Exterior sphere condition and time optimal control for differential inclusions

TL;DR

This paper extends the regularity analysis of the minimum time function from smooth control systems to differential inclusions, establishing an exterior sphere condition under weaker hypotheses, which implies regularity without requiring Lipschitz continuity.

Contribution

It generalizes the exterior sphere condition for the minimum time function to differential inclusions using weaker assumptions like continuity and an inner ball property.

Findings

The hypograph of the minimum time function satisfies an exterior sphere condition locally.

This geometric property implies regularity results similar to semiconcavity without Lipschitz assumptions.

The analysis broadens the understanding of time optimal control regularity for differential inclusions.

Abstract

The minimum time function $T(\cdot)$ of smooth control systems is known to be locally semiconcave provided Petrov's controllability condition is satisfied. Moreover, such a regularity holds up to the boundary of the target under an inner ball assumption. We generalize this analysis to differential inclusions, replacing the above hypotheses with the continuity of $T(\cdot)$ near the target, and an inner ball property for the multifunction associated with the dynamics. In such a weakened set-up, we prove that the hypograph of $T(\cdot)$ satisfies, locally, an exterior sphere condition. As is well-known, this geometric property ensures most of the regularity results that hold for semiconcave functions, without assuming $T(\cdot)$ to be Lipschitz.