  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.
