Conjugate times and regularity of the minimum time function with differential inclusions

TL;DR

This paper investigates the regularity properties of the minimum time function in control systems modeled by differential inclusions, establishing sensitivity relations and conditions for smoothness along optimal trajectories.

Contribution

It introduces a sensitivity relation for the minimum time function and proves its local $C^2$ regularity along optimal trajectories, excluding conjugate times.

Findings

Proves the propagation of the proximal subdifferential along optimal trajectories.

Establishes local $C^2$ regularity of the minimum time function.

Provides conditions to exclude conjugate times in the analysis.

Abstract

This paper studies the regularity of the minimum time function, $T(\cdot)$, for a control system with a general closed target, taking the state equation in the form of a differential inclusion. Our first result is a sensitivity relation which guarantees the propagation of the proximal subdifferential of $T$ along any optimal trajectory. Then, we obtain the local $C^2$ regularity of the minimum time function along optimal trajectories by using such a relation to exclude the presence of conjugate times.