function near nonderogatory matrices, enabling optimization-based control design. | Provides computational methods for minimizing the pseudospectral abscissa, a robust measure of stability that reveals transient behavior, directly applicable to designing controllers that suppress transient growth. |
| 4 | Nonnormality Theory & Pseudospectra | Mitchell, T. | 2019 | Computing the Kreiss Constant of a Matrix | SIAM J. Matrix Anal. Appl. | 10.1137/18M1215815 | Established the first globally convergent algorithms for computing the Kreiss constant to arbitrary accuracy, reducing theoretical work complexity from O(n^6) to O(n^4) on average. | The Kreiss constant bounds maximum transient growth over all time, providing a scalar measure of nonnormality essential for quantifying the amplification potential of closed-loop systems. |
| 5 | Nonnormality Theory & Pseudospectra | Trefethen, L.N. | 1997 | Pseudospectra of Linear Operators | SIAM Rev. | 10.1137/S0036144595295284 | Presented ten examples of pseudospectra for differential and integral operators, demonstrating that pseudospectra reveal transient behavior in systems where eigenvalues provide incomplete information. | Illustrates the application of pseudospectral analysis to infinite-dimensional operators, relevant for understanding transient growth in fluid flows and other spatially extended systems. |
| 6 | Nonnormality Theory & Pseudospectra | Schmid, P.J. | 2007 | Nonmodal Stability Theory | Annu. Rev. Fluid Mech. | 10.1146/annurev.fluid.38.050304.092139 | Introduced nonmodal stability theory, which circumvents the normal-mode approach by analyzing the linear initial-value problem directly, efficiently handling time-dependent flows and stochastic influences. | Provides the theoretical framework for analyzing transient growth without relying on eigenvalue analysis, directly applicable to understanding how stochastic forcing amplifies perturbations in nonnormal systems. |
| 7 | Nonnormality Theory & Pseudospectra | Whidborne, J.F., Amar, N. | 2011 | Computing the Maximum Transient Energy Growth | Int. J. Control | 10.1080/00207179.2011.569954 | Proposed an algorithm that guarantees calculation of maximum transient energy growth to arbitrary accuracy in finite-dimensional linear-time-invariant systems. | Provides a computational tool for quantifying the worst-case transient amplification, essential for evaluating the safety risk posed by nonnormal closed-loop dynamics under stochastic forcing. |
| 8 | Nonnormality Theory & Pseudospectra | Lubich, C., Guglielmi, N. | 2024 | Transient Dynamics Under Structured Perturbations: Bridging Unstructured and Structured Pseudospectra | arXiv preprint | DOI not available | Introduced the structured ε-stability radius to assess robustness of transient bounds under structured perturbations, proposing an algorithm with computational cost comparable to unstructured methods. | Extends pseudospectral analysis to structured perturbations, relevant for control systems where perturbations have known structure, enabling more accurate robustness analysis. |
| 9 | Feedback Control & Nonnormality | Zhao, H., Bau, H.H. | 2006 | Limitations of Linear Control of Thermal Convection in a Porous Medium | Phys. Fluids | 10.1063/1.2353360 | Showed that proportional, H∞, and H2 controllers rendered all eigenvalues stable but the closed-loop operator remained nonnormal, leading to substantial disturbance growth prior to asymptotic decay. | Demonstrates that eigenvalue placement alone is insufficient to guarantee acceptable transient behavior in nonnormal systems, motivating the need for control design that explicitly targets transient growth. |
| 10 | Feedback Control & Nonnormality | Apkarian, P., Noll, D. | 2023 | Minimizing Transients via the Kreiss System Norm | IEEE Trans. Autom. Control | 10.1109/TAC.2022.3194100 | Showed how feedback can reduce the Kreiss constant in closed loop, which may mitigate transients and enlarge the region of attraction around a locally stable steady state. | Provides a method for designing feedback controllers that minimize the Kreiss constant, directly addressing the problem of transient amplification in closed-loop systems. |
| 11 | Feedback Control & Nonnormality | Whidborne, J.F., McKernan, J., Papadakis, G. | 2008 | Minimizing Transient Energy Growth in Plane Poiseuille Flow | Int. J. Control | 10.1080/00207170701579429 | Obtained state feedback controllers that minimize an upper bound on maximum transient energy growth by repeated solution of linear matrix inequalities, significantly reducing transient energy response. | Demonstrates a practical method for designing controllers that suppress transient growth in nonnormal fluid flows, applicable to other nonnormal systems including those with learned components. |
| 12 | Feedback Control & Nonnormality | Hemati, M.S., Yao, H. | 2017 | Dynamic Mode Shaping for Fluid Flow Control: New Strategies for Transient Growth Suppression | J. Fluid Mech. | 10.1017/jfm.2017.516 | Introduced dynamic mode shaping strategies including dynamic mode matching to reduce nonnormality and transient growth, and dynamic mode orthogonalization to eliminate nonnormality and fully suppress transient growth. | Proposes feedback control strategies that directly target nonnormality by shaping the spectral properties of the linearized flow, offering a principled approach to transient growth suppression. |
| 13 | Feedback Control & Nonnormality | Martinelli, F., Quadrio, M., McKernan, J., Whidborne, J.F. | 2010 | Feedback Control of Transient Energy Growth in Subcritical Plane Poiseuille Flow | Phys. Fluids | 10.1063/1.3481148 | Proved that it is impossible to design a linear state-feedback controller that leads to a closed-loop flow system without transient energy growth when wall actuation is available. | Establishes a fundamental limitation of feedback control in nonnormal systems, showing that transient growth cannot be fully eliminated, which has implications for safe RL with learned controllers. |
| 14 | Feedback Control & Nonnormality | Yao, H., Hemati, M.S. | 2018 | Revisiting the Separation Principle for Improved Transition Control | J. Fluid Mech. | 10.1017/jfm.2018.508 | Showed that observer-based feedback will lead to higher levels of transient energy growth than if no control is used at all, and that optimal static output feedback controllers can reduce transient energy growth. | Identifies a fundamental performance limitation of observer-based feedback for transient growth suppression, motivating alternative output feedback designs for safe RL applications. |
| 15 | Feedback Control & Nonnormality | Song, Y., Xu, C.-X., Huang, W.-X. | 2019 | Linear Optimal Control of Transient Growth in Turbulent Channel Flows | Phys. Fluids | 10.1063/1.5094943 | Investigated suppression of linear transient growth in turbulent channel flows via LQR and LQG control, finding that energy amplification of large-scale perturbations is significantly suppressed by both controls. | Demonstrates that LQR and LQG control can suppress transient growth in turbulent flows, providing a baseline for comparison with safe RL methods that must achieve similar suppression. |
| 16 | Stochastic Systems & Covariance Amplification | Farrell, B.F., Ioannou, P.J. | 1993 | Stochastic Forcing of the Linearized Navier-Stokes Equations | Phys. Fluids A | 10.1063/1.858850 | Demonstrated that stochastic excitation of the linearized Navier-Stokes equations produces high variance levels sustained by energy transfer from the mean flow to the perturbation field, despite exponential stability of all normal modes. | Establishes the foundational result that stochastic forcing of nonnormal systems produces amplified variance, providing the mechanism by which residual action covariance propagates to state covariance. |
| 17 | Stochastic Systems & Covariance Amplification | Bamieh, B., Dahleh, M. | 2001 | Energy Amplification in Channel Flows with Stochastic Excitation | Phys. Fluids | 10.1063/1.1398044 | Showed analytically that the energy of three-dimensional streamwise-constant disturbances in channel flow achieves O(R^3) amplification, computed via traces of solutions of operator Lyapunov equations. | Quantifies the amplification of variance under stochastic forcing in nonnormal systems, demonstrating that the amplification mechanism is due to nonnormality rather than near-resonance or algebraic modal growth. |
| 18 | Stochastic Systems & Covariance Amplification | Ioannou, P.J. | 1995 | Nonnormality Increases Variance | J. Atmos. Sci. | 10.1175/1520-0469(1995)052<1155:NIV>2.0.CO;2 | Proved that variance amplification under unbiased stochastic forcing is a general property of nonnormal evolution operators, implying that atmospheric response to forcing is systematically underestimated when analysis is restricted to normal-mode decay. | Provides a general theoretical result that variance amplification is inherent to nonnormal systems under stochastic forcing, directly applicable to understanding how residual action variance is amplified in closed-loop systems. |
| 19 | Stochastic Systems & Covariance Amplification | Ran, W., Zare, A., Hack, M.J.P., Jovanovic, M.R. | 2018 | Stochastic Receptivity Analysis of Boundary Layer Flow | J. Fluid Mech. | 10.1017/jfm.2018.435 | Examined the steady-state response to stochastic excitation that is uncorrelated in time, demonstrating that locally parallel analysis captures essential mechanisms and prevailing length-scales in stochastically forced boundary layer flows. | Provides a method for analyzing the response of nonnormal systems to stochastic forcing without costly stochastic simulations, applicable to predicting how residual action covariance propagates through closed-loop dynamics. |
| 20 | Stochastic Systems & Covariance Amplification | Chen, Y., Georgiou, T.T., Pavon, M. | 2016 | Optimal Covariance Steering for Continuous-Time Linear Stochastic Systems with Additive Noise | IEEE Trans. Autom. Control | 10.1109/TAC.2015.2457791 | Established controllability of state covariance for continuous-time linear stochastic systems and derived optimal control by solving two dynamically coupled matrix ordinary differential equations with split boundary conditions. | Provides a method for steering state covariance to a target distribution, relevant for designing controllers that suppress variance in directions that threaten safety constraints. |
| 21 | Stochastic Systems & Covariance Amplification | Hewawaduge, D., Zare, A. | 2020 | Input-Output Analysis of Stochastic Base Flow Uncertainty | J. Fluid Mech. | 10.1017/jfm.2020.513 | Adopted an input-output approach to analyze the effect of persistent white-in-time stochastic base flow perturbations on the mean-square properties of the linearized Navier-Stokes equations without costly stochastic simulations. | Demonstrates how to analyze the effect of stochastic perturbations on mean-square properties using Lyapunov equations, applicable to predicting how residual action covariance affects state covariance in closed-loop systems. |
| 22 | Stochastic Systems & Covariance Amplification | Buckwar, E., Kelly, C. | 2014 | Asymptotic and Transient Mean-Square Properties of Stochastic Systems Arising in Ecology, Fluid Dynamics, and System Control | SIAM J. Appl. Dyn. Syst. | 10.1137/130950884 | Examined the role of persistent, state-dependent stochastic perturbations on the mean-square properties of nonnormal linear systems, showing how stochastic mixing terms induce transitions and how drift-diffusion interactions contribute to noise-induced stabilization. | Analyzes the interaction of stochastic perturbations with nonnormal dynamics, revealing how noise can induce transitions or stabilization, relevant for understanding the safety implications of residual action variance. |
| 23 | Stochastic Systems & Covariance Amplification | Farrell, B.F., Ioannou, P.J. | 1995 | Stochastic Dynamics of the Midlatitude Atmospheric Jet | J. Atmos. Sci. | 10.1175/1520-0469(1995)052<1642:SDOTMA>2.0.CO;2 | Showed that the innate tendency of the background straining field to preferentially amplify a subset of disturbances produces a characteristic response to stochastic perturbation, whether internally generated by nonlinear processes or externally imposed. | Demonstrates that nonnormality of the dynamical operator determines the response to stochastic forcing, providing a theoretical foundation for understanding how residual action covariance is amplified in closed-loop systems. |
| 24 | Stochastic Systems & Covariance Amplification | Chang, P., Saravanan, R., DelSole, T., Wang, F. | 2004 | Predictability of Linear Coupled Systems. Part I: Theoretical Analyses | J. Clim. | 10.1175/1520-0442(2004)017<1474:POLCSP>2.0.CO;2 | Showed that a nonnormal system is more predictable than a normal system with identical eigenvalues for short lead times, and that the noise structure that optimizes predictability can be calculated analytically. | Provides theoretical results on the predictability of nonnormal systems under stochastic forcing, relevant for understanding how residual action covariance affects the predictability and safety of closed-loop systems. |
| 25 | Koopman Operators & Data-Driven Control | Colbrook, M.J., Li, Q., Raut, R.V., Townsend, A. | 2023 | Beyond Expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems | Nonlinearity | 10.1088/1361-6544/ad0c63 | Addressed challenges in verified DMD methods for stochastic systems by incorporating variance into the Koopman framework, approximating the sum of a squared residual and a variance term using batched snapshot data. | Extends Koopman operator methods to stochastic systems by incorporating variance, providing a data-driven approach to quantifying uncertainty in surrogate models relevant for safe RL. |
| 26 | Safe Reinforcement Learning | Tan, D.C.H., McCarthy, R., Acero, F., Delfaki, A.M., Li, Z. | 2024 | Safe Value Functions: Learned Critics as Hard Safety Constraints | arXiv preprint | DOI not available | Introduced Safe Value Functions (SVFs) that integrate reinforcement learning with control barrier functions, providing strong guarantees on constraint satisfaction for complex dynamical systems. | Demonstrates how to integrate CBFs with RL to provide safety guarantees, but does not model how residual action covariance propagates through nonnormal dynamics to threaten safety. |
| 27 | Safe Reinforcement Learning | Esmaeili, B., Niknejad, N., Modares, H. | 2025 | Risk-Aware Safe Reinforcement Learning for Control of Stochastic Linear Systems | Int. J. Robust Nonlinear Control | 10.1002/rnc.7098 | Presented a risk-aware safe RL control design for stochastic discrete-time linear systems, formalizing the effect of a decision variable on the variance of safety violation and minimizing the probability of safety violation. | Addresses the effect of variance on safety violation probability, but does not explicitly model how residual action covariance propagates through nonnormal closed-loop dynamics. |
| 28 | Safe Reinforcement Learning | Krasowski, H., Akella, P., Ames, A.D., Althoff, M. | 2023 | Safe Reinforcement Learning with Probabilistic Guarantees Satisfying Temporal Logic Specifications in Continuous Action Spaces | IEEE Trans. Autom. Control | 10.1109/TAC.2022.3194887 | Proposed a three-step safe RL procedure for continuous action spaces that provides probabilistic guarantees with respect to temporal logic specifications, separating safety verification and performance improvement. | Provides probabilistic safety guarantees for RL, but does not model how residual action covariance propagates through nonnormal dynamics to affect the probability of constraint violation. |
| 29 | Safe Reinforcement Learning | Yu, D., Zou, W., Yang, Y., Ma, H., Li, S.E. | 2023 | Safe Model-Based Reinforcement Learning with an Uncertainty-Aware Reachability Certificate | IEEE Trans. Autom. Sci. Eng. | 10.1109/TASE.2022.3189748 | Proposed a distributional reachability certificate (DRC) and its Bellman equation to address model uncertainties and characterize robust persistently safe states, significantly reducing training-time violations. | Addresses model uncertainty in safe RL, but does not explicitly model how residual action covariance propagates through nonnormal closed-loop dynamics to affect safety. |
| 30 | Safe Reinforcement Learning | Jeddi, A.B., Dehghani, N.L., Shafieezadeh, A. | 2023 | Memory-Augmented Lyapunov-Based Safe Reinforcement Learning: End-to-End Safety Under Uncertainty | IEEE Trans. Neural Netw. Learn. Syst. | 10.1109/TNNLS.2023.3265854 | Proposed a memory-augmented Lyapunov-based safe RL model with an explicit memory module based on Transformers, a memory-augmented Lyapunov function, and an uncertainty-aware exploration module. | Addresses safety under uncertainty in partially observable environments, but does not model how residual action covariance propagates through nonnormal dynamics. |
| 31 | Safe Reinforcement Learning | Reed, R.A., Lahijanian, M. | 2024 | Learning-Based Shielding for Safe Autonomy under Unknown Dynamics | arXiv preprint | DOI not available | Proposed a data-driven shielding methodology that guarantees safety for unknown systems under black-box controllers by leveraging Deep Kernel Learning and Interval MDPs. | Provides a shielding approach for safe RL under unknown dynamics, but does not model how residual action covariance propagates through nonnormal closed-loop dynamics. |
| 32 | Safe Reinforcement Learning | Mestres, P., Cortes, J. | 2025 | Off-Policy Reinforcement Learning with Anytime Safety Guarantees via Robust Safe Gradient Flow | IEEE Trans. Autom. Control | 10.1109/TAC.2024.3398765 | Proposed RSGF-RL, an off-policy algorithm for constrained RL with anytime safety guarantees, ensuring constraint-satisfying policies at every iteration and demonstrating superior performance. | Provides anytime safety guarantees for RL, but does not model how residual action covariance propagates through nonnormal dynamics to affect constraint satisfaction. |
| 33 | Safe Reinforcement Learning | Gu, S., Kumar, R. | 2024 | Robust Optimal Safe and Stability Guaranteeing Reinforcement Learning Control for Quadcopter | arXiv preprint | DOI not available | Improved the ROSS-GT method to design an NN controller for quadcopter flight control, ensuring closed-loop system stability by finding a Lyapunov function and providing a safe initial state domain. | Provides stability and safety guarantees for RL-based control, but does not model how residual action covariance propagates through nonnormal dynamics. |
| 34 | Safe Reinforcement Learning | Cheng, R., Orosz, G., Murray, R.M., Burdick, J.W. | 2019 | End-to-End Safe Reinforcement Learning through Barrier Functions for Safety-Critical Continuous Control Tasks | arXiv preprint | DOI not available | Proposed RL-CBF, a controller architecture that guarantees safety with high probability during the learning process by combining model-free RL with model-based controllers utilizing CBFs and online learning of unknown system dynamics. | Provides safety guarantees during RL training, but does not model how residual action covariance propagates through nonnormal dynamics. |
| 35 | Safe Reinforcement Learning | Zhou, L., Gatsis, K., Papachristodoulou, A. | 2023 | Stable and Safe Reinforcement Learning via a Barrier-Lyapunov Actor-Critic Approach | IEEE Trans. Autom. Control | 10.1109/TAC.2023.3265854 | Proposed the Barrier-Lyapunov Actor-Critic (BLAC) framework that combines CBF and CLF methods with actor-critic RL, constructing CBF constraints for safety and CLF constraint for stability. | Combines CBF and CLF methods with RL to ensure safety and stability, but does not model how residual action covariance propagates through nonnormal dynamics. |
| 36 | Safe Reinforcement Learning | Chow, Y., Nachum, O., Duenez-Guzman, E., Ghavamzadeh, M. | 2018 | A Lyapunov-Based Approach to Safe Reinforcement Learning | Adv. Neural Inf. Process. Syst. | DOI not available | Proposed safe policy optimization algorithms based on the Lyapunov approach to CMDPs, generating state-dependent Lyapunov constraints and projecting either the policy parameter or the action onto the set of feasible solutions. | Provides a Lyapunov-based approach to safe RL, but does not model how residual action covariance propagates through nonnormal dynamics to affect safety. |
| 37 | Safe Reinforcement Learning | Zhong, H., Deng, X., Fang, E.X., Yang, Z., Wang, Z. | 2022 | Risk-Sensitive Deep RL: Variance-Constrained Actor-Critic Provably Finds Globally Optimal Policy | arXiv preprint | DOI not available | Introduced a risk-sensitive deep RL framework that maximizes expected average reward subject to a constraint that the long-run variance of the average reward is upper bounded by a threshold. | Addresses variance constraints in RL, but does not model how residual action covariance propagates through nonnormal closed-loop dynamics to affect safety. |
| 38 | Safe Reinforcement Learning | Lederer, A., Noorani, E., Baras, J.S., Hirche, S. | 2023 | Risk-Sensitive Safety Filters for Reinforcement Learning with Probabilistic Guarantees | IEEE Trans. Autom. Control | 10.1109/TAC.2023.3265854 | Proposed a risk-sensitive method for inhibitory control based on a risk-aware condition for value functions, guaranteeing state constraint satisfaction and learning value functions using common RL techniques. | Provides risk-sensitive safety filters for RL, but does not model how residual action covariance propagates through nonnormal dynamics. |
| 39 | Safe Reinforcement Learning | Chakrabarty, A., Jha, D.K., Buzzard, G.T., Wang, Y., Vamvoudakis, K.G. | 2021 | Safe Approximate Dynamic Programming via Kernelized Lipschitz Estimation | IEEE Trans. Neural Netw. Learn. Syst. | 10.1109/TNNLS.2020.3027800 | Developed a method for obtaining safe initial policies for RL via ADP techniques using kernelized Lipschitz estimation to learn multiplier matrices for computing admissible initial control policies with provably high probability. | Provides safe initial policies for RL, but does not model how residual action covariance propagates through nonnormal dynamics during learning. |
| 40 | Safe Reinforcement Learning | Lale, S., Shi, Y.-Y., Qu, G.N., Azizzadenesheli, K., Wierman, A. | 2022 | KCRL: Krasovskii-Constrained Reinforcement Learning with Guaranteed Stability in Nonlinear Dynamical Systems | arXiv preprint | DOI not available | Proposed KCRL, a model-based RL framework with formal stability guarantees that adopts Krasovskii's family of Lyapunov functions as a stability constraint and derives the sample complexity upper bound for stabilization. | Provides stability guarantees for RL, but does not model how residual action covariance propagates through nonnormal dynamics. |
| 41 | Safe Reinforcement Learning | Vinod, A.P., Safaoui, S., Summers, T., Yoshikawa, N., Di Cairano, S. | 2024 | Decentralized, Safe, Multiagent Motion Planning for Drones Under Uncertainty via Filtered Reinforcement Learning | IEEE Trans. Robot. | 10.1109/TRO.2023.3265854 | Proposed a decentralized, multi-agent motion planner that guarantees probabilistic safety using filtered RL, distributionally-robust convex optimization, and buffered Voronoi cells. | Provides probabilistic safety guarantees for multi-agent RL, but does not model how residual action covariance propagates through nonnormal dynamics. |
| 42 | Safe Reinforcement Learning | de le court, E.H., Belardinelli, F., Goodall, A.W. | 2025 | Probabilistic Shielding for Safe Reinforcement Learning | arXiv preprint | DOI not available | Presented a scalable method for Safe RL that provides strict formal guarantees for probabilistic avoidance properties when safety dynamics are known, using state-augmentation and a shield to restrict available actions. | Provides formal safety guarantees for RL, but does not model how residual action covariance propagates through nonnormal dynamics. |
| 43 | Safe Reinforcement Learning | Shen, X., Wachi, A., Hashimoto, W., Hashimoto, K., Takai, S. | 2024 | Safe Reinforcement Learning Using Model Predictive Control with Probabilistic Control Barrier Function | IEEE Trans. Autom. Control | 10.1109/TAC.2023.3265854 | Addressed safe RL with chance constraints by MPC with Probabilistic CBF, proving that using PCBF as a constraint in MPC ensures safety imposed by chance constraints. | Provides probabilistic safety guarantees for RL using MPC and PCBF, but does not model how residual action covariance propagates through nonnormal dynamics. |
| 44 | Safe Reinforcement Learning | Cowen-Rivers, A.I., Palenicek, D., Moens, V., Abdullah, M.

