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

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.