Optimization problems for a differential inclusion with convex-valued right-hand side are considered. The initial state of the inclusion is not given exactly; it is either a random vector or, more generally, an element of some random closed set. It is necessary to find a trajectory of the inclusion that minimizes the expected value of a function of the inclusion's terminal state or, more generally, a similar trajectory that minimizes the Choquet integral. Assumptions are given under which the Choquet integral is reduced to the usual integral with respect to a probability measure. The obtained results are used to derive sufficient optimality conditions in the mentioned optimization problems; these conditions are also necessary in certain cases. A prototype of such problems is the problem of controling an ensemble of trajectories. On the other hand, the necessity of studying control problems with random data in the form of sets arises in the solution of motion correction problems, where a random set appears naturally at the observation stage. Examples are given.