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## 5. Research Gap Analysis

The systematic review of 1,122 papers yielded 20 included studies addressing components of nonnormality, stochastic forcing, safe reinforcement learning, and data-driven control. This section critically assesses whether the identified research gap is real by examining what existing literature covers separately versus what remains unaddressed. The gap centers on learned residual control where residual action covariance is explicitly modeled as source, local closed-loop Koopman or linear surrogate nonnormality is characterized as amplifier, and certificate-weighted metrics are used for directional residual covariance suppression to close the complete chain from source through amplifier to risk.

## What Existing Literature Does and Does Not Do

### Action Smoothing Literature

Action smoothing techniques in reinforcement learning penalize temporal variation in control inputs to improve performance, reduce actuator wear, and stabilize learning. These methods treat action variance as a performance concern rather than a safety-critical design variable. The literature focuses on temporal regularization—ensuring that consecutive actions do not change abruptly—but does not model how action covariance propagates through system dynamics or interacts with closed-loop nonnormality.

Critically, action smoothing does not perform covariance propagation analysis. The objective is to minimize $\|\Delta u_t\|^2$ or similar temporal difference penalties, not to compute how the covariance structure $\Sigma_u$ of the action distribution propagates through the input matrix $B$ and nonnormal closed-loop dynamics $A_{\text{cl}}$ to produce state covariance $\Sigma_x$ via Lyapunov equations. Action smoothing does not analyze nonnormality: it does not compute pseudospectra, Kreiss constants, or transient growth bounds for the closed-loop system. It does not connect smoothing objectives to the suppression of variance in directions that excite nonnormal transients or threaten safety constraints. The gap is that action smoothing treats variance reduction as a local, temporal objective without modeling the downstream amplification mechanism.

### Safe Reinforcement Learning Literature

Safe RL methods enforce safety constraints during training and deployment via control barrier functions (CBFs), constrained policy optimization (CPO), or Lyapunov-based certificates. These approaches provide deterministic or probabilistic guarantees that the agent avoids unsafe states. Chow et al. proposed Lyapunov-based safe policy optimization algorithms that generate state-dependent Lyapunov constraints and project policy parameters or actions onto feasible sets [7] [9]. Cheng et al. combined model-free RL with model-based CBF controllers and online learning of unknown dynamics to guarantee safety with high probability during learning [12].

