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