Optimal control of a parabolic equation with a nonlocal nonlinearity

TL;DR

This paper develops an optimal control framework for a parabolic PDE with a nonlocal nonlinear term, establishing existence, regularity, and optimality conditions for solutions.

Contribution

It introduces a novel control problem for a nonlocal nonlinear parabolic equation and proves existence, uniqueness, regularity, and optimality conditions for solutions.

Findings

Existence and uniqueness of solutions to the controlled system

Regularity results for the control-to-state operator and adjoint state

First and second-order optimality conditions established

Abstract

This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity. The system is described by a parabolic equation involving a nonlinear term that depends on the solution and its integral over the domain. We prove the existence and uniqueness of the solution to the system and the boundedness of the solution. Regularity results for the control-to-state operator, the cost functional and the adjoint state are also proved. We show the existence of optimal solutions and derive first-order necessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimality are established.