Semiconcavity results and sensitivity relations for the sub-Riemannian distance

Regularity properties are investigated for the value function of the Bolza optimal control problem with affine dynamic and end-point constraints. In the absence of singular geodesics, we prove the local semiconcavity of the sub-Riemannian distance from a compact set Gamma subset of R-n. Such a regularity result was obtained by the second author and L. Rifford in Cannarsa and Rifford (2008) when Gamma is a singleton. Furthermore, we derive sensitivity relations for time optimal control problems with general target sets Gamma, that is, without imposing any geometric assumptions on Gamma. (C) 2019 Elsevier Ltd. All rights reserved