Optimal control of differential inclusions in infinite-dimensional spaces

This paper studies a general optimal control problem of Bolza for nonconvex differential inclusions with endpoint constraints in reflexive and separable Banach space. First, we construct a sequence of discrete approximation problem for the original Bolza problem and prove that optimal solutions to discrete approximations strongly converge in W^1,2 to a given intermediate relaxed local minimizer (in particular, to a strong minimizer) for the original continuous-time problem. Then, based on generalized differentiation, necessary optimality conditions are obtained for the discrete approximation problems under fairly general assumptions. Finally, the established stability of discrete approximations and advanced tools of variational analysis in infinite dimensions allow us to derive necessary optimality conditions in the Euler-Lagrange form for the constrained differential inclusions under consideration. The results obtained are expressed in terms of nonconvex normal cones, subdifferentials, and coderivatives of the initial nonsmooth data.