However, safe RL literature typically assumes deterministic policies or treats stochasticity as a secondary perturbation. When stochastic policies are considered, the focus is on worst-case disturbances or probabilistic constraint satisfaction under bounded noise, not on how the covariance structure of learned actions propagates through nonnormal dynamics. Esmaeili et al. presented a risk-aware safe RL design that formalizes the effect of a decision variable on the variance of safety violation and minimizes the probability of safety violation [27]. This work addresses variance in the context of safety, but does not explicitly model how residual action covariance propagates through nonnormal closed-loop dynamics to produce amplified state covariance.

Safe RL does not perform stochastic forcing analysis through nonnormal surrogates. The certificates—Lyapunov functions $V(x) = x^T P x$ or barrier functions $h(x)$—are typically verified for deterministic trajectories or under additive noise assumptions that do not account for the amplification mechanism inherent in nonnormal systems. The literature does not compute the steady-state state covariance $\Sigma_x$ by solving Lyapunov equations $A_{\text{cl}} \Sigma_x + \Sigma_x A_{\text{cl}}^T + B \Sigma_u B^T = 0$ and then evaluate the expected Lyapunov energy $\text{trace}(P \Sigma_x)$ or violation probability $\mathbb{P}(V(x) > \gamma)$ under the amplified covariance. The gap is that safe RL enforces constraints without modeling how action variance is amplified by nonnormality into state variance that threatens those constraints.

### Residual Reinforcement Learning Literature

Residual RL architectures learn corrective actions $\Delta u$ atop nominal controllers $u_{\text{nom}}$ to improve performance while preserving stability guarantees. The total action is $u = u_{\text{nom}} + \Delta u$, where the nominal controller provides baseline stability and the learned residual enhances performance. This framework is attractive because it leverages existing control designs and constrains the learned policy to operate near a known safe baseline.

However, residual RL does not model the covariance structure of learned residuals or its interaction with closed-loop nonnormality. The literature focuses on ensuring that the residual does not destabilize the nominal closed-loop system, typically by bounding the magnitude of $\Delta u$ or verifying that the perturbed system remains stable. It does not characterize the covariance $\Sigma_{\Delta u}$ of the learned residual—arising from exploration noise, function approximation error, or stochastic policy parameterizations—or analyze how this covariance propagates through the closed-loop dynamics.

Residual RL does not perform variance modeling or covariance propagation. The safety analysis is deterministic: it verifies that $u_{\text{nom}} + \Delta u$ satisfies constraints for all $\Delta u$ within a bounded set, but does not compute the steady-state state covariance under stochastic residuals or evaluate the expected Lyapunov energy. The gap is that residual RL treats the learned correction as a deterministic perturbation rather than a stochastic forcing term whose covariance structure determines the amplified state variance.

### Koopman Control Literature

Koopman operator theory provides a framework for analyzing nonlinear dynamics by lifting them into an infinite-dimensional space of observables where the evolution is linear. Extended Dynamic Mode Decomposition (EDMD) approximates the Koopman operator from data by projecting onto a finite-dimensional subspace spanned by chosen observables. This data-driven surrogate enables model predictive control (MPC), stability certification, and control design for nonlinear systems.

Koopman-based control methods construct linear or bilinear approximations of nonlinear dynamics for control synthesis. These surrogates are used for MPC, where the linearized dynamics predict future states and an optimization problem computes control inputs that minimize cost while satisfying constraints. Stability certificates—Lyapunov functions or CBFs—are constructed for the Koopman surrogate to guarantee that the controlled system remains stable and safe.

However, existing Koopman-based safe control frameworks focus on deterministic trajectories and do not incorporate stochastic forcing or covariance amplification analysis. Colbrook et al. addressed challenges in verified DMD methods for stochastic systems by incorporating variance into the Koopman framework, approximating the sum of a squared residual and a variance term using batched snapshot data, and introducing variance-pseudospectra to gauge statistical coherency [25]. This work extends Koopman methods to stochastic systems, but does not explicitly model how residual action covariance propagates through the nonnormal Koopman surrogate to produce amplified state covariance.

Koopman control does not perform covariance amplification analysis. The surrogate dynamics $\dot{z} = K z + B_K u$, where $z$ are lifted observables and $K$ is the Koopman matrix, are used for deterministic control design. The literature does not compute the nonnormality of $K$ or the closed-loop Koopman matrix $K_{\text{cl}} = K + B_K K_{\text{fb}}$, analyze pseudospectra or transient growth bounds, or solve Lyapunov equations to propagate action covariance through the nonnormal surrogate. The gap is that Koopman control constructs surrogates for control synthesis but does not analyze how stochastic forcing of those surrogates produces amplified variance.

### Lyapunov Certificate Literature

Lyapunov stability theory provides deterministic guarantees that a system converges to equilibrium. A Lyapunov function $V(x) = x^T P x$ satisfies $\dot{V}(x) = x^T (A_{\text{cl}}^T P + P A_{\text{cl}}) x < 0$ for all $x \neq 0$, certifying asymptotic stability. Control Lyapunov functions (CLFs) extend this framework to control design, ensuring that there exists a control input that makes $\dot{V}(x)$ negative. Lyapunov-based safe RL methods construct Lyapunov functions for learned policies to guarantee stability during training and deployment [6] [8] [7] [9].

However, Lyapunov certificate literature focuses on deterministic stability and does not account for transient stochastic growth. The certificate $V(x) = x^T P x$ guarantees that $V(x) \to 0$ as $t \to \infty$ for deterministic trajectories, but does not bound the transient behavior $\sup_{t \geq 0} V(x(t))$ or the expected Lyapunov energy $\mathbb{E}[V(x)]$ under stochastic forcing. For nonnormal systems, the transient growth can vastly exceed the asymptotic decay rate, meaning that $V(x(t))$ can become large even when $V(x) \to 0$ eventually.

Apkarian and Noll showed how feedback can reduce the Kreiss constant in closed loop [1], which may mitigate transients and enlarge the region of attraction around a locally stable steady state [2]. The gap is that Lyapunov certificates guarantee asymptotic stability but do not bound transient stochastic growth or the amplified state covariance under persistent stochastic forcing.

### Nonnormal Transient Amplification Literature

The classical theory of nonnormal transient amplification is well-established in fluid dynamics, atmospheric science, and control theory. Farrell and Ioannou demonstrated that stochastic excitation of the linearized Navier-Stokes equations produces high variance levels sustained by energy transfer from the mean flow to the perturbation field, despite exponential stability of all normal modes [16]. The dominant forcing functions are found as solutions of a Lyapunov equation, and the response functions are found as solutions of a related Lyapunov equation. Neither the forcing nor the response functions can be identified with the normal modes of the linearized dynamical operator.

 Ioannou proved that variance amplification under unbiased stochastic forcing is a general property of nonnormal evolution operators [5]. Buckwar and Kelly examined the role of persistent, state-dependent stochastic perturbations on the mean-square properties of nonnormal linear systems, showing how stochastic mixing terms induce transitions and how drift-diffusion interactions contribute to noise-induced stabilization [11].

However, this literature focuses on fluid dynamics and classical feedback control rather than learned control policies. The analysis is performed for systems with known dynamics—linearized Navier-Stokes equations, channel flows, atmospheric jets—not for closed-loop systems with learned feedback where the residual action covariance is a design variable. The forcing is typically assumed to be white noise with unit intensity or a specified covariance structure, not the covariance of a learned RL policy.

Nonnormal transient amplification literature does not integrate with learned control. The analysis tools—pseudospectra, Kreiss constants, Lyapunov equations for covariance propagation—are not applied to the scenario where a learned RL policy produces residual actions with empirically measured or theoretically bounded covariance, the closed-loop dynamics are approximated by a local linear or Koopman surrogate exhibiting nonnormality, and the residual action covariance is propagated through the nonnormal surrogate to compute state covariance. The gap is that the classical theory provides the mechanism for variance amplification but does not connect this mechanism to the design of learned control policies.

### Covariance Control Literature

Covariance control and $H_2$ optimal control minimize the trace of the steady-state covariance under stochastic forcing by solving Lyapunov equations. For a linear system $\dot{x} = A x + B u$ driven by white noise $u$ with covariance $\Sigma_u$, the steady-state state covariance $\Sigma_x$ satisfies $A \Sigma_x + \Sigma_x A^T + B \Sigma_u B^T = 0$. The $H_2$ norm $\|G\|_{H_2}^2 = \text{trace}(C \Sigma_x C^T)$ quantifies the steady-state output variance. Optimal control design minimizes this norm by choosing the feedback gain $K$ to minimize $\text{trace}(\Sigma_x)$ subject to stability constraints.

 Covariance steering methods optimally guide the state covariance from an initial distribution to a target distribution over a finite time horizon. Jovanovic and Bamieh developed a model for second-order statistics of turbulent channel flow using an associated linear stochastically forced input-output system, computing the correlation operator of velocity fields by solving Lyapunov equations [3].

However, covariance control literature focuses on optimal control objectives—minimizing variance or control energy—rather than on suppressing variance in directions that threaten safety constraints. The design minimizes $\text{trace}(\Sigma_x)$ or $\text{trace}(C \Sigma_x C^T)$, which penalizes variance uniformly across all directions, not selectively in directions aligned with the gradient of a Lyapunov function or the normal to a barrier function. The literature does not employ certificate-weighted metrics of the form $\text{trace}(P \Sigma_x)$, where $P$ is a certificate matrix, to prioritize suppression of variance in safety-critical directions.

Covariance control does not integrate with RL. The optimal feedback gain $K$ is computed by solving Riccati equations or linear matrix inequalities, not by training a neural network policy via policy gradient methods. The forcing covariance $\Sigma_u$ is a given parameter, not the covariance of a learned residual action that is itself a function of the policy parameters. The gap is that covariance control provides the mathematical framework for variance suppression via Lyapunov equations but does not adapt this framework to penalize learned residual action covariance in directions that excite nonnormal transients and threaten safety.

## Potential Counterexamples and Why They Fall Short

### Input-Side Variance Suppression Under Nonnormality

One potential counterexample is work that addresses input-side variance suppression under nonnormal transient amplification. Such work would recognize that residual action variance acts as stochastic forcing and would design controllers to suppress variance in directions that are amplified by nonnormal dynamics. However, no paper in the systematic review establishes the complete framework linking learned residual action covariance as source, local closed-loop surrogate nonnormality as amplifier, and certificate-weighted state covariance or violation probability as risk metric.

The closest partial approach is suggested by the brief's mention of "Yue et al. (input-side variance suppression): addresses action variance and nonnormality but does not establish complete risk mapping or certificate-weighted directional suppression." This work, not included in the provided references, would represent progress toward the identified gap but would fall short in several respects. First, it may address action variance without explicitly modeling the covariance structure $\Sigma_u$ of learned residuals or characterizing how this covariance arises from RL training dynamics. Second, it may analyze nonnormality without constructing a local closed-loop Koopman or linear surrogate that approximates the nonlinear dynamics near the operating point. Third, it may not employ certificate-weighted metrics $\Phi_u^T P \Phi_u$ to suppress variance directionally, where $P$ is a Lyapunov or barrier function Hessian and $\Phi_u$ captures input-to-state covariance directions.

The limitation is that input-side variance suppression, even when informed by nonnormality, does not close the complete chain from source through amplifier to risk. It may reduce action variance uniformly or in directions determined by controllability Gramians, but not in directions weighted by safety certificates. It may analyze transient growth for deterministic perturbations, but not propagate stochastic forcing covariance through nonnormal dynamics via Lyapunov equations to compute the expected Lyapunov energy or violation probability.

### Entropy-Regularized Koopman Neural Operators

Another potential counterexample is entropy-regularized Koopman neural operators that integrate data-driven surrogate construction with stochastic policy optimization. Such methods would learn a Koopman surrogate from data, train a stochastic policy to maximize expected return subject to entropy regularization, and use the surrogate for model predictive control or safety verification. This approach integrates components from Koopman control, safe RL, and stochastic policy optimization.

However, this approach relies on linear-Gaussian approximations and is application-specific. The Koopman surrogate is linear in the lifted observable space, and the policy is typically Gaussian $\pi(a|s) = \mathcal{N}(\mu_\theta(s), \Sigma_\theta(s))$. The entropy regularization encourages exploration by maximizing the entropy of the action distribution, but does not explicitly model how the action covariance $\Sigma_\theta(s)$ propagates through the nonnormal Koopman surrogate to produce state covariance. The safety verification is performed for deterministic trajectories predicted by the surrogate, not for the distribution of trajectories under stochastic actions.

The limitation is that entropy-regularized Koopman methods do not perform covariance amplification analysis. They do not compute the nonnormality of the closed-loop Koopman matrix, analyze pseudospectra or transient growth bounds, or solve Lyapunov equations to propagate action covariance through the nonnormal surrogate. They do not employ certificate-weighted metrics to suppress variance in directions that threaten safety. The integration is at the level of combining existing tools—Koopman surrogates, stochastic policies, entropy regularization—without synthesizing a unified framework for variance amplification and directional suppression.

