In the paper, we study a differential inclusion with a given continuous convex multivalued mapping. For a prescribed finite time interval, it is required to construct a solution to the differential inclusion, which satisfies the prescribed initial and final conditions and minimizes the integral functional. By means of support functions, the original problem is reduced to minimizing some functional in the space of partially continuous functions. When the support function of the multivalued mapping is continuously differentiable with respect to the phase variables, this functional is Gateaux differentiable. In the study, the Gateaux gradient is determined and the necessary conditions for the minimum of the functional are obtained. Based on these conditions, the method of steepest descent is applied to the original problem. The numerical examples illustrate the implementation of the constructed algorithm.