Abstract: This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form llu(t)ll L1(omega) >= G for t E (0, T ). This limits the total control that can be applied to the system at any instant of time. The L1-norm of the constraint leads to sparsity of the control in space, for the time instants when the constraint is active. Due to the non-smoothness of the constraint, the analysis of the control problem requires new techniques. Existence of a solution, first and second order optimality conditions, and regularity of the optimal control are proved. Further, stability of the optimal controls with respect to G is investigated on the basis of different second order conditions.

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