# American Institute of Mathematical Sciences

May  2009, 11(3): 629-653. doi: 10.3934/dcdsb.2009.11.629

## On Bolza optimal control problems with constraints

 1 Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma 2 Université Pierre et Marie Curie (Paris 6) case 189 - Combinatoire et Optimisation, 4 place Jussieu, 75252 Paris cedex 05, France 3 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy

Received  January 2008 Revised  September 2008 Published  March 2009

We provide sufficient conditions for the existence and Lipschitz continuity of solutions to the constrained Bolza optimal control problem

$\text{minimize}\quad \int_0^T L(x(t),u(t))\dt + l(x(T))$

over all trajectory / control pairs $(x,u)$, subject to the state equation

x'(t)=$f(x(t),u(t))$ for a.e. $t\in [0,T]$
$u(t)\in U$ for a.e. $t\in [0,T]$
$x(t)\in K$ for every $t\in [0,T]$
$x(0)\in Q_0\.$

The main feature of our problem is the unboundedness of $f(x,U)$ and the absence of superlinear growth conditions for $L$. Such classical assumptions are here replaced by conditions on the Hamiltonian that can be satisfied, for instance, by some Lagrangians with no growth. This paper extends our previous results in Existence and Lipschitz regularity of solutions to Bolza problems in optimal control to the state constrained case.

Citation: Piermarco Cannarsa, Hélène Frankowska, Elsa M. Marchini. On Bolza optimal control problems with constraints. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 629-653. doi: 10.3934/dcdsb.2009.11.629
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