Lyapunov Stability

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2 papers in the last four weeks, with none the four weeks before. 0.0% of all new papers.

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Latest papers 16

Oct 1, 2026cs.LG

Port-Hamiltonian Neural Networks for Systems with Multiple Asymptotically Stable Equilibria

Stable port-Hamiltonian neural networks certify asymptotic stability by construction. Yet, their Hamiltonian is a global Lyapunov function with a single global minimum, so they can represent only dynamic systems with one attractor. We demonstrate that this excludes even simple systems with energy landscapes forming a double well, and we overcome the restriction by parametrising the Hamiltonian as a product of Bregman divergences generated by one input-convex network. We prove that the resulting model is locally Lyapunov stable, that the coexistence of stable equilibria forces additional non-asymptotically-stable equilibria to exist, that all equilibria lie in a bounded region, and under a hyperbolicity assumption that almost-everywhere stability holds. On three systems our approach is able to recover the energy surface characteristics and improve the convergence speed by 1.8

\times

-8.5

\times

.

Simon Heilig, Jens Püttschneider, Mohammad Itani +2

Faculty of Computer Science, Ruhr University Bochum, Germany · Institute of Control Systems, Hamburg University of Technology, Germany

Dynamical Systems Lyapunov Stability Port-Hamiltonian Systems

Sep 16, 2026math.AP

Learning Lyapunov Operators for Nonlinear Systems

Constructing Lyapunov functions for nonlinear dynamical systems is a central problem in stability analysis, yet remains challenging. Lyapunov functions are commonly characterized as solutions to first-order partial differential equations (PDEs), but these solutions are typically obtained for single systems, limiting their reuse across systems. In this paper, we study the Lyapunov solution operator that maps a vector field to the corresponding Lyapunov function defined by a dissipation-based Lyapunov PDE. We establish that, on compact subsets of the domain of attraction and under exponential stability assumptions, this operator is well-defined, unique, and continuous with respect to perturbations of both the vector field and the dissipation function. These results provide a theoretical foundation for approximating Lyapunov functions uniformly over families of nonlinear systems. Building on these theoretical foundations, we employ Fourier Neural Operators (FNOs) as a data-driven approximation of the Lyapunov solution operator. Numerical experiments demonstrate that a single trained operator can accurately approximate the numerical Lyapunov functions across parameterized families of dynamics. This illustrates the potential of neural operators for approximating Lyapunov functions.

Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández +1

Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Dynamical Systems Lyapunov Stability

Jul 30, 2026cs.PF

Reflected UAS: Corrected Deterministic Stability and Direct CTMC Drift Calculation

We analyze Reflected UAS routing for heterogeneous multi-server queues at fixed parameters under subcritical load. The deterministic surrogate is a reflected ODE on the nonnegative orthant, not the unconstrained drift equation. This reflected ODE has a unique boundary equilibrium characterized by a scalar consistency equation and a convex-potential representation; all trajectories converge to it. The older argument lifting deterministic Lyapunov descent to CTMC stability fails: the exact generator applied to the deterministic potential produces a boundary term absent from the reflected-ODE descent identity. We give a direct Foster-Lyapunov drift inequality for the CTMC using a weighted-quadratic function, bypassing the failed lift. At the benchmark parameter point, the boundary equilibrium matches the numerical attractor to machine precision, and the default Reflected UAS policy has lower mean queue length than UAS and JSSQ across independent seed blocks.

Krishna Subedi

Neryva

Dynamical Systems Lyapunov Stability

Jul 20, 2026nlin.CD

Beyond the Edge of Chaos: Stability-Expressivity Transfer in Reservoir Forecasting

The edge-of-chaos heuristic has long served as a guiding principle for designing reservoir computers, yet its relevance to machine performance remains elusive. Here, taking the spectral radius of the reservoir network as the control parameter, we show that the radius yielding the best forecasting performance does not coincide with the Lyapunov edge of the isolated, teacher-forced, or closed-loop generative reservoir. By analyzing the collective dynamics of the teacher-forced reservoir, we find that the target dynamics are represented mainly by stable Lyapunov modes whose finite-time stability is strongly modulated by the input. This finding motivates a stability-expressivity transfer index, which balances the stability of these modes against their expressivity in representing the target. Across chaotic and quasiperiodic targets, and for both asymmetric and symmetric reservoirs, this index accurately identifies the optimal spectral radius for autonomous forecasting.

