In the paper we study a differential inclusion with a given continuous convex multivalued mapping. For a given finite time interval, it is required to construct a solution of the differential inclusion that satisfies the given initial and the final conditions. With the help of support functions, the original problem is reduced to minimizing some functional in the space of partially continuous functions. In the case of continuous differentiability of the support function of a multivalued mapping with respect to the phase variables, this functional is Gateaux differentiable. In the paper, Gateaux gradient is found, necessary and sufficient conditions for the minimum of the given functional are obtained. On the basis of these conditions, the method of the steepest descent is applied to the original problem. Numerical examples illustrate the method realization.