Variational Analysis for the Bilateral Minimal Time Function

Abstract

In this paper, we derive formulas for the Fréchet (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 Fréchet normals to the sub-level sets of $T$ and to its epigraph. Finally, we show that the Fréchet 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.

The paper was accepted to publish on the Journal of Convex Analysis