The paper explores the problem of an optimal (in Lagrange sense) control of a 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 minimum of the finite number of continuously differentiable (in phase coordinates) functions. It is required to find a solution of an inclusion that satisfies the given boundary conditions and delivers minimum to an integral functional. Herewith, the integrand of the functional is supposed to be subdifferentiable. One practical problem where such a statement arises is given. The initial problem is reduced to a variational one. It is proved that the functional constructed is subdifferentiable. The minimum conditions in terms of subdifferential are obtained. On the basis of this conditions the subdifferential descent method is used to minimize the functional under consideration. A numerical example is given illustrating the method constructed.