  This paper addresses a new class of optimal control problems for perturbed
sweeping processes with measurable controls in additive perturbations of the
dynamics and smooth controls in polyhedral moving sets. We develop a
constructive discrete approximation procedure that allows us to strongly
approximate any feasible trajectory of the controlled sweeping process by
feasible discrete trajectories and also establish a $W^{1,2}$-strong
convergence of optimal trajectories for discretized control problems to a given
local minimizer of the original continuous-time sweeping control problem of the
Bolza type. Employing advanced tools of first-order and second-order
variational analysis and generalized differentiation, we derive necessary
optimality conditions for discrete optimal solutions under fairly general
assumptions formulated entirely in terms of the given data. The obtained
results give us efficient suboptimality ("almost optimality") conditions for
the original sweeping control problem that are illustrated by a nontrivial
numerical example.
