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In this paper, we first present formulas for computing the Fr&#x00E9;chet subdifferentials and Fr&#x00E9;chet singular subdifferentials of the minimal time function <inline-formula> <tex-math notation="LaTeX">$\mathscr {T}$ </tex-math></inline-formula> for a differential inclusion in <inline-formula> <tex-math notation="LaTeX">$\mathbb {R}^{n}$ </tex-math></inline-formula> with a general target <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>. These formulas are characterized in terms of Fr&#x00E9;chet normal cones to a sub-level set of <inline-formula> <tex-math notation="LaTeX">$\mathscr {T}$ </tex-math></inline-formula> 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.

Fréchet Analysis and Sensitivity Relations for the Optimal Time Problem