Lyapunov characterization of boundedness of reachability sets for infinite-dimensional systems

TL;DR

This paper establishes a Lyapunov-based framework to characterize the boundedness of reachability sets in infinite-dimensional control systems, extending classical results to broader classes including semi-linear evolution equations and ODEs.

Contribution

It provides a converse Lyapunov theorem for boundedness of reachability sets in infinite-dimensional systems, including a new result for forward completeness in ODEs without input restrictions.

Findings

Lyapunov functions characterize boundedness of reachability sets

Applicable to semi-linear evolution equations

Converse theorem for forward completeness in ODEs

Abstract

We prove a converse Lyapunov theorem for boundedness of reachability sets for a general class of control systems whose flow is Lipschitz continuous on compact intervals with respect to trajectory-dominated inputs. We show that this condition is satisfied by many semi-linear evolution equations. For ordinary differential equations, as a consequence of our results, we obtain a converse Lyapunov theorem for forward completeness, without a priori restrictions on the magnitude of inputs.