Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems

We present optimality conditions for bilevel optimal control problems where the upper level, to be solved by a leader, is a scalar optimal control problem and the lower level, to be solved by several followers, is a multiobjective convex optimal control problem. Multiobjective optimal control problems arise in many application areas where several conflicting objectives need to be considered. Minimize several objective functionals leads to solutions such that none of the objective functional values can be improved further without deteriorating another. The set of all such solutions is referred to as efficient (also called Pareto optimal, noninferior, or nondominated) set of solutions. The lower level of the semivectorial bilevel optimal control problems can be considered to be associated to a ”grande coalition” of a p-player cooperative differential game, every player having its own objective and control function. We consider situations in which these p-?players react as ”followers” to every decision imposed by a ”leader” (who acts at the so-called upper level). The best reply correspondence of the followers being in general non uniquely determined, the leader cannot predict the followers choice simply on the basis of his rational behavior. So, the choice of the best strategy from the leader point of view depends of how the followers choose a strategy among his best responses. In this paper, we will consider two (extreme) possibilities: (i) the optimistic situation, when for every decison of the leader, the followers will choose a strategy amongst the efficient controls which minimizes the (scalar) objective of the leader; in this case the leader will choose a strategy which minimizes the best he can obtain amongst all the best responses of the followers: (ii) the pessimistic situation, when the followers can choose amongst the efficient controls one which maximizes the (scalar) objective of the leader; in this case the leader will choose a strategy which minimizes the worst he could obtain amongst all the best responses of the followers. This paper continues the research initiated in [17] where existence results for these problems have been obtained.