In this work, the least pointwise upper and/or lower bounds on the state variable on aspecified subdomain of a control system under piecewise constant control action are sought. This resultsin a non-smooth optimization problem in function spaces. Introducing a Moreau-Yosida regularizationof the state constraints, the problem can be solved using a superlinearly convergent semi-smoothNewton method. Optimality conditions are derived, convergence of the Moreau-Yosida regularizationis proved, and well-posedness and superlinear convergence of the Newton method is shown. Numericalexamples illustrate the features of this problem and the proposed approach.