Semiconcavity for optimal control problems with exit time

Piermarco Cannarsa; Cristina Pignotti; Carlo Sinestrari

doi:10.3934/dcds.2000.6.975

Article Contents

2000, Volume 6, Issue 4: 975-997. Doi: 10.3934/dcds.2000.6.975

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

Received: January 31, 2000

Revised: June 30, 2000

Published: September 2000

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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.

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Mathematics Subject Classification: 49L20, 49K30.

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