ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV)

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ESAIM: COCV
DOI: 10.1051/cocv:2008018

Interior Sphere Property for level sets of the value function of an exit time problem

Marco Castelpietra

Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Roma, Italy; castelpi@mat.uniroma2.it

(Received November 25, 2006. Revised September 18, 2007. Published online March 6, 2008.)

Abstract
We consider an optimal control problem for a system of the form $\dot{x}$ = f (x , u), with a running cost L. We prove an interior sphere property for the level sets of the corresponding value function V. From such a property we obtain a semiconcavity result for V, as well as perimeter estimates for the attainable sets of a symmetric control system.

Mathematics Subject Classification. 93B03, 49L20, 49L25

Key words: Control theory, interior sphere property, value function, semiconcavity, perimeter

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