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On Bolza optimal control problems with constraints

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  • 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.

    Mathematics Subject Classification: 49J15, 49J30, 49K15, 49K30.


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