Abstract: Optimal control problems for semilinear parabolic equations with control costs involving the total bounded variation seminorm are analyzed. This choice of control cost favors optimal controls which are piecewise constant and it penalizes the number of jumps. It is an appropriate choice if a simple structure of the optimal controls is desired, which, however, is still sufficiently flexible so that good tracking properties can be maintained. Existence of optimal controls, necessary and sufficient optimality conditions, and sparsity properties of the derivatives are obtained. Convergence of a finite element approximation is analyzed and numerical examples illustrating structural properties of the optimal controls are provided.

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