Nonconvex control systems and differential inclusions

We study optimal and boundary trajectories of nonsmooth, nonconvex control systems and differential inclusions. Optimization problems which we consider are of the following form. For a given g : Rn ?? R : minimize g(x(T)) (1.1) over the set of all absolutly continuous functions x(??), x(??) : [0,T] ?? Rn, satisfying for given K and M the end-points constraints: x(0) ?? K, x(T) ?? M, (1.2) and the constraint: x(t)= f(t,x(t), u(t)), u(t) ?? U, a.e. t ?? [0,T], (1.3) where the dynamics f(??, ??, ??) and the control set U are given. In the second part we consider the case when the control system (1.3) is replaced by a differential inclusion: x(t) ?? F(t,x(t)), a.e. t u(t) ?? [0,T], (1.4) for a given set-valued mapping F(??,??). The goal is to find necessary conditions for solutions of such problems. Necessary conditions can be usually obtained if one can characterize, so called, boundary trajectories of the system (1.3) or respectively the inclusion (1.4).