NECESSARY CONDITIONS OF OPTIMALITY FOR EVOLUTION EQUATIONS WITH DISCONTINUOUS VECTOR FIELDS

Abstract

We consider a general class of semilinear evolution equations on Banach spaces with discontinuous vector fields. The solutions are (generalized) functions of time with values from the space of finitely additive measures on the state space which is a Banach space. The admissible controls are also (generalized) functions with values from the space of finitely additive measures on another Banach space. We present necessary conditions of optimality from this class of relaxed controls for nonconvex control problems. This is based on some recent results on existence of optimal relaxed controls for the same class of systems. We consider also state constraint problems including time optimal control.