Regional optimal control problem for semilinear parabolic systems with boundary controls

Abstract

This work concentrates on the regional optimal control problem for a semilinear parabolic system defined over a spatial domain Ω ⊂ ℝ<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup>. The system is driven by a multiplcation of a time control and a semilinear state function at the boundary ∂Ω of Ω. And then, it focuses on minimizing the deviation on a subregion ω of Ω between the desired state and the final one at time T , the tracking term all over the time interval [0, T ] and the term energy. Next, we demonstrate the existence of a boundary optimal control and characterize it as the solution of an associated optimality system. Moreover, the necessary conditions for optimality are expressed in the form of a variational inequality. Additionally, we explore applications to different sets of controls. An algorithm is derived from this approach and its performance is demonstrated using numerical simulations.