lyapunov stability analysis

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.

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.

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 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.

Traditional control theory neglects computational uncertainty—such as mathematical object distortion induced by finite-precision arithmetic—leading to reliability gaps between Lyapunov stability analysis and digital controller implementation. Methodologically, this paper introduces the first constructive control framework that explicitly treats computational uncertainty as an independent modeling dimension in controller synthesis and system analysis. Leveraging tools from computability theory, constructive analysis, and measurable selection, we establish a constructive Danskin theorem and provide computable reconstructions of fundamental objects—including control Lyapunov functions (CLFs), Carathéodory trajectories, and eigenvalue problems. Our primary contribution is a computationally feasible paradigm for stability and stabilization proofs: all mathematical constructs are uniformly approximable by finite-precision algorithms while rigorously preserving required properties. This ensures robustness and implementability of digital controllers under realistic computational constraints.

研究通过构建Lyapunov解算子并使用Fourier神经算子来近似解决非线性系统稳定性分析中Lyapunov函数难以构建的问题。

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.

This work addresses the problem of feedback motion planning for continuous-time stochastic nonlinear systems under Signal Temporal Logic (STL) specifications by proposing a novel framework that integrates predicate erosion with probabilistic reachable tubes. Predicate erosion is employed to transform stochastic STL constraints into tightened deterministic ones, while probabilistic reachable tubes quantify the deviation of stochastic trajectories from their nominal counterparts. Leveraging contraction theory, a tracking controller is designed to establish a closed-loop planning pipeline. The proposed approach significantly reduces the conservatism inherent in conventional methods, achieving high STL satisfaction probability without compromising planning performance. Simulations and real-world experiments on a quadrupedal robot demonstrate that the method outperforms baseline approaches in both STL satisfaction rate and computational efficiency.

This study addresses the challenge of transient instability induced by finite-time disturbances in nonlinear stochastic flight dynamics, where conventional asymptotic or mean-square stability criteria fail to ensure short-term safety. The work presents the first extension of logarithmic norms to nonlinear Itô stochastic systems, establishing a finite-time transient stability analysis framework based on matrix measures of Lipschitz nonlinear mappings. By leveraging Itô calculus, it derives explicit bounds on the growth of state mean and variance, elucidating the fundamental distinction between expected-value stability and sample-path stability. Furthermore, it introduces a trade-off mechanism between estimation consistency and transient robustness under data injection. Experiments on lunar lander–like telemetry data demonstrate that trajectories with identical mean behavior can exhibit markedly different transient responses, with mission failure strongly correlated to cumulative transient instability during critical short intervals, thereby offering a novel finite-time probabilistic stability metric for autonomous systems.

This study addresses the topological obstruction that prevents continuous feedback from achieving globally asymptotically stable navigation on non-convex domains and manifolds. To overcome this theoretical limitation, the project introduces stochastic noise into the control channel and constructs stochastic feedback laws by integrating local Lyapunov stability with positive recurrence criteria. The proposed methodology is further extended to strongly convex optimization problems in Euclidean spaces with obstacles and on boundaryless manifolds. Numerical simulations conducted on circular obstacle domains and two-dimensional spheres validate the effectiveness of the approach while revealing metastability phenomena under non-convex obstacles. Ultimately, this work establishes a novel stochastic control theoretical framework for system navigation and optimization within environments characterized by complex topologies.