Approximate Penalty Method in Optimal Control Problems for Nonsmooth Singular Systems

We consider an abstract optimal control problem with additional constraints and nonsmooth terms, but without the requirement that both the state equation on the set of admissible controls and the extremum problem be solvable. We use the approximate penalty method proposed here to find an approximate (in the weak sense) solution of the problem. As an example, we consider the optimal control problem for a singular nonlinear elliptic type equation.