Optimality conditions for bilevel optimal control problems with non-convex quasi-variational inequalities

¹ Laboratory LaSMA, Department of Mathematics, Sidi Mohammed Ben Abdellah University, Fes, Morocco
² School of Science and Engineering, Al Akhawayn University in Ifrane, PO Box104, Ifrane 53000, Morocco
³ Laboratoire LMMPA, ENS, Department of mathematics, Sidi Mohamed Ben Abdellah University, Fez, Morocco

^* Corresponding author: E.Kalmoun@aui.ma

Received: 18 December 2023
Accepted: 6 March 2024

Abstract

We establish Pontryagin optimality conditions for a generalized bilevel optimal control problem in which the leader is subject to a pure state inequality constraint, while the follower is governed by a non-convex quasi-variational inequality parameterized by the final state. To simplify the problem at hand, we convert it into a single-level optimal control problem by mapping the solution set of the quasi-variational inequality to a parametric optimization problem and employing the value function reformulation. Furthermore, we introduce certain regularity conditions to ensure that the derived maximum principle remains non-degenerate. Finally, we provide an illustrative example to elucidate our research findings.