Carlo Mariconda, Non-occurrence of the Lavrentiev gap for a Bolza type optimal control problem with state constraints and no end cost, Vol. 2023 (2023), Article ID 12, pp. 1-18

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DOI: 10.23952/cot.2023.12

Received September 13, 2022; Accepted November 29, 2022; Published March 16, 2023

Abstract. Let $\Lambda:[t,T]\times{\mathbb R}^n\times {\mathbb R}^m\to [0, +\infty[\cup\{+\infty\}$ be Borel. We consider the problem ( $\mathcal P$ ) of minimizing an integral functional $F$ of the form $F(y, u):=\int_I\Lambda(s, y(s), u(s))\, ds$ in the set of admissible pairs $(y,u)$ such that $F(y,u)<+\infty$ and satisfy the following linear controlled dynamics, state and control constraints:
$y\in W^{1,1}([t,T];{\mathbb R}^n),$
$y'=b(y)u\text{ a.e. in } [t,T], \,y(t)=X\in{\mathbb R}^n,$
$u\in L^1(I;{\mathbb R}^m),\, u(s)\in \mathcal U\text{ a.e. } s\in [t,T],\, y(s)\in \mathcal S\,\,\forall s\in [t,T].$
We prove that if $\Lambda$ is radially convex on the control variable, locally Lipschitz in the time variable and a mild boundedness assumption (satisfied if $\Lambda$ is locally bounded where it is finite), then there is a minimizing sequence of admissible pairs with bounded controls. In the calculus of variations ( $b=1$ ) this corresponds to the non-occurrence of the Lavrentiev phenomenon for the problem with an initial constraint.

How to Cite this Article:
C. Mariconda, Non-occurrence of the Lavrentiev gap for a Bolza type optimal control problem with state constraints and no end cost, Commun. Optim. Theory 2023 (2023) 12.