# American Institute of Mathematical Sciences

October  2000, 6(4): 975-997. doi: 10.3934/dcds.2000.6.975

## Semiconcavity for optimal control problems with exit time

 1 Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy 2 Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy 3 Dipartimento di Matematica, Università di Roma, Via della Ricerca Scientifica 1, 00133 Roma, Italy

Received  February 2000 Revised  July 2000 Published  August 2000

In this paper a semiconcavity result is obtained for the value function of an optimal exit time problem. The related state equation is of general form

$\dot y(t)=f(y(t),u(t))$,  $y(t)\in\mathbb R^n$, $u(t)\in U\subset \mathbb R^m$.

However, suitable assumptions are needed relating $f$ with the running and exit costs.
The semiconcavity property is then applied to obtain necessary optimality conditions, through the formulation of a suitable version of the Maximum Principle, and to study the singular set of the value function.

Citation: Piermarco Cannarsa, Cristina Pignotti, Carlo Sinestrari. Semiconcavity for optimal control problems with exit time. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 975-997. doi: 10.3934/dcds.2000.6.975
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