Yao Du, Xingang Wang

School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China

Lyapunov Stability Reservoir Computing Neural Network Expressivity Chaotic Time-Series Prediction

Jul 9, 2026eess.SY

Adaptive MPPI with Online Disturbance Covariance Estimation: Provable Stability Tightening via Spatial Smoothing

We study Model Predictive Path Integral (MPPI) control for nonlinear systems with additive process disturbances whose covariance is unknown, spatially varying, and slowly time-varying. A mismatched disturbance covariance produces a persistent penalty in closed-loop stability certificates, while online estimation can reduce this penalty as data are collected. We propose a cell-wise recursive covariance estimator with spatial diffusion and prove a finite-horizon error bound that separates stochastic-approximation error, spatial-smoothing bias, and temporal-drift effects. The diffusion kernel is chosen to be reversible with respect to the stationary visitation measure, making the diffusion operator dissipative in the weighted Lyapunov analysis. We then substitute the resulting covariance estimate into the MPPI sampling distribution and derive an adaptive stability certificate with an explicit learning penalty. The main result is a payoff theorem: after a computable crossover time, the adaptive controller achieves a strictly tighter certified stability bound than any fixed covariance choice whose mismatch exceeds the residual smoothing and drift allowance. Numerical experiments illustrate the estimator convergence and the resulting stability-tightening effect.

Hyung-Jin Yoon, Hunmin Kim

Department of Mechanical and Nuclear Engineering, Tennessee Technological University, Cookeville, TN, USA

Lyapunov Stability Stochastic Approximation MPPI Control Covariance Estimation Adaptive Control

Jul 4, 2026math.OC

Finite-Sample Closed-Loop Stability of Model Predictive Path Integral Control for Linear Time-Invariant Systems

We establish finite-sample closed-loop stability guarantees for Model Predictive Path Integral (MPPI) control applied to discrete-time Linear Time-Invariant (LTI) systems with additive Gaussian process disturbances. The key observation is that, for unconstrained LTI/quadratic systems with the DARE terminal cost, the exact finite-horizon MPC law has the same first control action as the infinite-horizon LQR law for every planning horizon. Thus, finite-sample MPPI can be analyzed as a stochastic perturbation of LQR. First, we show that the MPPI control law approximates the LQR feedback with high probability. The approximation error decomposes into a Monte Carlo term that decreases with the sample count and an infinite-sample temperature bias that persists at finite temperature but vanishes as the temperature is reduced. The resulting constants are written in terms of the horizon-dependent stacked cost matrices, making explicit that the finite-sample certificate is parametrized by the selected planning horizon. Second, we use a Lyapunov perturbation argument to prove practical exponential stability in expectation. On sample paths that remain in a compact Lyapunov sublevel set over a finite operating horizon, the expected state norm decays exponentially up to three residual floors: a process-noise floor, an MPPI approximation floor, and a confidence floor from the per-step sampling failure probability. The sufficient sample threshold is explicit and computable from the DARE solution, LQR stability margin, MPPI sampling parameters, temperature, and planning horizon. In the joint limit of infinite samples and vanishing temperature bias, the result recovers the stochastic LQR stability bound.

Hyung-Jin Yoon, Hunmin Kim

Department of Mechanical and Nuclear Engineering, Tennessee Technological University, Cookeville, TN, USA

Dynamical Systems Stochastic Linear Dynamical Systems Lyapunov Stability Stochastic Optimal Control MPPI Control

Jun 26, 2026eess.SY

Reachability Guarantees for Cart-Pole Swing-Up and Stabilization

The cart-pole swing-up is a canonical benchmark for nonlinear control of underactuated systems, yet an end-to-end guarantee linking the global swing-up maneuver to the local stabilizer is seldom formalized. We present a reachability analysis of a switched energy-based/LQR controller that certifies convergence to the upright equilibrium from a compact set of initial conditions. The swing-up law is derived from an energy-error Lyapunov function; canceling the autonomous conservative term yields a strictly sign-definite Lyapunov derivative, and convergence follows from LaSalle's invariance principle. We also propose an augmented Lyapunov function to regulate the steady-state cart velocity to zero, for which we establish almost-global convergence. For the controller handoff, a switching region is designed to lie strictly within the LQR region of attraction, formally certifying the swing-up-to-stabilization transition. Numerical simulations corroborate the theoretical analysis.

