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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Semiconcavity for optimal control problems with exit time

Pages: 975 - 997, Volume 6, Issue 4, October 2000

doi:10.3934/dcds.2000.6.975       Abstract        Full Text (258.3K)       Related Articles

Piermarco Cannarsa - Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy (email)
Cristina Pignotti - Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy (email)
Carlo Sinestrari - Dipartimento di Matematica, Università di Roma, Via della Ricerca Scientifica 1, 00133 Roma, Italy (email)

Abstract: 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.

Keywords:  Optimal control problems, exit time problems, dynamic programming, optimality conditions, semiconcavity.
Mathematics Subject Classification:  49L20, 49K30.

Received: February 2000;      Revised: July 2000;      Published: August 2000.