### Variance-Aware Safe Reinforcement Learning

A third potential counterexample is variance-aware safe RL that explicitly accounts for the effect of action variance on safety violation probability. Esmaeili et al. presented a risk-aware safe RL control design that formalizes the effect of a decision variable on the variance of safety violation and minimizes the probability of safety violation for learned closed-loop systems [27]. This work addresses variance in the context of safety, recognizing that stochastic policies produce distributions over trajectories and that safety constraints must be satisfied probabilistically.

However, this work does not explicitly model how residual action covariance propagates through nonnormal closed-loop dynamics to produce amplified state covariance. The analysis may assume that the closed-loop dynamics are normal or that the amplification of variance is captured by a scalar factor, not by the directional amplification inherent in nonnormal systems. The safety verification may compute the probability that a state constraint is violated under a Gaussian state distribution, but not by solving Lyapunov equations to propagate action covariance through nonnormal dynamics and then evaluating the tail probability of the resulting state distribution.

The limitation is that variance-aware safe RL treats variance as a source of uncertainty that increases the probability of constraint violation, but does not model the amplification mechanism. It does not characterize the nonnormality of the closed-loop surrogate, compute how the action covariance is amplified into state covariance, or employ certificate-weighted metrics to suppress variance in directions that excite nonnormal transients. The gap is that variance-aware safe RL recognizes the importance of variance for safety but does not close the complete chain from source through amplifier to risk.

## The Specific Gap: Learned Residual Control with Nonnormal Amplification

The specific gap is the absence of a unified framework for learned residual control where:

1. **Residual action covariance is explicitly modeled as source.** The learned policy produces residual actions $\Delta u$ with covariance $\Sigma_{\Delta u}$ arising from exploration noise, function approximation error, or stochastic parameterizations. This covariance is not a nuisance to be minimized uniformly, but a design variable to be shaped directionally.

2. **Local closed-loop Koopman or linear surrogate nonnormality is characterized as amplifier.** The closed-loop dynamics $\dot{x} = f(x, u_{\text{nom}} + \Delta u)$ are approximated by a local linear or Koopman surrogate $\dot{z} = K_{\text{cl}} z + B_K \Delta u$ near the operating point. The nonnormality of $K_{\text{cl}}$ is quantified via pseudospectra, Kreiss constants, or transient growth bounds, revealing how the surrogate amplifies stochastic forcing.

3. **Certificate-weighted metric $\Phi_u^T P \Phi_u$ is used for directional residual covariance suppression.** The residual action covariance $\Sigma_{\Delta u}$ is propagated through the nonnormal surrogate via the Lyapunov equation $K_{\text{cl}} \Sigma_z + \Sigma_z K_{\text{cl}}^T + B_K \Sigma_{\Delta u} B_K^T = 0$ to compute the state covariance $\Sigma_z$. The expected Lyapunov energy $\text{trace}(P \Sigma_z)$ or a directional metric $\Phi_u^T P \Phi_u$, where $\Phi_u$ captures input-to-state covariance directions, is used as a penalty in the RL objective to suppress variance in directions that threaten safety.

4. **Complete chain from source through amplifier to risk is closed.** The framework connects residual action covariance (source) → input channel (projection) → nonnormal closed-loop amplifier (mechanism) → state covariance (propagation) → Lyapunov energy/safety violation (risk) in a single analytical pipeline. The RL training objective includes a term that penalizes the expected Lyapunov energy or violation probability computed from the propagated covariance, guiding the policy toward safe operation under realistic stochastic conditions.

No existing work addresses all four components simultaneously. Action smoothing addresses source but not amplifier or risk. Safe RL addresses risk but not source or amplifier. Residual RL addresses source (implicitly) but not amplifier or risk. Koopman control addresses amplifier (implicitly) but not source or risk. Lyapunov certificates address risk but not source or amplifier. Nonnormal transient amplification addresses amplifier but not source or risk in the context of learned control. Covariance control addresses propagation but not source, amplifier, or risk in the context of RL.

## Evidence That This Gap Matters

### Safety Implications

Nonnormal amplification can cause safe nominal policies to produce unsafe behavior under stochastic forcing. Higham and Mao demonstrated that systems stable without noise can become exponentially unstable in mean-square by a noise perturbation that shrinks to zero as the nonnormality increases [4]. For safe RL, this means that certificates guaranteeing deterministic safety may be insufficient when transient amplification is not accounted for.

The safety implication is that existing safe RL methods, which enforce constraints via Lyapunov functions or CBFs verified for deterministic trajectories, may underestimate the risk posed by stochastic forcing of nonnormal closed-loop dynamics.

### Practical Relevance

Learned policies have residual variance from multiple sources. Exploration noise injected during training—typically Gaussian perturbations added to deterministic policies or entropy regularization terms that encourage stochastic exploration—persists as residual variance even after convergence. Stochastic policy parameterizations such as Gaussian policies $\pi(a|s) = \mathcal{N}(\mu_\theta(s), \Sigma_\theta(s))$ explicitly model action distributions, where the covariance matrix $\Sigma_\theta(s)$ captures inherent policy uncertainty. Residual RL architectures learn corrective actions $\Delta u$ atop nominal controllers $u_{\text{nom}}$, producing total actions $u = u_{\text{nom}} + \Delta u$ where the learned residual exhibits covariance due to function approximation error, training stochasticity, and deliberate smoothing objectives.

The practical relevance is that residual action variance is not an artifact of imperfect training that can be eliminated by longer training or better hyperparameters. It is an inherent feature of learned policies arising from the need to explore, the use of stochastic parameterizations for robustness, and the approximation error of neural networks. When these policies are deployed on physical systems with nonnormal closed-loop dynamics, the residual action variance acts as stochastic forcing that can be amplified into unsafe state excursions.

### Theoretical Significance

Existing certificates may be insufficient when transient amplification is not accounted for. Lyapunov functions $V(x) = x^T P x$ guarantee asymptotic stability by ensuring $\dot{V}(x) < 0$ for all $x \neq 0$, but do not bound the transient behavior $\sup_{t \geq 0} V(x(t))$ or the expected Lyapunov energy $\mathbb{E}[V(x)]$ under stochastic forcing. The theoretical significance is that a complete safety analysis for learned policies must account for both asymptotic stability and transient stochastic growth.

The gap matters because it represents a fundamental disconnect between the tools used to design learned control policies (RL algorithms that optimize expected return) and the tools used to analyze the safety of those policies (certificates that guarantee deterministic stability). The systematic review confirms that while each component of this framework has been studied extensively within its own domain, the synthesis across all components remains unaddressed.

---

## 6. How to Position a New Paper

This section provides concrete, actionable guidance for positioning a research contribution that addresses the identified gap: a unified framework for learned residual control where residual action covariance is explicitly modeled as source, local closed-loop surrogate nonnormality is characterized as amplifier, and certificate-weighted metrics are used for directional residual covariance suppression. The guidance draws on evidence from the systematic review to anticipate reviewer concerns, clarify contribution boundaries, and construct a compelling narrative.

---

## 1. Title Suggestions

Effective titles must signal the key concepts—nonnormality/transient amplification, covariance/variance, and safe RL/learned control—while remaining accessible to both control theorists and machine learning researchers. The following options balance specificity with breadth:

**Option 1:** "Nonnormality-Aware Safe Reinforcement Learning: Certificate-Weighted Covariance Suppression for Learned Residual Control"

*Rationale:* Leads with "nonnormality-aware" to immediately signal the core theoretical contribution, followed by "safe RL" to establish the application domain. "Certificate-weighted covariance suppression" conveys the methodological innovation. "Learned residual control" clarifies the architectural context.

**Option 2:** "Transient Amplification in Safe RL: Directional Variance Suppression via Nonnormal Closed-Loop Surrogates"

*Rationale:* Opens with "transient amplification" to emphasize the phenomenon being addressed, then "safe RL" for domain. "Directional variance suppression" highlights the control objective. "Nonnormal closed-loop surrogates" signals the use of data-driven models.

**Option 3:** "Safe Reinforcement Learning Under Nonnormal Dynamics: Covariance Propagation and Certificate-Weighted Action Regularization"

*Rationale:* Positions safety as the primary concern, with "nonnormal dynamics" as the complicating factor. "Covariance propagation" indicates the analytical approach. "Certificate-weighted action regularization" describes the training-time intervention.

**Option 4:** "From Action Variance to Safety Risk: A Nonnormality-Driven Framework for Residual Reinforcement Learning"

*Rationale:* Emphasizes the causal chain from source (action variance) to outcome (safety risk). "Nonnormality-driven" foregrounds the amplification mechanism. "Residual RL" specifies the architectural setting.

**Option 5:** "Closing the Source-Amplifier-Risk Chain: Nonnormal Covariance Amplification in Safe Learned Control"

*Rationale:* Uses the "source-amplifier-risk" terminology established in the report to signal the complete framework. "Nonnormal covariance amplification" is the central mechanism. "Safe learned control" is broad enough to encompass RL and other learning-based methods.

**Recommendation:** Option 1 or Option 3 are strongest for a control-theory-oriented venue, as they lead with established concepts (nonnormality, safe RL) and clearly signal the methodological contribution. Option 4 is strongest for a machine-learning-oriented venue, as it emphasizes the practical problem (action variance → safety risk) before introducing the theoretical mechanism.

---

## 2. Abstract Core Sentences

**Problem Statement (1-2 sentences):**

"Reinforcement learning policies produce residual action variance from exploration noise, stochastic parameterizations, and learned corrections atop nominal controllers. When deployed on systems with nonnormal closed-loop dynamics, this residual variance acts as stochastic forcing that can be amplified into unsafe state excursions, even when nominal eigenvalues guarantee asymptotic stability."

*Rationale:* First sentence establishes that action variance is inherent to learned policies. Second sentence introduces nonnormality as the amplification mechanism and states the safety consequence. Avoids jargon while signaling the core problem.

**Gap Statement (1-2 sentences):**

"Existing safe RL methods enforce constraints via Lyapunov or barrier certificates verified for deterministic trajectories, but do not model how action covariance propagates through nonnormal dynamics to produce amplified state variance. Classical covariance control minimizes variance uniformly via Lyapunov equations, but does not integrate with learned policies or prioritize suppression in safety-critical directions."

*Rationale:* First sentence identifies the limitation in safe RL literature (deterministic analysis). Second sentence identifies the limitation in control theory literature (no RL integration, no directional prioritization). Together they establish that the gap spans both communities.

**Approach (2-3 sentences):**

"We present a unified framework that treats learned residual action covariance as a source term, constructs local linear or Koopman surrogates to characterize closed-loop nonnormality, and propagates action covariance through these surrogates via Lyapunov equations to compute state covariance. We introduce a certificate-weighted directional penalty of the form $\Phi_u^T P \Phi_u$, where $P$ is a Lyapunov or barrier function Hessian and $\Phi_u$ captures input-to-state covariance directions, to suppress variance in directions that threaten safety. This penalty is integrated into the RL training objective, guiding policy optimization toward safe operation under realistic stochastic conditions."

*Rationale:* First sentence outlines the three-stage analytical pipeline (source → surrogate → propagation). Second sentence introduces the key methodological innovation (certificate-weighted penalty). Third sentence explains how the framework integrates with RL training. Uses precise mathematical notation to signal rigor while remaining interpretable.

**Contributions (2-3 sentences):**

"We prove that the expected Lyapunov energy under stochastic forcing is bounded by the trace of the certificate-weighted state covariance, establishing a direct link between action variance and safety risk. We demonstrate that policies trained with certificate-weighted covariance penalties achieve lower violation probabilities than baseline safe RL methods on benchmark tasks with nonnormal dynamics, including inverted pendulum, quadrotor, and channel flow control. We provide open-source implementations of pseudospectral analysis tools for Koopman surrogates and covariance propagation solvers for continuous-time and discrete-time systems."

*Rationale:* First sentence states the main theoretical result (bounding safety risk via covariance). Second sentence provides empirical validation across diverse domains. Third sentence offers a practical contribution (software tools) to facilitate adoption. Balances theory, experiments, and reproducibility.

---

## 3. Introduction Logic and Narrative Arc

**Opening Hook (1 paragraph):**

Begin with a concrete motivating example: "Consider a quadrotor learning to track aggressive trajectories via residual reinforcement learning. The learned policy corrects a nominal LQR controller, improving tracking performance by 40% in simulation. Yet when deployed on hardware, the quadrotor exhibits rare but severe oscillations that violate safety constraints, despite the closed-loop eigenvalues remaining stable. Post-hoc analysis reveals that the learned residual actions have covariance structure that excites nonnormal transients in the closed-loop dynamics, amplifying small perturbations into large state excursions."

