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

(

H H

is a Hilbert space) and a convex constant dynamics

F\subset H F ⊂ H

we study the Lipschitz (or, in general, Hölder) regularity of the (unique) point

\pi_{C}^{F}\left( x\right) π C F ( x )

in

C C

achieved from

x x

for a minimal time. As a consequence, smoothness of the value function is proved, and an explicit formula for its derivative is given.