# Semivectorial Bilevel Convex Optimal Control Problems

Bonnel, Henry

### Abstract

We consider a bilevel optimal control problem where the upper level is a scalar optimal control problem and the lower level is a multi-objective convex optimal control problem. We deal with the so called optimistic case, i.e. when the follower cooperates with the leader. First we state a relationship between the (weakly or properly) efficient set of the multi-objective problem and the solution 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 semivectorial bilevel problem as a classical bilevel problem where upper and lower level are scalar optimization problem and the lower level admits a unique solution. Finally we present some existence results.

Based on a paper written jointly with Jacqueline Morgan from the University of Naples Federico II.

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