Bilinear optimal control for a fractional diffusive equation

Abstract

We consider a bilinear optimal control for an evolution equation involving the fractional Laplace operator of order 0 < s < 1. We first give some existence and uniqueness results for the considered evolution equation. Next, we establish some weak maximum principle results allowing us to obtain more regularity of our state equation. Then, we consider an optimal control problem which consists to bring the state of the system at final time to a desired state. We show that this optimal control problem has a solution and we derive the first and second order optimality conditions. Finally, under additional assumptions on the initial datum and the given target, we prove that local uniqueness of optimal solutions can be achieved.

Bilinear optimal control for the fractional Laplacian: error estimates on Lipschitz domains

“…To the best of our knowledge, the only work available in the literature that provides an advance concerning the parabolic version of (1.1)- (1.3) is the very recent manuscript [19]. In this work, the authors provide an analysis for the continuous parabolic problem including existence of solutions and first and second order optimality conditions; second order sufficiency requiring an additional assumption on the problem data (see also [20,Remark 2.21]).…”