Fréchet Analysis and Sensitivity Relations for the Optimal Time Problem

In this paper, we first present formulas for computing the Fréchet subdifferentials and Fréchet singular subdifferentials of the minimal time function
$\mathscr {T}$
for a differential inclusion in
$\mathbb {R}^{n}$
with a general target
$K$
. These formulas are characterized in terms of Fréchet normal cones to a sub-level set of
$\mathscr {T}$
and a level set of the minimized Hamiltonian associated with the differential inclusion. Our results extend, improve and complement to existing results. An application to sensitivity analysis for a class of time optimal control problems is also given.