Necessary and Sufficient Conditions for the Nonincrease of Scalar Functions Along Solutions to Constrained Differential Inclusions

TL;DR

This paper establishes necessary and sufficient conditions for scalar functions to be nonincreasing along solutions of constrained differential inclusions, aiding stability and safety analysis without solving the system.

Contribution

It provides a comprehensive set of infinitesimal conditions applicable under various regularity assumptions, advancing the theoretical understanding of Lyapunov and barrier functions.

Findings

Infinitesimal conditions for nonincrease are derived.

Conditions applicable to lower semicontinuous, Lipschitz, and differentiable functions.

Results do not require explicit solutions of the differential inclusions.

Abstract

In this paper, we propose necessary and sufficient conditions for a scalar function to be nonincreasing along solutions to general differential inclusions with state constraints. The problem of determining if a function is nonincreasing appears in the study of stability and safety, typically using Lyapunov and barrier functions, respectively. The results in this paper present infinitesimal conditions that do not require any knowledge about the solutions to the system. Results under different regularity properties of the considered scalar function are provided. This includes when the scalar function is lower semicontinuous, locally Lipschitz and regular, or continuously differentiable.