Bang-bang and multiple valued optimal solutions of control problems related to quasi-linear elliptic equations

This paper investigates two optimal control problems in which the state equation is a quasi-linear elliptic partial differential equation. In the first problem the admissible set comprises functions with range in the interval [0, 1] with prescribed integral. We show the optimal solution is unique and it is of bang-bang type. In the second problem the admissible set is a particular class of simple functions. We will show again the optimal solution is unique and derive the corresponding optimality condition.