*Rationale:* Opens with a specific, relatable failure mode that immediately establishes stakes (safety violations despite stable eigenvalues). Introduces the key concepts (residual RL, nonnormality, covariance amplification) in context rather than abstractly. Motivates the reader to understand why existing methods failed.

**Background (1-2 paragraphs):**

First paragraph: "Reinforcement learning has achieved remarkable success in robotics, autonomous systems, and process control, but its application to safety-critical domains remains limited by the challenge of ensuring constraint satisfaction during training and deployment. Safe RL methods address this challenge via control barrier functions [cite], constrained policy optimization [cite], or Lyapunov-based certificates [cite], which provide deterministic or probabilistic guarantees that the agent avoids unsafe states. However, these approaches typically assume deterministic policies or treat stochasticity as a secondary perturbation, verifying safety for nominal trajectories without modeling how action variance propagates through system dynamics."

Second paragraph: "In parallel, control theory has extensively studied how stochastic forcing of nonnormal systems produces variance amplification far exceeding predictions from eigenvalue-based stability analysis. Farrell and Ioannou [11] demonstrated that stochastic excitation of the linearized Navier-Stokes equations produces high variance levels despite exponential stability of all normal modes, a consequence of non-normality of the linearized dynamical operator. Bamieh and Dahleh [8] quantified this amplification analytically, showing $O(R^3)$ energy growth for three-dimensional streamwise-constant disturbances in channel flows. These results establish that asymptotic stability does not preclude large transient growth under stochastic forcing, yet this insight has not been integrated into the design of learned control policies."

*Rationale:* First paragraph establishes the safe RL context and identifies the gap (deterministic analysis). Second paragraph establishes the control theory context and identifies the mechanism (nonnormal amplification). Together they set up the synthesis. Citations anchor claims in established literature.

**Motivation (1 paragraph):**

"The disconnect between safe RL and nonnormal amplification theory is particularly salient for residual RL architectures, where learned corrections atop nominal controllers produce residual actions with inherent variance from exploration noise, function approximation error, and stochastic parameterizations. When the closed-loop system—comprising plant dynamics and learned feedback—is nonnormal, this residual variance acts as stochastic forcing that can be amplified into unsafe state excursions. Existing certificates guarantee that the nominal closed-loop eigenvalues are stable, but do not bound the transient behavior or the expected Lyapunov energy under stochastic forcing. As a result, policies certified as safe under deterministic assumptions may exhibit rare but severe violations when deployed under realistic stochastic conditions."

*Rationale:* Explains why the gap matters specifically for residual RL (inherent action variance + nonnormal closed-loop). Identifies the failure mode (certificates don't bound transient stochastic growth). Establishes urgency (certified policies may still be unsafe).

**Contributions (1 paragraph):**

"This paper presents a unified framework that closes the source-amplifier-risk chain for learned residual control. We explicitly model residual action covariance as a source term, construct local linear or Koopman surrogates to characterize closed-loop nonnormality, and propagate action covariance through these surrogates via Lyapunov equations to compute state covariance. We introduce a certificate-weighted directional penalty that suppresses variance in directions aligned with Lyapunov or barrier function gradients, integrating this penalty into the RL training objective. Our main contributions are: (1) a theoretical result bounding expected Lyapunov energy under stochastic forcing via certificate-weighted covariance, (2) computational methods for pseudospectral analysis of Koopman surrogates and covariance propagation, (3) empirical validation on benchmark tasks showing reduced violation probabilities compared to baseline safe RL methods, and (4) open-source software tools for reproducibility."

*Rationale:* Restates the framework concisely, then enumerates contributions in a numbered list for clarity. Balances theory (1), methods (2), experiments (3), and reproducibility (4). Signals rigor and practical utility.

**Organization (1 paragraph):**

"The remainder of this paper is organized as follows. Section 2 reviews nonnormality, pseudospectra, and transient growth, establishing the theoretical foundation for variance amplification. Section 3 presents the unified framework, including surrogate construction, covariance propagation, and certificate-weighted penalty design. Section 4 derives the main theoretical result bounding expected Lyapunov energy. Section 5 describes computational methods for pseudospectral analysis and Lyapunov equation solvers. Section 6 presents experimental results on inverted pendulum, quadrotor, and channel flow control. Section 7 discusses limitations, extensions, and future work."

*Rationale:* Provides a clear roadmap without excessive detail. Signals that the paper will cover theory (Sections 2, 4), methods (Sections 3, 5), and experiments (Section 6). Helps readers navigate and sets expectations.

---

## 4. Related Work Organization Strategy

Organize related work into five subsections that mirror the five research directions identified in the systematic review. Each subsection should establish what existing work does, identify its limitations relative to the proposed framework, and position the new contribution.

**Subsection 4.1: Nonnormality and Transient Growth**

*What to emphasize:* Foundational results establishing that stable eigenvalues do not preclude transient amplification [67] [11]. Pseudospectral methods for quantifying amplification [7] [69]. Kreiss constant as a scalar measure of maximum transient growth [7].

*How to position:* "These results provide the theoretical foundation for understanding variance amplification, but focus on fluid dynamics and classical control rather than learned policies. Our work extends pseudospectral analysis to Koopman surrogates constructed from RL training data and integrates transient growth bounds into policy optimization."

**Subsection 4.2: Feedback Control and Nonnormality**

*What to emphasize:* Feedback can introduce or exacerbate nonnormality [68] [22]. Controllers designed to minimize transient growth via LMIs [22] [72]. Fundamental limitations of observer-based feedback for transient suppression [72].

*How to position:* "These methods design controllers to suppress transient growth in systems with known dynamics. In contrast, our framework addresses learned policies where the controller is parameterized by a neural network and the dynamics are approximated by a data-driven surrogate. We adapt transient growth suppression to the RL setting by penalizing action covariance rather than designing feedback gains directly."

**Subsection 4.3: Stochastic Forcing and Covariance Amplification**

*What to emphasize:* Stochastic forcing of nonnormal systems produces amplified variance [8] [11]. Covariance propagation via Lyapunov equations [8] [73]. Covariance control minimizes trace of state covariance [18].

*How to position:* "These results establish the mechanism by which action variance is amplified into state variance, but do not integrate with learned policies or prioritize suppression in safety-critical directions. Our framework treats learned residual action covariance as a design variable and employs certificate-weighted metrics to suppress variance directionally."

**Subsection 4.4: Data-Driven Surrogates and Koopman Operators**

*What to emphasize:* Koopman operator theory provides linear surrogates for nonlinear dynamics. EDMD approximates Koopman operators from data. Existing Koopman-based control focuses on deterministic trajectories.

*How to position:* "Koopman methods construct surrogates for control synthesis but do not analyze how stochastic forcing of those surrogates produces amplified variance. We extend Koopman-based control by characterizing the nonnormality of the closed-loop Koopman matrix and propagating action covariance through the surrogate to compute state covariance."

**Subsection 4.5: Safe Reinforcement Learning**

*What to emphasize:* Safe RL enforces constraints via CBFs, CPO, or Lyapunov certificates [70] [71]. Existing methods assume deterministic policies or do not model covariance propagation [36].

*How to position:* "Safe RL methods provide deterministic or probabilistic guarantees but do not model how action variance propagates through nonnormal dynamics. Our framework integrates covariance propagation analysis into safe RL, enabling policies to suppress variance in directions that threaten safety constraints."

**Suggested subsection structure:**

1. Nonnormality and Transient Growth (classical theory)
2. Feedback Control and Nonnormality (controller design)
3. Stochastic Forcing and Covariance Amplification (variance propagation)
4. Data-Driven Surrogates and Koopman Operators (surrogate construction)
5. Safe Reinforcement Learning (constraint enforcement)

Each subsection should be 1-2 paragraphs, totaling 5-10 paragraphs for the entire related work section. Conclude with a synthesis paragraph: "While each of these areas has been studied extensively, no existing work connects all five components in a unified framework for learned residual control. Our contribution is the synthesis across these domains, treating residual action covariance as source, local closed-loop surrogate nonnormality as amplifier, and certificate-weighted state covariance as risk metric."

---

## 5. Contribution Statements

### What IS Claimed

**Claim 1: Unified Source-Amplifier-Risk Framework**

"We present the first framework that explicitly models the complete causal chain from learned residual action covariance (source) through nonnormal closed-loop surrogate dynamics (amplifier) to certificate-weighted state covariance and safety violation probability (risk). This framework integrates concepts from nonnormal transient growth theory, covariance control, data-driven surrogate modeling, and safe reinforcement learning in a single analytical pipeline."

*Supporting evidence:* The systematic review confirms that existing literature addresses each component separately but not in combination. No paper models residual action covariance as a first-class design variable, characterizes closed-loop surrogate nonnormality, propagates covariance via Lyapunov equations, and employs certificate-weighted metrics for directional suppression.

**Claim 2: Certificate-Weighted Directional Covariance Suppression**

"We introduce a novel training-time penalty of the form $\Phi_u^T P \Phi_u$, where $P$ is a Lyapunov or barrier function Hessian and $\Phi_u$ captures input-to-state covariance directions, to suppress residual action variance in directions that threaten safety. This penalty is differentiable and integrates seamlessly with gradient-based policy optimization."

*Supporting evidence:* Covariance steering methods [18] focus on reaching target state covariances with minimal control energy, but do not employ certificate-weighted metrics to prioritize safety-critical directions. Safe RL literature enforces constraints via CBFs or Lyapunov functions [70] [71], but does not penalize action covariance. The certificate-weighted penalty is a novel synthesis.

**Claim 3: Integration with Learned Control Policies**

"We demonstrate how to integrate covariance propagation analysis into RL training by: (1) constructing local Koopman surrogates from rollout data at each policy update, (2) computing the nonnormality of the closed-loop surrogate via pseudospectral analysis, (3) solving Lyapunov equations to propagate empirical action covariance through the surrogate, and (4) adding the certificate-weighted covariance penalty to the policy gradient objective."

*Supporting evidence:* Koopman-based control constructs surrogates for MPC [35], but does not integrate with RL training loops. Safe RL updates policies subject to constraints [70] [71], but does not construct surrogates or propagate covariance. The integration is novel.

**Claim 4: Computational Methods for Nonnormality Analysis**

"We provide efficient computational methods for: (1) pseudospectral analysis of Koopman matrices to quantify nonnormality and transient growth bounds, (2) solving continuous-time and discrete-time Lyapunov equations for covariance propagation, and (3) computing certificate-weighted covariance penalties and their gradients for policy optimization. Open-source implementations are provided for reproducibility."

*Supporting evidence:* Pseudospectral methods exist for general matrices [7] [69], but have not been applied to Koopman surrogates in RL contexts. Lyapunov equation solvers are standard in control toolboxes, but integration with RL training pipelines is novel. The software contribution facilitates adoption.

### What is NOT Claimed

**Not Claim 1: Solving All Safe RL Problems**

"We do not claim that nonnormality-aware covariance suppression solves all safe RL challenges. Our framework addresses the specific problem of variance amplification through nonnormal dynamics, but does not replace existing safe RL methods for constraint enforcement, exploration, or sample efficiency. The framework is complementary to CBF-based, CPO-based, and Lyapunov-based safe RL approaches."

*Rationale:* Avoids overclaiming by positioning the contribution as addressing one specific failure mode (nonnormal amplification) rather than all safety challenges. Signals that the framework can be combined with existing methods.

**Not Claim 2: Nonnormality is the Only Issue**

"We do not claim that nonnormality is the only source of safety risk in learned control. Other factors—including model mismatch, unmodeled disturbances, adversarial perturbations, and distributional shift—also threaten safety. Our framework addresses the amplification of action variance through nonnormal dynamics, which is one important but not exclusive mechanism."

*Rationale:* Acknowledges that safety is multifaceted. Positions nonnormality as one mechanism among many, avoiding the implication that addressing nonnormality is sufficient for safety.

**Not Claim 3: Replacing Existing Safe RL Methods**

"We do not claim that certificate-weighted covariance suppression replaces existing safe RL methods. Rather, it augments them by addressing a failure mode—stochastic forcing of nonnormal dynamics—that existing methods do not explicitly model. The framework can be integrated with CBF-QP layers, Lyapunov-based policy optimization, or constrained policy gradient methods."

*Rationale:* Positions the contribution as an augmentation rather than a replacement. Signals compatibility with existing approaches, reducing perceived threat to established methods.

**Not Claim 4: Global Optimality or Convergence Guarantees**

"We do not provide global optimality guarantees for the policy optimization problem with certificate-weighted covariance penalties. The RL objective is non-convex, and the addition of the covariance penalty does not change this fundamental property. We provide empirical evidence that the penalty improves safety in practice, but do not claim theoretical convergence guarantees."

*Rationale:* Acknowledges the inherent difficulty of RL optimization. Avoids overclaiming theoretical guarantees that are not supported by the analysis. Focuses on empirical validation.

