Compositionally Verifiable Vector Neural Lyapunov Functions for Stability Analysis of Interconnected Nonlinear Systems

While there has been increasing interest in using neural networks to compute Lyapunov functions, verifying that these functions satisfy the Lyapunov conditions and certifying stability regions remain challenging due to the curse of dimensionality. In this paper, we demonstrate that by leveraging the compositional structure of interconnected nonlinear systems, it is possible to verify neural Lyapunov functions for high-dimensional systems beyond the capabilities of current satisfiability modulo theories (SMT) solvers using a monolithic approach. Our numerical examples employ neural Lyapunov functions trained by solving Zubov's partial differential equation (PDE), which characterizes the domain of attraction for individual subsystems. These examples show a performance advantage over sums-of-squares (SOS) polynomial Lyapunov functions derived from semidefinite programming.

I. Introduction

One of the longstanding challenges in nonlinear control is the construction of Lyapunov functions for stability analysis and controller design. Since Lyapunov's seminal paper over a century ago [25], there has been an ongoing quest for constructive methods for the design of Lyapunov functions. Both analytical [15], [33] and computational [11], [12] strategies have been explored.