Mohamed Khalid M Jaffar

Department of Aerospace Engineering University of Maryland, College Park, USA

Reachability Analysis Lyapunov-Based Control Lyapunov Stability

Jun 24, 2026cs.AI

Feedback-Coupled Memory Systems in Continuous Time

The Feedback-Coupled Memory Systems (FCMS) architecture formalizes closed-loop coordination through four abstract operators, two of which - the agent update operator

f_i

and the environmental update operator

Ψ

- are left axiomatically undefined in the original framework. To address this,

f_i

is defined by Mechanism-Based Intelligence (MBI), where agents update locally through a decentralized price mechanism and economic principles, and

Ψ

is defined by the Coupled Memory Graph Process (CMGP), a non-Markovian framework where the environment is treated as a physical substrate that records and responds to trajectory history coherently without external forcing. The resulting continuous-time FCMS instantiation achieves Lyapunov global dissipativity governed by the computable threshold

4β^2 < 2ημγ^2

. This generalizes both the discrete FCMS stability condition

4ηβ^2 < γ

and CMGP's physical bifurcation threshold

α_c = 1/K

, confirming that memory dissipation must outpace feedback gain as a universal organizing principle. Numerical simulation with

N=2

agents and mean-field validation at

N=10^6

confirm the stability threshold and the self-reinforcing coordination cascade that emerges when it is violated.

Stefano Grassi

Bangkok University, Phahonyothin Rd, Khlong Nueng, Khlong Luang District, Pathum Thani 12120, Thailand

Dynamical Systems Multi-Agent Coordination Lyapunov Stability Feedback Control

Jun 7, 2026cs.LG

Between Amnesia and Chaos: A Memory Stability Expressivity Trilemma for Trainable Dissipative Oscillator Networks

Physical reservoir computing harnesses nonlinear mechanical dynamics but, by convention, freezes the substrate and trains only a linear readout, presuming the substrate is not usefully trainable. We revisit that premise for networks of nonlinear oscillators whose mass, damping, and stiffness are learned end-to-end through a symplectic integrator. Our central result is a trilemma: memory horizon, gradient stability, and dynamical expressivity cannot be simultaneously maximized, because all three are governed by the damping. The backward gradient decays at a rate set by the damping, capping how far back credit can propagate, while forward sensitivities grow exponentially in the largest Lyapunov exponent, so usable gradients require damping above a stability floor. Since the Lyapunov exponent falls as damping rises while the memory ceiling falls as the horizon grows, stable training is confined to a band that contracts with horizon and closes at a critical point. We test every step on a twenty-oscillator network. A damping sweep finds the largest Lyapunov exponent monotone and crossing zero at a well-defined stability floor, confirming the theorem's key assumption. A compute-matched comparison of learned versus frozen substrate on delayed recall across nine horizons shows the learned substrate dominating at short horizons and the advantage closing and reversing near a horizon of eleven steps, the predicted signature of band closure; trained models settle near the stability floor, seeking the edge of chaos unprompted. The analytic ceiling overestimates the empirical crossover roughly fivefold, a gap between detectable and learnable gradient that we report rather than tune away. The contribution is a confirmed account of when training a physical substrate beats freezing it.

Caleb Munigety

Independent Researcher

Dynamical Systems Lyapunov Stability Reservoir Computing Neural Network Training Dynamics

Jun 2, 2026cs.RO

Let the Dynamics Flow: Stable Flow Matching Dynamical Systems

Flow matching has recently emerged as a powerful approach for imitation learning, enabling scalable, expressive, and multimodal motion policies. However, when modeling these policies as dynamical systems, incorporating formal stability guarantees into these generative models is a prerequisite to ensure safe and generalizable robot behaviors, which remains a significant challenge. This paper introduces Stable Flow Matching Dynamical Systems (SFMDS), a novel framework that bridges the gap between highly expressive generative modeling and formal stability guarantees. SFMDS parametrizes dynamical systems via flow matching while constraining the model to satisfy positive invariance and/or Lyapunov stability conditions. We propose two variants: a soft constraint based on a penalty term, and a hard structural constraint embedded directly into the model architecture. We further extend both formulations to Lie groups to robustly handle orientation trajectories. Experiments on benchmark datasets, in simulation, and on a humanoid robot show that SFMDS learns stable, scalable, and multimodal dynamical systems in low- and high-dimensional state spaces, enabling safe and expressive robot motion generation. SFMDS matches or outperforms state-of-the-art methods on unimodal datasets, while substantially improving performance on multimodal datasets, where competing approaches fail to capture multi-modal behaviors. Accompanying source code and video are available at: https://let-the-dynamics-flow.github.io/SFMDS/.

