lyapunov function construction

This work addresses the finite-time convergence of stochastic iterative algorithms for fixed-point equations accessible only through a noisy oracle. The authors propose a norm-independent, unified Lyapunov function framework constructed via a generalized Moreau envelope, which integrates Lyapunov stability theory with stochastic approximation analysis. This framework accommodates complex settings such as Markovian noise, seminorm contractive operators, and dissipative operators, yielding sharp non-asymptotic convergence bounds in both high-probability and mean-square senses. As a result, it provides a unified and refined finite-time convergence guarantee for a broad class of algorithms, including stochastic gradient descent, linear stochastic approximation, Q-learning, and temporal difference learning.

Verifying stability of black-box nonlinear control systems is challenging when no prior dynamical model is available. Method: This paper proposes a model-free, data-driven stability verification method that directly learns the Lie derivative of a Lyapunov function—bypassing explicit system dynamics approximation. It integrates region-wise sampling-based validation with a counterexample-guided inductive synthesis (CEGIS) framework, underpinned by Lipschitz-based error bounds to ensure provably terminating synthesis. Contribution/Results: The approach certifies regional stability for 2D and 3D systems using only thousands of samples—requiring fewer than 0.01% of the samples needed by state-of-the-art black-box methods. It guarantees soundness and completeness within bounded regions, enables certified termination, and supports visualization of hard-to-verify stable regions.

This study unifies the intrinsic relationships among reactivity, contraction, and finite-time Lyapunov exponents (FTLE) in discrete-time dynamical systems. Method: We introduce a *p*-iteration system framework applicable to time-invariant, time-varying linear, and certain nonlinear maps, and rigorously establish equivalences and implication relations among the three properties. Based on this, we derive a *p*-iteration contraction criterion and extend the theory to synchronization stability analysis of coupled networks, leveraging matrix measures, operator norms, and finite-time stability theory. Contribution/Results: We prove that *p*-iteration contraction guarantees the existence of a globally asymptotically stable attractor (e.g., fixed point or limit cycle) in the original system. The proposed framework yields novel sufficient conditions for stable attractor existence and significantly improves both accuracy and applicability in synchronization analysis of coupled oscillator networks.

This work addresses stability analysis of nonlinear systems by proposing a Lyapunov function construction method that integrates physical priors with neural networks. Methodologically, it formulates the Zubov equation as a partial differential equation (PDE) constraint within a physics-informed neural network (PINN) framework and establishes theoretical guarantees for uniform approximation of the true region of attraction (ROA). It further introduces SMT-verifiable sufficient stability conditions—overcoming the scalability and conservatism limitations inherent in traditional sum-of-squares (SOS) and semidefinite programming (SDP) approaches. The method unifies PINN-based PDE solving, Zubov-type modeling, formal verification via SMT solvers (e.g., dReal, Barcelogic), and rigorous error convergence analysis. Experiments demonstrate that the proposed approach significantly improves ROA estimation tightness on multiscale and high-dimensional nonlinear systems, while achieving over an order-of-magnitude speedup in verification efficiency compared to SOS-SDP.

Estimating the maximal region of attraction (ROA) and constructing Lyapunov functions for nonlinear systems remains challenging due to the lack of scalable, certifiable methods. Method: This paper proposes the Taylor-Neural Lyapunov (TNL) framework, which synergistically combines local Taylor expansions with neural residual modeling to formulate maximal Lyapunov function learning as a verifiable, physics-informed neural network optimization problem. Crucially, TNL requires no simulation data and employs symbolic Lyapunov condition verification to provide formal convergence guarantees. Contribution/Results: TNL achieves end-to-end coupling between learning and rigorous control-theoretic robustness certification—the first method to do so. It generates strict numerical convergence certificates on multiple benchmark systems, matching the performance of sum-of-squares (SOS) and LyZNet. Moreover, it maintains high-accuracy ROA estimation even under zero-shot settings, significantly enhancing interpretability and trustworthiness in nonlinear stability analysis.

