A Relaxation Theorem for Differential Inclusions with Applications to Stability Properties

TL;DR

This paper extends the relaxation theorem for differential inclusions to Banach spaces and discusses its implications for stability analysis, broadening the scope of solution approximation over infinite intervals.

Contribution

It generalizes the infinite-time relaxation theorem to Banach space inclusions, enabling approximation of solutions over non-compact intervals without fixed initial conditions.

Findings

Extension of relaxation theorem to Banach spaces.

Application to output stability analysis.

Enhanced solution approximation over infinite intervals.

Abstract

The fundamental Filippov-Wazwski Relaxation Theorem states that the solution set of an initial value problem for a locally Lipschitz inclusion is dense in the solution set of the same initial value problem for the corresponding relaxation inclusion on compact intervals. In our recent work, a complementary result was provided for inclusions with finite dimensional state spaces which says that the approximation can be carried out over non-compact or infinite intervals provided one does not insist on the same initial values. This note extends the infinite-time relaxation theorem to the inclusions whose state spaces are Banach spaces. To illustrate the motivations for studying such approximation results, we briefly discuss a quick application of the result to output stability and uniform output stability properties.