Rodrigo Pérez-Dattari, Francisco Leiva, Andrea Testa +3

Department of Robotics, Perception, and Learning, KTH Royal Institute of Technology, Stockholm, Sweden. · Advanced Mining Technology Center (AMTC) and Department of Electrical Engineering, Universidad de Chile, Santiago, Chile. · Bosch Center for Artificial Intelligence, Renningen, Germany. +1

Flow Matching Lyapunov Stability Robot Motion Generation Robot Imitation Learning Flow-Based Generative Modeling

May 29, 2026cs.LG

Non-Asymptotic Convergence of Stochastic Iterative Algorithms: A Lyapunov Framework

We survey Lyapunov-based techniques for the finite-time analysis of stochastic iterative algorithms, also known as stochastic approximation (SA) algorithms, for solving fixed-point equations

\bar{F}(x)=x

, where the operator

\bar{F}(\cdot)

can only be accessed through a noisy oracle. We first focus on the standard setting in which

\bar{F}(\cdot)

is contractive with respect to some norm and the noise is i.i.d., and explain how generalized Moreau envelopes serve as universal Lyapunov functions, regardless of the underlying norm. We then show how this framework yields mean-square convergence guarantees and applies to stochastic gradient descent, linear SA, and value-based reinforcement learning algorithms such as Q-learning and temporal-difference learning. Finally, we discuss extensions to Markovian noise, seminorm-contractive operators, dissipative operators, and high-probability bounds, and conclude with open problems. The goal is to present a unified and self-contained roadmap for the finite-time analysis of SA and its applications, especially in reinforcement learning.

Zaiwei Chen, Siva Theja Maguluri

Purdue IE · Georgia Tech IE

Lyapunov Stability Stochastic Approximation Temporal-Difference Learning Convergence Analysis

May 15, 2026cs.LG

Sharp Spectral Thresholds for Logit Fixed Points

Softmax feedback systems are a common mathematical core of entropy-regularized reinforcement learning, logit game dynamics, population choice, and mean-field variational updates. Their central stability question is simple: when does a self-reinforcing softmax system produce a unique and globally predictable outcome? Classical theory gives a conservative answer. By treating softmax as a unit-scale response, it certifies stability only in a strongly randomized regime. We prove that the classical approach misses an entire stable regime and does not identify the point at which the qualitative change truly occurs. For finite-dimensional affine logit systems, the sharp dimension-free Euclidean threshold is

β\|ΠWΠ\|_{\mathcal T\to\mathcal T}<2,

rather than the previously used condition, which certifies stability only while the softmax system remains safely over-regularized. Our theorem fills the previously missing pre-bifurcation regime, extending stability guarantees for affine softmax feedback systems to reward-responsive yet globally predictable systems. It enlarges the certified stability boundary for these systems and identifies where the model genuinely undergoes a phase transition.

Tongxi Wang

Southeast University Nanjing, China

Dynamical Systems Lyapunov Stability Entropy Regularization

May 8, 2026math.OC

A Unified Lyapunov-IQC Framework for Uniform Stability of Smooth Quadratic First-Order Accelerated Optimizers

We develop a unified Lyapunov-integral quadratic constraint (IQC) framework for establishing uniform stability of first-order accelerated optimization algorithms in the

β

-smooth and

γ

-strongly convex regime. Classical analyses of uniform stability, such as the work of Hardt, Recht, and Singer for stochastic gradient descent (SGD), rely on direct coupling arguments and case-by-case control of iterate differences under random sampling. Extending such arguments to accelerated methods, such as Nesterov Accelerated Gradient (NAG), is complicated by the presence of higher-order state dynamics induced by momentum. We first extend this classical approach with the use of Lyapunov functions to provide a uniform stability bound for smooth quadratic NAG, and supplement this result with small-scale numerical experiments. We then extend this framework by modeling first-order accelerated optimizers as Lur'e-type feedback interconnections between a linear dynamical system and a (non-linear) gradient operator.

