[Paper Review] Piecewise structure of Lyapunov functions and densely checked decrease conditions for hybrid systems

Matteo Della Rossa, Rafal Goebel+2 more|arXiv (Cornell University)|Dec 2, 2020

Stability and Controllability of Differential Equations43 references22 citations

TL;DR

This paper introduces a class of piecewise-structured, locally Lipschitz Lyapunov functions for hybrid dynamical systems that enables stability analysis by verifying Lyapunov decrease conditions only on a dense subset of the flow set, avoiding pointwise checks at nondifferentiable points. The key contribution is a less restrictive alternative to Clarke generalized gradient conditions, proven effective even when standard conditions fail at boundaries.

ABSTRACT

We propose a class of locally Lipschitz functions with piecewise structure for use as Lyapunov functions for hybrid dynamical systems. Subject to some regularity of the dynamics, we show that Lyapunov inequalities can be checked only on a dense set and thus we avoid checking them at points of nondifferentiability of the Lyapunov function. Connections to other classes of locally Lipschitz or piecewise regular functions are also discussed and applications to hybrid dynamical systems are included.

Motivation & Objective

• To develop a class of locally Lipschitz Lyapunov functions with piecewise structure for hybrid systems.
• To enable stability verification by checking Lyapunov decrease conditions only on a dense subset of the flow set, avoiding points of nondifferentiability.
• To provide less restrictive stability conditions than classical Clarke gradient-based approaches.
• To establish sufficient conditions for uniform global asymptotic stability (UGAS) in hybrid systems using these functions.
• To demonstrate the method with explicit constructions of nonconvex and convex Lyapunov functions for a 2D system.

Proposed method

• Introduces a class of locally Lipschitz functions built from pointwise maxima and minima of continuously differentiable functions, generalizing piecewise quadratic forms.
• Establishes that Lyapunov decrease inequalities need only be verified on a dense subset of the flow set C, under regularity assumptions on the dynamics F.
• Uses regularization and convolution techniques to analyze nonsmooth behavior and support the dense-check condition.
• Applies the framework to hybrid systems with flow on C and jump on D, combining dense-check conditions on C with standard jump conditions on D.
• Relies on converse Lyapunov theorems and properties of Clarke generalized gradients to compare the new conditions with existing ones.
• Constructs explicit examples: max of two indefinite quadratics, mid of three quadratics, and a convexified version via tangent line patching.

Experimental results

Research questions

• RQ1 Can Lyapunov decrease conditions be verified on a dense subset of the flow set, avoiding points of nondifferentiability?
• RQ2 How do the proposed conditions compare in restrictiveness to Clarke generalized gradient conditions?
• RQ3 Can this approach be applied to hybrid systems with discontinuous dynamics and state-dependent switching?
• RQ4 Can nonconvex and convex Lyapunov functions be constructed that satisfy the dense-check condition?
• RQ5 Is it possible to patch Lyapunov functions at the boundary of the flow set while preserving stability verification without relying on Clarke conditions?

Key findings

• The proposed Lyapunov functions allow stability verification by checking decrease conditions only on a dense subset of the flow set C, bypassing nondifferentiable points.
• The method yields less restrictive conditions than Clarke gradient-based approaches, especially when standard conditions fail at boundaries.
• For the 2D system with ε=0.1, all three constructed Lyapunov functions—V_M, V_mid, and V_conv—satisfy the conditions of Corollary 1, proving UGAS of the origin.
• Even though Clarke conditions fail on the line x₁=0 (boundary of C), the proposed method still establishes UGAS, demonstrating robustness to boundary issues.
• The convex Lyapunov function V_conv is constructed by patching V_mid with a quadratic form tangent to the boundary, ensuring continuity and satisfying the dense-check condition.
• The results show that intuitive patching at boundaries can be validated without relying on Clarke conditions, resolving a known limitation in the literature.