Bilevel programming problems with simple convex lower level

Abstract

This article is dedicated to the study of bilevel optimal control problems equipped with a fully convex lower level of special structure. In order to construct necessary optimality conditions, we consider a general bilevel programming problem in Banach spaces possessing operator constraints, which is a generalization of the original bilevel optimal control problem. We derive necessary optimality conditions for the latter problem using the lower level optimal value function, ideas from DC-programming and partial penalization. Afterwards, we apply our results to the original optimal control problem to obtain necessary optimality conditions of Pontryagin-type. Along the way, we derive a handy formula, which might be used to compute the subdifferential of the optimal value function which corresponds to the lower level parametric optimal control problem.