Necessary optimality and controllability conditions for nonsmooth control systems

We consider nonconvex differential inclusions and obtain necessary optimality conditions for the general Bolza problem, a nonsmooth optimisation problem, and stronger resuits for its special Mayer case when f=0. In the case of Bolza problem we derive necessary conditions for a new concept of intermediate relaxed local minimum which takes an intermediate place between the classical weak and strong local minima. The main tools used are based on approximation techniques accompanied by proper robust constructions in nonsmooth analysis. Actually we use two approximation procedures. The first one is related to discrete approximations of the original infinite dimensionaI dynamic problem by a family of nonsmooth optimization problems in finite dimensions. Employing strong convergence results, we prove the Euler-Lagrange conditions for an intermediate relaxed local minimum in the Bolza problem. The second procedure deals with approximating the Mayer problem for constrained differential inclusions by a family of unconstrained Bolza problems involving Lipschitzian integrands. This leads to necessary optimality and controllability conditions for fully nonconvex problems with no relaxation