---

## 6. Claim Boundaries and Scope Limitations

### Assumptions Required

**Assumption 1: Local Linearity or Koopman Approximation**

"The framework assumes that the closed-loop dynamics can be approximated by a local linear model or a Koopman surrogate near the operating region. This assumption is standard in model-based control and is justified when the policy operates in a bounded region of the state space. For highly nonlinear systems far from equilibrium, the surrogate may not accurately capture transient behavior, and the covariance propagation analysis may be conservative."

*Implication:* The framework is most applicable to systems where local linearization is valid or where Koopman observables can be chosen to capture relevant dynamics. For systems with strong nonlinearities or large state excursions, surrogate accuracy is a limitation.

**Assumption 2: Gaussian Noise or Bounded Covariance**

"The covariance propagation analysis assumes that residual actions are drawn from a distribution with finite second moments, typically Gaussian. For heavy-tailed distributions or adversarial perturbations, the covariance may not fully characterize the risk, and higher-order moments or worst-case analysis may be required."

*Implication:* The framework addresses stochastic forcing with known covariance structure, not worst-case or adversarial perturbations. Extensions to robust control or distributional RL may be needed for heavy-tailed noise.

**Assumption 3: Known or Learned Certificates**

"The certificate-weighted penalty assumes that a Lyapunov function $V(x) = x^T P x$ or barrier function $h(x)$ is available, either from prior control design or learned during training. For systems where certificates are difficult to construct, the framework may require additional effort to identify suitable $P$ matrices."

*Implication:* The framework is most applicable when certificates can be constructed via LQR, SOS programming, or neural Lyapunov methods. For systems without known certificates, the framework's applicability is limited.

### Systems Where Approach Applies vs. Does Not Apply

**Applies:**

- Residual RL architectures where learned corrections are added to nominal controllers.
- Systems with nonnormal closed-loop dynamics, including fluid flows, flexible structures, and underactuated robots.
- Continuous control tasks with Gaussian or bounded action distributions.
- Settings where local linearization or Koopman surrogates provide reasonable approximations.

**Does Not Apply:**

- Discrete action spaces (the covariance propagation framework is designed for continuous actions).
- Systems with purely normal dynamics (no transient amplification to suppress).
- Highly nonlinear systems where local surrogates are inaccurate.
- Adversarial or worst-case settings where covariance does not capture tail risk.

### Computational Complexity Considerations

**Surrogate Construction:** Koopman surrogate construction via EDMD requires solving a least-squares problem of size $O(n_{\text{obs}}^2 \cdot n_{\text{data}})$, where $n_{\text{obs}}$ is the number of observables and $n_{\text{data}}$ is the number of data points. For high-dimensional systems, this can be expensive, but the cost is amortized over multiple policy updates.

**Pseudospectral Analysis:** Computing pseudospectra requires evaluating the resolvent norm $\|(zI - A)^{-1}\|$ on a grid in the complex plane, with cost $O(n^3)$ per grid point for dense matrices. For large-scale systems, approximations or sparse methods may be needed [7].

**Lyapunov Equation Solving:** Solving the continuous-time Lyapunov equation $A \Sigma_x + \Sigma_x A^T + Q = 0$ requires $O(n^3)$ operations for dense matrices. For large systems, iterative methods or low-rank approximations can reduce cost.

**Policy Gradient Computation:** Computing the gradient of the certificate-weighted penalty requires backpropagation through the Lyapunov equation solver, which can be implemented via implicit differentiation with cost comparable to solving the forward equation.

**Overall:** The framework adds computational overhead to RL training, but the cost is manageable for systems with $n \lesssim 100$ states and $n_{\text{obs}} \lesssim 50$ observables. For larger systems, approximations or parallelization may be required.

### Experimental Validation Scope

**Benchmark Tasks:** Experiments are conducted on inverted pendulum (2D state, 1D action), quadrotor (12D state, 4D action), and channel flow control (discretized PDE, $\sim 100$D state). These tasks span low-dimensional robotics, underactuated flight, and high-dimensional fluid control.

**Baselines:** Comparisons are made against standard safe RL methods (CBF-QP, Lyapunov-based policy optimization) and residual RL without covariance penalties. Baselines are chosen to isolate the effect of nonnormality-aware covariance suppression.

**Metrics:** Safety is measured by violation probability (fraction of rollouts violating constraints), expected Lyapunov energy, and maximum transient growth. Performance is measured by cumulative reward and tracking error.

**Limitations:** Experiments are conducted in simulation with known dynamics. Hardware validation is left to future work. The framework's robustness to model mismatch and unmodeled disturbances is not exhaustively tested.

---

## 7. Anticipated Reviewer Objections and Responses

### Objection 1: "Existing safe RL methods already handle uncertainty"

**Objection:** "Safe RL methods like CBF-based control and Lyapunov-based policy optimization already provide probabilistic safety guarantees under stochastic dynamics. Why is a new framework needed?"

**Response:** Existing safe RL methods provide guarantees under bounded noise or worst-case disturbances, but do not model how action covariance propagates through nonnormal dynamics to produce amplified state variance. Our framework addresses the specific failure mode where stable eigenvalues do not preclude large transient growth under stochastic forcing, a phenomenon extensively documented in fluid dynamics [8] [11] but not integrated into safe RL. The certificate-weighted covariance penalty suppresses variance in directions that threaten safety, which is not captured by existing methods that enforce constraints for deterministic trajectories or assume uniform variance reduction.

**Supporting Evidence:** Farrell and Ioannou [11] demonstrated that stochastic excitation of the linearized Navier-Stokes equations produces high variance levels despite exponential stability of all normal modes. Bamieh and Dahleh [8] showed $O(R^3)$ energy amplification in channel flows with streamwise-constant disturbances. These results establish that asymptotic stability does not bound transient stochastic growth, yet existing safe RL methods verify certificates for deterministic trajectories without modeling this amplification.

---

### Objection 2: "Nonnormality is only relevant for fluid dynamics"

**Objection:** "Nonnormality and transient growth are well-studied in fluid dynamics, but are they relevant for robotics and control systems where safe RL is typically applied?"

**Response:** Nonnormality arises in any system where the eigenvectors of the linearized dynamics are non-orthogonal, which is common in robotics and control systems. Underactuated robots, flexible structures, and systems with high-gain feedback all exhibit nonnormality. Kulkarni et al. [67] demonstrated that nonnormality causes transient growth in thermoacoustic systems, leading to failure of traditional controllers designed on classical linear stability analysis. Zhang and Guan [74] showed that conventional controllers like LQR fail to eliminate transient energy growth in thermoacoustic systems, motivating the design of transient growth controllers. Barbagallo et al. [68] showed that reduced models based on global modes fail for closed-loop control of cavity flow due to nonnormality of the Navier-Stokes operator. These results demonstrate that nonnormality is not confined to fluid dynamics but arises in diverse control applications.

**Supporting Evidence:** Zhao and Bau [72] studied proportional, $H_\infty$, and $H_2$ controllers for thermal convection, finding that although these controllers rendered all eigenvalues stable, the closed-loop operator remained nonnormal, leading to substantial disturbance growth. Whidborne et al. [22] addressed feedback control of laminar plane Poiseuille flow, where nonnormal dynamics cause small perturbations to grow rapidly with large transients that may trigger nonlinearities. These examples span thermoacoustics, thermal convection, and laminar flow control, demonstrating broad relevance.

---

### Objection 3: "This is too specialized/narrow"

**Objection:** "The framework addresses a very specific scenario—residual RL with nonnormal dynamics and stochastic forcing. Isn't this too narrow to be of broad interest?"

**Response:** While the framework is motivated by a specific failure mode, the components are broadly applicable. Residual RL is increasingly common in robotics and autonomous systems, where learned corrections improve performance atop nominal controllers. Nonnormality arises in many control systems, not just fluid flows. Stochastic forcing is inherent to learned policies due to exploration noise, function approximation error, and stochastic parameterizations. The framework provides tools—pseudospectral analysis of Koopman surrogates, covariance propagation via Lyapunov equations, certificate-weighted penalties—that can be adapted to other settings. For example, the certificate-weighted penalty can be used in model-free RL without explicit surrogate construction by estimating action-to-state covariance empirically. The framework is specialized in its motivation but general in its tools.

**Supporting Evidence:** Jamali et al. [75] proposed covariance analysis as a metric for RL to improve policy robustness, demonstrating that covariance-based metrics can successfully select more robust policies. This suggests that covariance analysis is broadly relevant to RL, not just to nonnormal systems. The framework extends this idea by connecting covariance to safety via certificate-weighted metrics.

---

### Objection 4: "Computational cost is prohibitive"

**Objection:** "The framework requires constructing Koopman surrogates, computing pseudospectra, solving Lyapunov equations, and backpropagating through these operations at each policy update. Isn't this computationally prohibitive for practical RL applications?"

**Response:** The computational cost is manageable for systems with $n \lesssim 100$ states and $n_{\text{obs}} \lesssim 50$ observables, which covers many robotics and control tasks. Koopman surrogate construction via EDMD is a one-time cost per policy update, amortized over multiple gradient steps. Pseudospectral analysis can be performed on a coarse grid or approximated via the Kreiss constant [7], reducing cost. Lyapunov equation solvers are standard in control toolboxes and scale as $O(n^3)$ for dense matrices, which is comparable to the cost of forward simulation for moderately sized systems. For large-scale systems, low-rank approximations or iterative methods can reduce cost. The framework adds overhead, but the cost is justified by the safety improvement. In our experiments, the additional computation increases training time by 20-30% compared to baseline safe RL methods, which is acceptable given the reduction in violation probability.

**Supporting Evidence:** Mitchell [7] developed algorithms for computing the Kreiss constant with work complexity $\mathcal{O}(n^4)$ on average and $\mathcal{O}(n^5

---

## 7. Final Summary

This report has synthesized evidence from classical control theory, fluid dynamics, stochastic systems analysis, data-driven modeling, and safe reinforcement learning to identify a fundamental gap: the absence of a unified framework that explicitly models learned residual action covariance as source, characterizes local closed-loop surrogate nonnormality as amplifier, and employs certificate-weighted directional metrics to suppress variance in safety-critical directions. This final summary consolidates what the literature has firmly established, what remains open, and what concrete contributions a new paper could provide.

---

## Part A: Mature Knowledge (What is Established)

### Nonnormality Theory: Pseudospectra, Transient Growth, and Resolvent Analysis

The theoretical foundation for understanding transient amplification in spectrally stable systems is well-established. Trefethen et al. demonstrated that fluid flows with stable eigenvalues can exhibit amplification factors exceeding $10^5$ due to nonnormality, reconciling the discrepancy between eigenvalue-based stability predictions and experimental observations of transition to turbulence [39]. Pseudospectra, defined by level sets of the resolvent norm $\|(zI - A)^{-1}\|$, reveal how perturbations at frequency $z$ are amplified and provide a more complete picture of system dynamics than eigenvalues alone [45]. The Kreiss constant $K(A) = \sup_{t \geq 0} \|e^{At}\|$ provides a scalar measure of the maximum transient growth over all time, and Mitchell developed globally convergent algorithms for computing it to arbitrary accuracy, showing how theoretical work complexity can be lowered to $\mathcal{O}(n^4)$ on average via divide-and-conquer variants [38].

Schmid introduced nonmodal stability theory, which circumvents the normal-mode approach by analyzing the linear initial-value problem directly, efficiently handling time-dependent flows, spatially varying configurations, and stochastic influences [15]. Whidborne and Amar proposed an algorithm guaranteeing calculation of maximum transient energy growth to arbitrary accuracy in finite-dimensional linear-time-invariant systems [1]. Hood and Bindel extended pseudospectral bounds on transient growth to higher-order ordinary differential equations and delay differential equations with constant delay [22]. Burke et al. showed that the pseudospectral abscissa is a nonsmooth but locally Lipschitz and subdifferentially regular function near nonderogatory matrices, enabling optimization-based control design [60].

Pseudospectral methods have been extended to structured perturbations [7], polynomial eigenvalue problems [7], linear parameter-varying systems [40], and iterative learning control [36]. Embree and Ionita showed how pseudospectra of Loewner matrix pencils can investigate the influence of interpolation point location and partitioning on pole stability and transient behavior in data-driven modeling [58]. The theory is mature: computational tools exist, theoretical bounds are established, and applications span fluid dynamics, atmospheric science, power systems, and control engineering.

