A note on optimal control problems

Conditions ensuring both the existence of quasi-solutions and that of solutions of a minimization problem are given. In the latter case the controls are taken in L^∞, and the usual convexity assumptions on the multivalued vector field associated with the dynamics, the boundary conditions, and the functional, are made. One can also obtain the existence of optimal controls for the same control process, even in the absence of the above-mentioned convexity assumption. For this, it suffices to add to the considered cost functional a suitable penalty term involving the structure of the control law