Geometric Conditions for Regularity in a Time-Minimum Problem with Constant Dynamics

Continuing earlier research [``Neighbourhood retractions of nonconvex sets in a Hilbert space via sublinear functionals'', J. Convex Analysis 18 (2011) 1--36] on local well-posedness of a time-minimum problem associated to a closed target set $C\subset H$ ($H$ is a Hilbert space) and a convex constant dynamics $F\subset H$ we study the Lipschitz (or, in general, H\"{o}lder) regularity of the (unique) point $\pi_{C}^{F}\left( x\right)$ in $C$ achieved from $x$ for a minimal time. As a consequence, smoothness of the value function is proved, and an explicit formula for its derivative is given.