Journal of Convex Analysis 19 (2012), No. 3, 631--669
Copyright Heldermann Verlag 2012
Geometric Conditions for Regularity in a Time-Minimum Problem with Constant Dynamics
Vladimir V. Goncharov
CIMA-UE, Dep. de Matemática, Universidade de Évora, Rua Romăo Ramalho 59, 7000-671 Évora, Portugal
goncha@uevora.pt
Fátima F. Pereira
CIMA-UE, Dep. de Matemática, Universidade de Évora, Rua Romăo Ramalho 59, 7000-671 Évora, Portugal
fmfp@uevora.pt
[Abstract-pdf]
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.
Keywords: Time-minimum problem, Hoelder continuity, proximal, Frechet subdifferential, Clarke subdifferential, duality mapping, curvature, proximal smoothness.
MSC: 49J52, 49N15
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