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.
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