  This paper deals with optimal control problems described by a controlled
version of Moreau's sweeping process governed by convex polyhedra, where
measurable control actions enter additive perturbations. This class of
problems, which addresses unbounded discontinuous differential inclusions with
intrinsic state constraints, is truly challenging and underinvestigated in
control theory while being highly important for various applications. To attack
such problems with constrained measurable controls, we develop a refined method
of discrete approximations with establishing its well-posedness and strong
convergence. This approach, married to advanced tools of first-order and
second-order variational analysis and generalized differentiations, allows us
to derive adequate collections of necessary optimality conditions for local
minimizers, first in discrete-time problems and then in the original
continuous-time controlled sweeping process by passing to the limit. The new
results include an appropriate maximum condition and significantly extend the
previous ones obtained under essentially more restrictive assumptions. We
compare them with other versions of the maximum principle for controlled
sweeping processes that have been recently established for global minimizers in
problems with smooth sweeping sets by using different techniques. The obtained
necessary optimality conditions are illustrated by several examples.
