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

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.

*minimal state*, the existence of a regular global attractor is proved.

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