An optimal control problem in L/infinity

We consider an optimal control problem inL infin, where the cost functional has a penalty term which involves the structure of the control law. This type of penalization allows us to restrict our attention only to piecewise constant controls, with assigned switching times, i.e., the controls belong to finite-dimensional control spaces. This fact and our assumptions on the dynamics give as a consequence the compactness of the minimizing sequences inC([0, 1], RopfPrime) ×L infin([0, 1], Ropf). The existence of a minimum of the cost functional is then obtained by a direct method. This result allows us to avoid the usual convexity assumption on the cost functionalC and on the multivalued vector field associated to the dynamics when we have to consider the controls in all ofL infin.