Bilevel optimal control problems with pure state constraints and finite-dimensional lower level (Q2789613)

The problem is NEWLINE\[NEWLINE\min_{u, x, y}F(x(T), y),NEWLINE\]NEWLINE where \(x(T)\) is the final point of the solution of NEWLINE\[NEWLINE x'(t) = \phi(t, x(t), u(t)) \quad (0 \leq t \leq T),\quad x(0)=x_0.NEWLINE\]NEWLINE There are state constraints and control constraints: NEWLINE\[NEWLINE G(t, x(t))\leq 0,\quad u(t)\in\mathcal U.NEWLINE\]NEWLINE This is the upper level system or leader. It is connected with the lower level system or follower through \(y\), which must satisfy \(y \in \mathcal Y\), where \(\mathcal Y\) is the solution set of the problem NEWLINE\[NEWLINE \min_y f(x(T), y),\quad g(x(T), y) \leq 0.NEWLINE\]NEWLINE The object of this paper is to reduce the problem to a single level problem where a version of Pontryagin's maximum principle can be proved. Using a reformulation of the optimal value, the authors obtain a single level problem. Then, under additional assumptions on the constraints, they obtain a system for which the maximum principle is nondegenerate. The results are illustrated with an example.