Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems

Abstract

We present optimality conditions for bilevel optimal control problems where the upper level is a scalar optimal control problem to be solved by a leader and the lower level is a multiobjective convex optimal control problem to be solved by several followers acting in a cooperative way inside the greatest coalition and choosing amongst efficient optimal controls. We deal with the so-called optimistic case, when the followers are assumed to choose the best choice for the leader amongst 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. This paper continues the research initiated in Bonnel (SIAM J. Control Optim. 50(6), 3224–3241, 2012) where existence results for these problems have been obtained

When Semivectorial Bilevel Optimization Reduces to Ordinary Bilevel Optimization

“…The definition of ϕ and the fact that (18) we get ϕ(x * ; λ * ) ≥ f (x * , y * ). Hence ϕ(x * ; λ * ) = f (x * , y * ), which implies immediately that (x * , λ * , y * ) is an optimal solution for problem (13).…”

“…The study of semivectorial bilevel optimization problems in Euclidean or Hilbert spaces was initiated in [16,10], and continued by several authors [2,21,31,33,50,48,28]. The case of semivectorial bilevel optimal control problems was considered in [17,18], and a study on Riemannian manifolds has been done in [19].…”

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“…Multiobjective bilevel programming problem, where one of the objective functions of the upper-level problem and/or the lower-level problem is a vector function, needs to be investigated intensively as it is demanded from the point-of-view of applications. They have been studied in the literature by authors such as Bouibed et al [8], Ye [36], Bonnel [5], Bonnel and Morgan [6,7], Tung [35], Luu and Mai [29] and others. Recently, Luu and Mai [29] developed necessary and sufficient efficiency conditions for multiobjective bilevel programming problem via convexifactors.…”