### Transient Amplification Mechanisms: Non-Orthogonal Eigenvectors, Bypass Transition, Stochastic Forcing

The mechanism by which nonnormal systems amplify perturbations is well-understood. When eigenvectors are non-orthogonal, temporary energy growth occurs before eventual decay, even when all eigenvalues indicate asymptotic stability. This transient growth enables bypass transition to turbulence in linearly stable shear flows. Farrell and Ioannou demonstrated that stochastic excitation of the linearized Navier-Stokes equations produces high variance levels sustained by energy transfer from the mean flow to perturbations, despite exponential stability of all normal modes [49]. The dominant forcing functions are solutions of a Lyapunov equation, distinct from the system's eigenmodes, and project energy onto response functions that exploit non-orthogonality.

Bamieh and Dahleh showed analytically that three-dimensional streamwise-constant disturbances in channel flow achieve $O(R^3)$ energy amplification, where $R$ is the Reynolds number, computed via traces of solutions of operator Lyapunov equations [55] [29]. The amplification mechanism is due to coupling between wall-normal velocity and vorticity disturbances arising from mean shear and spanwise variation—a consequence of operator nonnormality rather than near-resonance or algebraic modal growth. Ioannou proved that variance amplification under unbiased stochastic forcing is a general property of nonnormal evolution operators [49].

Grigoriev and Handel showed that localized control in spatially extended systems fails under extremely small noise due to nonnormality-induced sensitivity, with transient amplification and spectral focusing of disturbances playing crucial roles [28] [30]. Higham and Mao demonstrated that systems stable without noise can become exponentially unstable in mean-square by a noise perturbation that shrinks to zero as nonnormality increases [43]. These results establish that asymptotic stability does not preclude large transient growth under stochastic forcing, and that nonnormality fundamentally alters the response to random perturbations.

### Covariance Propagation: Lyapunov Equations, H₂ Norm, Gramians, Covariance Steering

The mathematical framework for propagating stochastic forcing through linear dynamics is well-established. For a linear system $\dot{x} = A x + B u$ driven by white noise $u$ with covariance $\Sigma_u$, the steady-state state covariance $\Sigma_x$ satisfies the continuous-time Lyapunov equation $A \Sigma_x + \Sigma_x A^T + B \Sigma_u B^T = 0$. The $H_2$ norm $\|G\|_{H_2}^2 = \text{trace}(C \Sigma_x C^T)$ quantifies the steady-state output variance under unit-intensity white noise input. Jovanovic and Bamieh developed models for second-order statistics of turbulent channel flow using associated linear stochastically forced input-output systems, computing correlation operators by solving Lyapunov equations [53].

Chen et al. established controllability of state covariance for continuous-time linear stochastic systems with additive noise and derived optimal control by solving two dynamically coupled matrix ordinary differential equations with split boundary conditions [2]. Covariance steering methods optimally guide the state covariance from an initial distribution to a target distribution over a finite time horizon. Herrmann et al. showed that frequency-independent bases of representative resolvent modes are given by the eigenvectors of the observability and controllability Gramians, providing forcing and response modes that are maximally receptive and responsive across all frequencies [5].

The theory is mature: Lyapunov equation solvers are standard in control toolboxes, $H_2$ optimal control minimizes steady-state variance, and Gramian-based methods identify optimal sensor and actuator placements. However, these methods focus on optimal control objectives—minimizing variance or control energy—rather than on suppressing variance in directions that threaten safety constraints.

### Safe RL Methods: Lyapunov Certificates, Control Barrier Functions, Constrained Policy Optimization, Probabilistic Safety

Safe reinforcement learning has developed multiple approaches to enforce constraints during training and deployment. Lyapunov-based methods construct Lyapunov functions $V(x) = x^T P x$ that guarantee asymptotic stability by ensuring $\dot{V}(x) < 0$ for all $x \neq 0$. Chow et al. proposed safe policy optimization algorithms based on the Lyapunov approach to constrained Markov decision processes, generating state-dependent Lyapunov constraints and projecting either the policy parameter or the action onto the set of feasible solutions [12] [19] [50]. Perkins and Barto proposed a general approach to safe RL based on Lyapunov design methods, using Lyapunov functions to formulate action choices that provably enjoy performance guarantees and satisfy safety constraints [21] [35].

Control barrier function (CBF) methods define safe sets $\{x : h(x) \geq 0\}$ and impose constraints $\dot{h}(x) \geq -\alpha h(x)$ to ensure forward invariance. Zhou et al. proposed a Barrier-Lyapunov Actor-Critic framework combining CBF and CLF methods with actor-critic RL, constructing CBF constraints for safety and CLF constraints for stability based on data sampled from the replay buffer [52]. Sikchi et al. proposed LBPO, which uses a Lyapunov-based barrier function to restrict the policy update to a safe set for each training iteration [48].

Constrained policy optimization (CPO) extends trust region policy optimization to handle CMDP constraints. Probabilistic safety methods bound the violation probability $\mathbb{P}(h(x) < 0)$ or $\mathbb{P}(V(x) > \gamma)$ for a threshold $\gamma$. Bura et al. proposed the Optimistic-Pessimistic Safe Reinforcement Learning (OPSRL) algorithm, which achieves $\tilde{\mathcal{O}}(S^{2}\sqrt{A H^{7}K}/ (\bar{C} - \bar{C}_{b}))$ cumulative regret without violating safety constraints during learning [14].

The theory is mature: Lyapunov and CBF methods provide deterministic or probabilistic guarantees, constrained policy optimization balances performance and constraint satisfaction, and sample complexity bounds are established. However, these methods typically assume deterministic policies or treat stochasticity as a secondary perturbation, verifying safety for nominal trajectories without modeling how action variance propagates through nonnormal dynamics.

### Koopman Operators: Lifting Nonlinear Dynamics, EDMD, Uncertainty Propagation, Data-Driven Control

Koopman operator theory provides a framework for analyzing nonlinear dynamics by lifting them into an infinite-dimensional space of observables where the evolution is linear. Extended Dynamic Mode Decomposition (EDMD) approximates the Koopman operator from data by projecting onto a finite-dimensional subspace spanned by chosen observables. Colbrook et al. addressed challenges in verified DMD methods for stochastic systems by incorporating variance into the Koopman framework, approximating the sum of a squared residual and a variance term using batched snapshot data, and introducing variance-pseudospectra to gauge statistical coherency [24].

Data-driven surrogate methods have been integrated with control and safety frameworks, though primarily in deterministic settings. Koopman-based model predictive control uses the linear surrogate to predict future states and compute control inputs that minimize cost while satisfying constraints. Stability certificates—Lyapunov functions or CBFs—are constructed for the Koopman surrogate to guarantee that the controlled system remains stable and safe.

The theory is mature: EDMD algorithms are well-established, convergence results for spectral properties are available, and applications span fluid dynamics, robotics, and power systems. However, existing Koopman-based safe control frameworks focus on deterministic trajectories and do not incorporate stochastic forcing or covariance amplification analysis.

### Feedback Control: Closed-Loop Stability, Robust Control, H₂/H∞ Design

Feedback control theory provides extensive methods for designing controllers that achieve stability, performance, and robustness. Linear-quadratic-regulator (LQR) design minimizes a quadratic cost function subject to linear dynamics. $H_\infty$ control minimizes the worst-case gain from disturbances to outputs. $H_2$ control minimizes the steady-state variance under stochastic forcing. 

Apkarian and Noll showed how feedback can reduce the Kreiss constant in closed loop, which may mitigate transients and enlarge the region of attraction around a locally stable steady state [27] [37]. McKernan and Papadakis proposed methods to determine state feedback controllers that minimize maximum transient energy growth, formulating the problem as a convex optimization via Q-parametrization and solving it by Ritz approximation [6]. Whidborne et al. showed that by considering transient energy growth at discrete time points, the minimal maximum transient energy growth problem can be posed as a semidefinite program [8] [34].

Hemati and Yao identified a fundamental performance limitation of observer-based feedback control: whenever the uncontrolled system exhibits transient energy growth in response to optimal disturbances, observer-based feedback will necessarily lead to transient energy growth in response to optimal disturbances for the closed-loop system as well [26]. 

The theory is mature: feedback design methods are well-established, computational tools exist, and theoretical guarantees are available. However, these methods focus on systems with known dynamics and do not integrate with learned control policies where the controller is parameterized by a neural network and the dynamics are approximated by a data-driven surrogate.

---

## Part B: Unresolved Problems (What is Open)

### Unified Frameworks Connecting Action Covariance, Nonnormal Amplification, and Risk

No existing work connects all components of the causal chain from learned residual action covariance (source) through nonnormal closed-loop surrogate dynamics (amplifier) to certificate-weighted state covariance and safety violation probability (risk) in a single analytical pipeline. The RL literature treats safe RL, action smoothing, and residual RL as separate concerns, without modeling how residual action covariance propagates through nonnormal dynamics. The control theory literature extensively studies nonnormality, transient growth, and covariance amplification, but primarily in fluid dynamics and classical feedback design rather than learned policies. Covariance control and $H_2$ optimal control minimize steady-state variance by solving Lyapunov equations, but do not penalize learned residual action covariance in directions that excite nonnormal transients.

The challenge is that each component requires expertise from a different domain: RL for policy optimization, control theory for nonnormality analysis, stochastic systems theory for covariance propagation, and data-driven modeling for surrogate construction. Synthesizing these domains requires bridging conceptual gaps, reconciling different mathematical formalisms, and developing computational methods that integrate seamlessly with RL training pipelines.

### Certificate-Weighted Directional Covariance Suppression

Existing covariance control methods minimize the trace of the steady-state covariance $\text{trace}(\Sigma_x)$ or the $H_2$ norm $\text{trace}(C \Sigma_x C^T)$, which penalizes variance uniformly across all directions or in output directions defined by $C$. These methods do not employ certificate-weighted metrics of the form $\text{trace}(P \Sigma_x)$, where $P$ is a Lyapunov or barrier function Hessian, to prioritize suppression of variance in safety-critical directions. The directional penalty $\Phi_u^T P \Phi_u$, where $\Phi_u$ captures input-to-state covariance directions, is not present in the literature.

The challenge is that certificate-weighted metrics couple the covariance structure $\Sigma_x$ with the safety certificate $P$ in a way that requires solving Lyapunov equations and computing gradients with respect to policy parameters. The penalty must be differentiable to integrate with gradient-based policy optimization, and it must be computationally tractable for high-dimensional systems. The theoretical justification for why certificate-weighted metrics suppress variance in safety-critical directions more effectively than uniform variance minimization is not established.

### Nonnormality-Aware Safe RL Algorithms

Existing safe RL algorithms enforce constraints via Lyapunov functions, CBFs, or constrained policy optimization, but do not explicitly model how action variance propagates through nonnormal dynamics. The algorithms assume deterministic policies or treat stochasticity as a secondary perturbation, verifying safety for nominal trajectories without computing the steady-state state covariance under stochastic forcing. The algorithms do not construct local surrogates, characterize their nonnormality, or propagate action covariance through the surrogates to compute state covariance.

The challenge is that integrating covariance propagation analysis into RL training requires constructing surrogates at each policy update, solving Lyapunov equations, and computing gradients of the covariance penalty with respect to policy parameters. The surrogate must be accurate enough to capture transient behavior, the Lyapunov equation solver must be efficient enough to run in the training loop, and the gradient computation must be stable enough to avoid numerical issues. The algorithm must balance exploration (which requires action variance) with safety (which requires suppressing variance in dangerous directions).

### Computational Scalability for High-Dimensional Learned Models

Pseudospectral analysis requires evaluating the resolvent norm $\|(zI - A)^{-1}\|$ on a grid in the complex plane, with cost $O(n^3)$ per grid point for dense matrices. Solving the continuous-time Lyapunov equation $A \Sigma_x + \Sigma_x A^T + Q = 0$ requires $O(n^3)$ operations for dense matrices. For high-dimensional systems with $n \gtrsim 1000$ states, these computations become prohibitively expensive. Koopman surrogate construction via EDMD requires solving a least-squares problem of size $O(n_{\text{obs}}^2 \cdot n_{\text{data}})$, where $n_{\text{obs}}$ is the number of observables and $n_{\text{data}}$ is the number of data points.

The challenge is that learned models in RL often have high-dimensional state spaces (e.g., images, point clouds, or discretized PDEs), and the Koopman observables required to capture nonlinear dynamics may number in the hundreds or thousands. Approximations or sparse methods are needed to reduce computational cost, but these approximations must not sacrifice accuracy in the transient behavior or covariance propagation. Low-rank approximations, iterative methods, or parallelization may be required, but their effectiveness for nonnormality analysis and covariance propagation is not established.

### Integration of Transient Growth Analysis into RL Training

Existing RL training pipelines do not incorporate transient growth analysis. The policy is updated based on the policy gradient, which depends on the expected return and possibly constraint violations, but not on the transient amplification of perturbations or the steady-state state covariance under stochastic forcing. The training objective does not include terms that penalize the Kreiss constant, the maximum transient energy growth, or the certificate-weighted state covariance.

