Computation and verification of contraction metrics for exponentially stable equilibria

TL;DR

This paper presents a combined computational and verification approach for contraction metrics to establish exponential stability of equilibria in dynamical systems, using Radial Basis Functions and CPA methods for rigorous validation.

Contribution

It introduces a novel method combining RBF-based approximation with CPA verification to rigorously compute and verify contraction metrics for stability analysis.

Findings

Successfully computes contraction metrics for given systems.

Provides a rigorous verification process for the computed metrics.

Demonstrates applicability through two example systems.

Abstract

The determination of exponentially stable equilibria and their basin of attraction for a dynamical system given by a general autonomous ordinary differential equation can be achieved by means of a contraction metric. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions decreases as time increases. The Riemannian metric can be expressed by a matrix-valued function on the phase space. The determination of a contraction metric can be achieved by approximately solving a matrix-valued partial differential equation by mesh-free collocation using Radial Basis Functions (RBF). However, so far no rigorous verification that the computed metric is indeed a contraction metric has been provided. In this paper, we combine the RBF method to compute a contraction metric with the CPA method to rigorously verify it. In particular, the computed contraction metric is interpolated by a continuous piecewise affine (CPA) metric at the vertices of a fixed triangulation, and by checking finitely many inequalities, we can verify that the interpolation is a contraction metric. Moreover, we show that, using sufficiently dense collocation points and a sufficiently fine triangulation, we always succeed with the construction and verification. We apply the method to two examples.