  This paper addresses a new class of generalized Bolza problems governed by
nonconvex integro-differential inclusions with endpoint constraints on
trajectories, where the integral terms are given in the general (with
time-dependent integrands in the dynamics) Volterra form. We pursue here a
threefold goal. First we construct well-posed approximations of continuous-time
integro-differential systems by their discrete-time counterparts with showing
that any feasible solution to the original system can be strongly approximated
in the $W^{1,2}$-norm topology by piecewise-linear extensions of feasible
discrete trajectories. This allows us to verify in turn the strong convergence
of discrete optimal solutions to a prescribed local minimizer for the original
problem. Facing intrinsic nonsmoothness of original integro-differential
problem and its discrete approximations, we employ appropriate tools of
generalized differentiation in variational analysis to derive necessary
optimality conditions for discrete-time problems (which is our second goal) and
finally accomplish our third goal to obtain necessary conditions for the
original continuous-time problems by passing to the limit from discrete
approximations. In this way we establish, in particular, a novel necessary
optimality condition of the Volterra type, which is the crucial result for
dynamic optimization of integro-differential inclusions.