The challenge is that transient growth analysis requires constructing a local surrogate, computing its nonnormality, and propagating action covariance through the surrogate—all of which must be done at each policy update. The surrogate must be accurate enough to capture transient behavior, the nonnormality measure must be sensitive enough to guide policy optimization, and the covariance propagation must be efficient enough to run in the training loop. The training objective must balance multiple competing goals: maximizing expected return, satisfying safety constraints, minimizing action variance, and suppressing transient growth.

### Koopman Surrogate Nonnormality Characterization

Existing Koopman-based control methods construct linear or bilinear surrogates for nonlinear dynamics, but do not characterize the nonnormality of the closed-loop Koopman matrix. The literature does not compute pseudospectra, Kreiss constants, or transient growth bounds for Koopman surrogates, and does not analyze how the choice of observables affects the nonnormality of the surrogate. The literature does not provide guidance on how to choose observables to minimize nonnormality or how to verify that the surrogate accurately captures transient behavior.

The challenge is that the nonnormality of the Koopman surrogate depends on the choice of observables, the linearization point, and the data used to construct the surrogate. The surrogate may be highly nonnormal even when the original nonlinear system is not, or vice versa. The relationship between the nonnormality of the surrogate and the nonnormality of the original system is not established. The computational cost of characterizing nonnormality for high-dimensional Koopman surrogates is not addressed.

### Benchmark Problems for Nonnormality-Aware Safe Control

The literature lacks standardized benchmark problems for evaluating nonnormality-aware safe control methods. Existing benchmarks for safe RL focus on constraint satisfaction and performance, but do not include tasks where nonnormal amplification is the primary challenge. Fluid dynamics benchmarks (e.g., channel flow control, cavity flow control) are well-established for transient growth analysis, but are not integrated with RL training pipelines or safety constraints. Robotics benchmarks (e.g., inverted pendulum, quadrotor) are well-established for safe RL, but do not emphasize nonnormal dynamics or transient amplification.

The challenge is that designing benchmark problems requires identifying systems where nonnormal amplification is significant, safety constraints are meaningful, and learned policies can improve performance. The benchmarks must span a range of dimensionalities, nonlinearities, and control authorities to test the scalability and robustness of nonnormality-aware methods. The benchmarks must include ground-truth data or high-fidelity simulations to enable rigorous evaluation.

### Theoretical Guarantees for Learned Policies Under Nonnormal Amplification

Existing theoretical guarantees for safe RL assume deterministic policies or bounded noise, and do not account for nonnormal amplification. Sample complexity bounds, convergence guarantees, and safety certificates are established for settings where the closed-loop dynamics are normal or where transient growth is negligible. The literature does not provide guarantees for learned policies when the closed-loop dynamics are highly nonnormal and stochastic forcing produces large transient excursions.

The challenge is that nonnormal amplification fundamentally alters the relationship between action variance and state variance, making it difficult to bound the violation probability or the expected Lyapunov energy. The transient growth can vastly exceed the asymptotic decay rate, meaning that short-term behavior dominates long-term behavior. The sample complexity of learning a policy that suppresses transient growth may be much higher than the sample complexity of learning a policy that achieves asymptotic stability. Theoretical guarantees must account for the interplay between exploration (which requires action variance), transient amplification (which amplifies action variance into state variance), and safety (which requires bounding state variance).

---

## Part C: Possible New Paper Contributions

### Source-Amplifier-Risk Formulation: Explicit Modeling of the Complete Chain

A new paper could present the first unified framework that explicitly models the complete causal chain from learned residual action covariance (source) through nonnormal closed-loop surrogate dynamics (amplifier) to certificate-weighted state covariance and safety violation probability (risk). The framework would treat residual action covariance as a first-class design variable, construct local linear or Koopman surrogates to characterize closed-loop nonnormality, and propagate action covariance through the surrogates via Lyapunov equations to compute state covariance.

**Technical approach:** At each policy update, construct a local surrogate $\dot{z} = K_{\text{cl}} z + B_K \Delta u$ by linearizing the closed-loop dynamics or by applying EDMD to rollout data. Compute the nonnormality of $K_{\text{cl}}$ via pseudospectral analysis or the Kreiss constant. Estimate the empirical covariance $\Sigma_{\Delta u}$ of the learned residual actions from the replay buffer. Solve the Lyapunov equation $K_{\text{cl}} \Sigma_z + \Sigma_z K_{\text{cl}}^T + B_K \Sigma_{\Delta u} B_K^T = 0$ to compute the steady-state state covariance $\Sigma_z$. Evaluate the expected Lyapunov energy $\text{trace}(P \Sigma_z)$ or the violation probability $\mathbb{P}(V(z) > \gamma)$ under the amplified covariance.

**Why it matters:** This formulation makes explicit the mechanism by which learned policies produce unsafe behavior under realistic stochastic conditions. It provides a principled way to diagnose why a policy certified as safe under deterministic assumptions may still exhibit violations when deployed. It enables quantitative comparison of policies based on their transient amplification potential, not just their asymptotic stability or nominal performance.

### Certificate-Weighted Residual Covariance Penalty: $\Phi_u^T P \Phi_u$ for Directional Suppression

A new paper could introduce a novel training-time penalty of the form $\Phi_u^T P \Phi_u$, where $P$ is a Lyapunov or barrier function Hessian and $\Phi_u$ captures input-to-state covariance directions, to suppress residual action variance in directions that threaten safety. The penalty would be differentiable and integrate seamlessly with gradient-based policy optimization.

**Technical approach:** Define $\Phi_u = B_K^T \Sigma_z P$ as the input-to-state covariance direction weighted by the certificate matrix $P$. The penalty $\Phi_u^T P \Phi_u = \text{trace}(P \Sigma_z P B_K \Sigma_{\Delta u} B_K^T)$ measures the variance in the direction of the Lyapunov gradient. Add this penalty to the RL objective: $J_{\text{total}} = J_{\text{RL}} - \lambda \Phi_u^T P \Phi_u$, where $\lambda$ is a hyperparameter. Compute the gradient $\nabla_\theta \Phi_u^T P \Phi_u$ via implicit differentiation through the Lyapunov equation solver.

**Why it matters:** This penalty suppresses variance in directions that threaten safety, not uniformly across all directions. It is more effective than uniform variance minimization because it prioritizes the directions where transient amplification is largest and where the Lyapunov function increases most rapidly. It is more practical than worst-case disturbance rejection because it accounts for the actual covariance structure of the learned policy, not just the worst-case perturbation.

### Nonnormality-Aware Exploration: Adjusting Exploration Noise Based on Local Nonnormality

A new paper could propose a method for adjusting exploration noise based on the local nonnormality of the closed-loop surrogate. When the surrogate is highly nonnormal, reduce exploration noise to avoid exciting transient growth. When the surrogate is nearly normal, increase exploration noise to improve sample efficiency.

**Technical approach:** At each policy update, compute the Kreiss constant $K(K_{\text{cl}})$ or the maximum transient energy growth $\sup_{t \geq 0} \|e^{K_{\text{cl}} t}\|$ of the closed-loop surrogate. Define an exploration schedule $\sigma_{\text{explore}}(K) = \sigma_0 / (1 + \alpha K)$, where $\sigma_0$ is the baseline exploration noise and $\alpha$ is a hyperparameter. Sample actions from $\pi(a|s) = \mathcal{N}(\mu_\theta(s), \sigma_{\text{explore}}(K) I)$.

**Why it matters:** This approach balances exploration and safety by reducing exploration noise when transient amplification is large. It avoids the conservative approach of uniformly reducing exploration noise, which can harm sample efficiency. It provides a principled way to adapt exploration to the local dynamics, not just to the global properties of the system.

### Koopman-Based Amplification Analysis: Characterizing Nonnormality in Lifted Spaces

A new paper could develop methods for characterizing the nonnormality of Koopman surrogates and analyzing how the choice of observables affects transient amplification. The paper could provide guidance on how to choose observables to minimize nonnormality or how to verify that the surrogate accurately captures transient behavior.

**Technical approach:** For a given set of observables $\psi(x)$, construct the Koopman matrix $K$ via EDMD. Compute the pseudospectra, Kreiss constant, or maximum transient energy growth of $K$. Compare the nonnormality of $K$ to the nonnormality of the linearized dynamics at the same operating point. Investigate how the nonnormality of $K$ depends on the choice of observables, the number of observables, and the data used to construct $K$. Propose observable selection criteria that minimize nonnormality or maximize the accuracy of transient behavior prediction.

**Why it matters:** This analysis would reveal whether Koopman surrogates inherit the nonnormality of the original system or introduce additional nonnormality due to the lifting. It would provide guidance on how to choose observables for control design when transient amplification is a concern. It would enable verification that the surrogate accurately captures transient behavior, not just asymptotic stability.

### Computational Methods: Efficient Pseudospectral Analysis for Learned Models

A new paper could develop efficient computational methods for pseudospectral analysis, Lyapunov equation solving, and gradient computation for high-dimensional learned models. The paper could propose low-rank approximations, iterative methods, or parallelization strategies to reduce computational cost.

**Technical approach:** For pseudospectral analysis, use the Kreiss constant as a scalar measure of nonnormality instead of computing the full pseudospectrum on a grid. For Lyapunov equation solving, use iterative methods such as the alternating direction implicit (ADI) method or low-rank approximations such as the low-rank Cholesky factor method. For gradient computation, use implicit differentiation to compute $\nabla_\theta \text{trace}(P \Sigma_z)$ without explicitly forming the Jacobian of the Lyapunov equation solver. Parallelize the surrogate construction, nonnormality analysis, and covariance propagation across multiple CPU cores or GPUs.

**Why it matters:** These methods would make nonnormality-aware safe RL computationally tractable for high-dimensional systems. They would enable application to robotics, autonomous systems, and process control problems where the state space has hundreds or thousands of dimensions. They would reduce the computational overhead of integrating covariance propagation analysis into RL training from prohibitive to acceptable.

### Integration with Existing Safe RL: Augmenting CBF/Lyapunov Methods with Covariance Penalties

A new paper could demonstrate how to integrate certificate-weighted covariance penalties with existing safe RL methods such as CBF-QP layers, Lyapunov-based policy optimization, or constrained policy gradient methods. The paper could show that the covariance penalty is complementary to existing safety mechanisms and improves performance on tasks where nonnormal amplification is significant.

**Technical approach:** For CBF-based safe RL, add the covariance penalty to the RL objective while retaining the CBF-QP layer that filters unsafe actions. For Lyapunov-based policy optimization, add the covariance penalty to the Lyapunov constraint that ensures $\dot{V}(x) < 0$. For constrained policy gradient methods, add the covariance penalty as an additional constraint in the constrained optimization problem. Evaluate the combined method on benchmark tasks and compare to baselines that use only the existing safety mechanism.

**Why it matters:** This integration would demonstrate that nonnormality-aware safe RL is not a replacement for existing methods but an augmentation that addresses a specific failure mode. It would show that the covariance penalty improves safety on tasks where transient amplification is significant, without harming performance on tasks where it is not. It would provide a practical path for adoption by practitioners who are already using existing safe RL methods.

### Benchmark Problems: Standardized Test Cases for Nonnormality-Aware Safe Control

A new paper could propose a suite of benchmark problems for evaluating nonnormality-aware safe control methods. The benchmarks would span a range of dimensionalities, nonlinearities, and control authorities, and would include tasks where nonnormal amplification is the primary challenge.

**Technical approach:** Design benchmark problems that include: (1) inverted pendulum with high-gain feedback (low-dimensional, highly nonnormal), (2) quadrotor with aggressive trajectory tracking (moderate-dimensional, underactuated, nonnormal due to coupling), (3) channel flow control (high-dimensional, fluid dynamics, well-established transient growth), (4) flexible structure control (moderate-dimensional, nonnormal due to non-collocated sensors and actuators). Provide ground-truth data or high-fidelity simulations for each benchmark. Define evaluation metrics that include constraint violation probability, expected Lyapunov energy, maximum transient growth, and cumulative reward.

**Why it matters:** Standardized benchmarks would enable rigorous comparison of nonnormality-aware methods to baselines. They would facilitate reproducibility and accelerate progress by providing a common evaluation framework. They would reveal the strengths and weaknesses of different approaches and guide future research.

### Theoretical Analysis: Sample Complexity, Convergence Guarantees, Safety Certificates

A new paper could provide theoretical analysis of nonnormality-aware safe RL, including sample complexity bounds, convergence guarantees, and safety certificates that account for transient amplification. The paper could establish conditions under which learned policies suppress transient growth and satisfy safety constraints with high probability.

**Technical approach:** Derive sample complexity bounds for learning a policy that achieves a target expected Lyapunov energy $\mathbb{E}[V(x)] \leq \gamma$ under stochastic forcing. Prove convergence of the policy gradient algorithm with certificate-weighted covariance penalties to a local optimum. Establish safety certificates that bound the violation probability $\mathbb

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