Approximation of Bilevel Optimization Problems Involving an Optimal Control Problem at the Lower Level

This work studies a regularization procedure whose goal is the approximation of a solution to a bilevel optimization problem where the lower-level problem is formulated as an optimal control problem. This is done by regularizing the lower-level problem in such a way that an optimal control to the regularized problem can be explicitly computed as a feedback of state and costate functions. Convergence between the regularized problems and the original one is addressed: in particular, it is shown that as the regularization parameter tends to zero, the upper-level objective function computed on the optimal control to the regularized problem converges to the value function associated with the original program. The analysis that is carried out partly relies on the hypothesis that the lower-level problem admits a unique solution.