  The paper explores the differential inclusion of a special form. It is
supposed that the support function of the set in the right-hand side of an
inclusion may contain the maximum of the finite number of continuously
differentiable (in phase coordinates) functions. It is required to find a
trajectory that would satisfy the differential inclusion with the boundary
conditions prescribed and simultaneously lie on the surface given. Such
problems arise while practical modeling of discontinuous systems and in other
applied problems. The initial problem is reduced to a variational one. It is
proved that the resulting functional to be minimized is superdifferentiable.
The necessary minimum conditions in terms of superdifferential are formulated.
The superdifferential (or the steepest) descent method in a classical form is
then applied in order to find stationary points of this functional. Herewith,
the functional is constructed in a such a way that one can verify whether the
stationary point constructed is indeed a global minimum point of the problem.
The convergence of the method proposed is proved. The method constructed is
illustrated by examples.
