On continuation and convex Lyapunov functions

TL;DR

This paper demonstrates that under convex Lyapunov functions, a continuous deformation (homotopy) exists between vector fields maintaining global asymptotic stability, with implications for learning stability certificates and control systems.

Contribution

It establishes the existence of homotopies between vector fields with convex Lyapunov functions that preserve stability, extending to geodesic convexity and control Lyapunov functions.

Findings

Homotopy exists between vector fields with convex Lyapunov functions maintaining stability.

Results extend to geodesic convexity and convexity on convex sets.

Implications for stability certification and control system design.

Abstract

Suppose that the origin is globally asymptotically stable under a set of continuous vector fields on Euclidean space and suppose that all those vector fields come equipped with -- possibly different -- convex Lyapunov functions. We show that this implies there is a homotopy between any two of those vector fields such that the origin remains globally asymptotically stable along the homotopy. Relaxing the assumption on the origin to any compact convex set or relaxing convexity to geodesic convexity does not alter the conclusion. Imposing the same convexity assumptions on control Lyapunov functions leads to a Hautus-like stabilizability test. These results ought to be of interest in the context of learning stability certificates, policy gradient methods and switched systems.