  The paper deals with systems of ordinary differential equations containing in
the right-hand side controls which are discontinuous in phase variables. These
controls cause the occurrence of sliding modes. If one uses one of the
well-known definitions of the solution of discontinuous systems, then the
motion of an object while being on some surface can be described in terms of
differential inclusions. With the help of the previously developed apparatus
for solving differential inclusions, a method is constructed for finding the
trajectories of a system moving in a such a mode. Since some of frequently used
discontinuous controls contain nonsmooth functions of phase variables, the
paper pays special attention to study the differential properties of such
systems. At the end of the paper controls of a slightly different, in contrast
to the classical, type are considered which have useful differential
properties, and a method is constructed for solving systems with such controls
considered both before hitting the required surface and moving in its vicinity.
