Regional Tracking problem for a reaction-diffusion equation with bilinear boundary controls

Abstract In this paper we investigate a regional optimal control problem of a reaction-diffusion equation evolving on a spatial domain Ω ⊂ R2 where controls act in bilinear manner on the boundary ∂Ω of Ω. It addresses the tracking of a desired state all over the time interval [0, T ] only on a subregion ω of Ω with minimum energy. This may be expressed as a minimization problem of which we discuss the existence and characterization of an optimal control. Then under a sufficient condition the uniqueness of such control is established. The obtained results lead to a computational algorithm that we illustrate by a two-dimensional fish diffusion model.