β

-Smoothness and

γ

-strong convexity are encoded a sector IQC inequality. Under this representation, uniform stability is certified via the existence of a quadratic Lyapunov function satisfying a finite-dimensional linear matrix inequality (LMI) in the form of a feasibility problem, which can be solved via semi-definite programming (SDP). We instantiate this framework for NAG and show how classical uniform stability bounds can be recovered via this framework. These results underscore a structural connection between optimization dynamics and robust control theory, providing a modular methodology for reliable and reproducible numerical certification of uniform stability and generalization behavior of first-order methods via convex optimization tools that is adaptable to increasingly complex optimization algorithms.

Don Li, Dacian Daescu

Department of Mathematics & Statistics, Portland State University, Portland, OR 97201, USA

Lyapunov Stability Algorithmic Stability Gradient Descent Accelerated Gradient Methods

May 3, 2026eess.SY

Stability of Control Lyapunov Function Guided Reinforcement Learning

Reinforcement learning (RL) has become the de facto method for achieving locomotion on humanoid robots in practice, yet stability analysis of the corresponding control policies is lacking. Recent work has attempted to merge control theoretic ideas with reinforcement learning through control guided learning. A notable example of this is the use of a control Lyapunov function (CLF) to synthesize the reinforcement learning rewards, a technique known as CLF-RL, which has shown practical success. This paper investigates the stability properties of optimal controllers using CLF-RL with the goal of bridging experimentally observed stability with theoretical guarantees. The RL problem is viewed as an optimal control problem and exponential stability is proven in both continuous and discrete time using both core CLF reward terms and the additional terms used in practice. The theoretical bounds are numerically verified on systems such as the double integrator and cart-pole. Finally, the CLF guided rewards are implemented for a walking humanoid robot to generate stable periodic orbits.

Zachary Olkin, William D. Compton, Aaron D. Ames

The authors are with the Department of Control and Dynamical Systems at the California Institute of Technology.

RL Control Lyapunov Stability RL for Legged Locomotion Optimal Control

Apr 28, 2026math.NA

State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization

We study gradient descent for rank-1 matrix factorization through a state-dependent Lyapunov perspective. The central object is a parameterized quadratic certificate

I(δ;\,\cdot)

whose boundary-inward property induces a monotone state parameter

δ_t

, thereby certifying that the trajectory is confined to a shrinking family of level sets. For certified initializations below the critical step size, this mechanism proves convergence to global minimizers. Above the critical step size, the same monotone-state mechanism instead leads to a balanced terminal regime; for a range of post-critical step sizes, the reduced dynamics exhibit period-2 behavior consistent with edge-of-stability phenomena. We further show that the scalar certificate is not an ad hoc algebraic construction: under structural axioms and a natural state-parameter normalization, it is uniquely determined by the monotonicity mechanism. Numerical experiments suggest that this state-dependent Lyapunov mechanism persists beyond the proved cases, including two-dimensional rank-1 approximation and quartic augmentations of scalar factorization.

Jaehong Moon

Industrial & Enterprise Systems Engineering University of Illinois at Urbana-Champaign Urbana, IL 61801

Edge of Stability Lyapunov Stability Low-Rank Matrix Decomposition Matrix Optimization Matrix Factorization Gradient Descent Dynamics

Feb 2, 2026cs.LG

Local exponential stability of mean-field Langevin descent-ascent and associated particle system

We study the mean-field Langevin descent-ascent (MFL-DA), a coupled optimization dynamics on the space of probability measures for entropically regularized two-player zero-sum games, together with its associated interacting particle system. For general nonconvex-nonconcave payoffs, Wang and Chizat (COLT 2024) asked whether the original single-timescale MFL-DA converges to the mixed Nash equilibrium and, if so, at what rate. We prove a local affirmative answer in Wasserstein space: if the initial datum is sufficiently close to the mixed Nash equilibrium, then the mean-field dynamics converges to it exponentially fast at a quantitative rate. We further show that the finite-

N

particle system inherits this stability up to times exponential in

N

, with an

N

-independent exponential rate modulo a finite-particle error floor. Combined with the recent counterexample of Mourrat and Pillaud-Vivien for MFL-DA, which shows that global convergence cannot hold in general, our theorem completes the positive local counterpart of the Wang-Chizat question: the mixed Nash equilibrium has a robust basin of attraction, stable under both the mean-field flow and its finite-particle approximation.

Geuntaek Seo, Minseop Shin, Pierre Monmarché +1

Nash Equilibrium Lyapunov Stability Game Theory Wasserstein Gradient Flows Zero-Sum Games Mean-Field Theory+1