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.
Institute of Mathematics of the Romanian Academy, Report 7/ May 2009.
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