Differentiability properties of the minimum time function for normal linear systems

Abstract

A result by Sussmann [32], implying that the minimum time function T for normal linear systems is analytic out of a locally finite union of analytic submanifolds, is revisited. The original proof relies on classical properties of subanalytic sets. A constructive and simpler proof is given of a part of it: we show that the singular, i.e., the nondifferentiability, set of T is contained in a lower dimensionally rectifiable set which can be identified through suitable properties of the switching function. Our approach extends to large times the strategy initiated by Hájek in [23]. Furthermore, a result on the propagation of singularities of non-Lipschitz type is proved. Finally, we give explicit formulas for the propagation of first and second order partial derivatives of T along optimal trajectories and we prove that normal cones to the epigraph of T at each (x,T(x)) and to the sublevel of T at x have the same dimension.