Semivectorial Bilevel Convex Optimal Control Problems: Existence Results

Abstract

We consider a bilevel optimal control problem 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. We deal with the so-called optimistic case, when the followers are assumed to choose a best choice for the leader among their best responses, as well with the so-called pessimistic case, when the best response chosen by the followers can be the worst choice for the leader. First, the strategy of the leader fixed, we state a relationship between the (weakly or properly) efficient set of the follower's multicriteria problem and the solution set of the problem scalarized via a convex combination of objectives through a vector of parameters (weights). Then we establish that (i) the solution of the scalarized (parametric) problem for any given parameter vector is unique and (weakly or properly) efficient and (ii) for each solution in the (weakly or properly) efficient set, there exists at least one corresponding parameter vector for the scalarized problem yielding the same solution. Therefore, the set of all parametric solutions (obtained by solving the scalarized problem) is equal to the efficient set. Thus we are able to rewrite the optimistic and pessimistic semivectorial bilevel control problems as bilevel problems where the lower level is a scalar optimization problem which always admits a unique solution. Finally, we present sufficient conditions on the data for existence of solutions to both the optimistic and pessimistic optimal control problems.