  The main goal of this paper is developing the method of discrete
approximations to derive necessary optimality conditions for a class of
constrained sweeping processes with nonsmooth perturbations. Optimal control
problems for sweeping processes have been recently recognized among the most
interesting and challenging problems in modern control theory for discontinuous
differential inclusions with irregular dynamics and implicit state constrained,
while deriving necessary optimality conditions for their local minimizers have
been significantly based on the smoothness of controlled dynamic perturbations.
To overcome these difficulties, we use the method of discrete approximations
and employ advanced tools of second-order variational analysis. This approach
allows us to obtain new necessary optimality conditions for nonsmooth and
nonconvex discrete-time problems of the sweeping type. Then we employ the
obtained conditions and the strong convergence of discrete approximations to
establish novel results for original nonsmooth sweeping control problems that
include extended Euler-Lagrange and maximization conditions for local
minimizers. Finally, we present applications of the obtained results to solving
a controlled mobile robot model with a nonsmooth sweeping dynamics that is of
some practical interest.
