Variational analysis and regularity of the minimum time function for differential inclusions

TL;DR

This paper investigates the regularity and geometric properties of the minimum time function in optimal control problems for differential inclusions, providing new insights into its subdifferential propagation and convexity near targets.

Contribution

It introduces novel representations of subgradients and establishes the first nonlinear b4-convexity result for the epigraph of the minimum time function in any dimension.

Findings

Proximal horizontal subgradients of b4 are characterized.

Relationships between normal cones to sublevel sets and epigraph are established.

The epigraph of the minimum time function is shown to be b4-convex near the target.

Abstract

We study the time optimal control problem for differential inclusions with a general closed target. We first give the representation of the proximal horizontal subgradients of the minimum time function $\mathcal{T}$ and then, together with the representation of the proximal subgradients, we obtain some relationships between the normal cones to the sublevel of $\mathcal{T}$ and the normal cones to its epigraph. The relationships allow us to get the propagation of the proximal subdifferential as well as of the proximal horizontal subdifferential of $\mathcal{T}$ along optimal trajectories. Finally, we show, under suitable assumptions, that the epigraph of $\mathcal{T}$ is $\varphi$-convex near the target. This is the first nonlinear $\varphi$-convexity result valid in any dimension.