This work addresses the challenge of convergence failure in inverse parallel solvers for nonlinear systems of equations, which often arises due to oscillatory or chaotic dynamics. To enhance stability, the authors propose an adaptive stabilization mechanism based on the local maximum Lyapunov exponent (LLE). By estimating the LLE via k-nearest neighbors and integrating it with sliding-window micro-time-series analysis, the method enables real-time detection of unstable phases along the solution trajectory. A Lyapunov-guided parameter control strategy is then developed to dynamically adjust solver parameters, thereby reinforcing numerical stability. Experimental results demonstrate strong agreement between theoretical stability diagrams and empirical Lyapunov profiles, confirming that the proposed approach significantly improves the robustness and convergence performance of solvers under perturbed initial conditions.

This work addresses the lack of verifiable error guarantees in existing physics-informed neural networks (PINNs) when solving Lyapunov and Hamilton-Jacobi-Bellman (HJB) equations—specifically, whether small PDE residuals imply small solution errors remains unclear. To bridge this gap, we develop the first theoretical framework that provides rigorous, verifiable error bounds for PINN approximations of these critical partial differential equations arising in nonlinear system analysis and control. Our approach converts residual bounds into relative error bounds and a posteriori estimates for the true solution, proves that one-sided residual bounds suffice to guarantee the PINN approximation itself constitutes a valid Lyapunov function, and delivers computable upper and lower bounds on the optimal value function along with quantified optimality gaps for feedback policies in HJB problems. Numerical experiments demonstrate the effectiveness and practical utility of the proposed methodology.

This work addresses the lack of machine-verifiable foundations in control theory for cyber-physical systems by developing an open-source formal library within the Lean interactive theorem prover. The library formalizes Lyapunov stability theory and the small-gain theorem, supporting continuous, discrete, and hybrid dynamical systems. A key contribution is a unified formulation of Lyapunov’s theorem applicable to both points and sets, alongside a relational definition of input–output systems that avoids well-posedness assumptions, enabling a fully formalized proof of the small-gain theorem. Leveraging mathematical tools such as neighborhood filters, the project establishes a scalable verification framework for control theory, laying the groundwork for trustworthy, machine-checked validation of cyber-physical systems.

This work addresses the lack of a unified framework in existing optimizer design, which often relies on heuristic modifications and struggles to balance stability and generalization. The authors propose the first systematic approach that integrates control theory with Riemannian geometry, modeling the optimization process as a discrete-time controlled dynamical system on a Riemannian manifold. By introducing normally attracting invariant manifolds (NAIMs) and strict Lyapunov functions, they establish a theoretically grounded framework for generating optimizers with provable convergence guarantees. This framework not only recovers classical algorithms but also yields novel optimizers that achieve state-of-the-art performance on large-scale benchmarks. Geometric diagnostics further validate the method’s efficacy, offering a stable, interpretable, and theoretically rigorous toolkit for optimizer design.

This study addresses the problem of ensuring algorithmic convergence in complex dynamic systems—spanning physics, social sciences, and engineering—subject to external disturbances, stochastic noise, and coupled interactions. To overcome the limitation of existing theories in characterizing disturbance robustness, we systematically introduce the converse Lyapunov theorem into algorithmic convergence analysis for the first time, establishing a unified theoretical framework that jointly guarantees stability and quantifies convergence rates under perturbations. Our method integrates converse Lyapunov theory, nonlinear stability analysis, and quantitative robustness modeling, yielding explicit, computable bounds on convergence via quantitative perturbation inequalities. The framework is successfully applied to three domains: modeling communication constraints in distributed learning, analyzing generalization sensitivity in machine learning, and designing differential privacy mechanisms with calibrated noise injection. The results provide a verifiable, quantifiable theoretical foundation for dynamic algorithm design across disciplines.