Measure Control of a Semilinear Parabolic Equation with a Nonlocal Time Delay

TL;DR

This paper investigates a control problem for a semilinear parabolic equation involving a measure-based control with nonlocal time delay, establishing theoretical properties and numerical convergence results.

Contribution

It introduces a novel control framework with measure controls for nonlocal delays, providing existence, regularity, optimality conditions, and discretization convergence analysis.

Findings

Existence and uniqueness of solutions established.

First order optimality conditions derived.

Numerical discretization converges to the continuous solution.

Abstract

We study a control problem governed by a semilinear parabolic equation. The control is a measure that acts as the kernel of a possibly nonlocal time delay term and the functional includes a non-differentiable term with the measure-norm of the control. Existence, uniqueness and regularity of the solution of the state equation, as well as differentiability properties of the control-to-state operator are obtained. Next, we provide first order optimality conditions for local solutions. Finally, the control space is suitably discretized and we prove convergence of the solutions of the discrete problems to the solutions of the original problem. Several numerical examples are included to illustrate the theoretical results.