Control problems with mixed constraints and application to an optimal investment problem

Abstract We discuss two optimal control problems of parabolic equations, with mixed state and control constraints, for which the standard qualification condition does not hold. Our first example is a bottleneck problem, and the second one is an optimal investment problem where a utility type function is to be minimized. By an adapted penalization technique, we derive optimality conditions from which useful information of the solution can be derived. In the case of a control entering linearly in the state equation and cost function, we obtain generalized bang-bang properties.