Differentiability properties of the minimum time function for normal linear systems

A result by Sussmann [32], implying that the minimum time function TT 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 TT 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 TT along optimal trajectories and we prove that normal cones to the epigraph of TT at each (x,T(x))(x,T(x)) and to the sublevel of TT at x have the same dimension.