Variational Analysis for the Bilateral Minimal Time Function

TL;DR

This paper develops formulas for the Fréchet subdifferentials of the bilateral minimal time function in differential inclusion systems, linking normals to sub-level sets and epigraphs, and analyzing their geometric properties.

Contribution

It introduces new formulas for Fréchet subdifferentials of the bilateral minimal time function and explores their geometric relationships in the context of differential inclusions.

Findings

Formulas for Fréchet subdifferentials of T

Connection between normals to sub-level sets and epigraph

Equality of dimensions of Fréchet normal cones

Abstract

In this paper, we derive formulas for the Fr\'echet (singular) subdiferentials of the bilateral minimal time function $T:\mathbb{R}^n \times \mathbb{R}^n \to [0,+\infty]$ associated with a system governed by differential inclusions. As a consequence, we give a connection between the Fr\'echet normals to the sub-level sets of $T$ and to its epigraph. Finally, we show that the Fr\'echet normal cones to the sub-level set of $T$ at a point $(\alpha,\beta)$ and to epi($T$) at $((\alpha,\beta),T(\alpha,\beta))$ have the same dimension.