On a problem of control of trajectory bundles

Various problems of control of trajectory bundles constitute an important object of study in modern mathematical control theory. Such problems arise, for example, in studying the motion of a flow of charged particles, and also in the presence of incomplete information about the initial state of the controlled system. In the present article, for a nonlinear controlled object of a quite general form on a fixed time interval $[0,T]$, the problem of control of trajectory bundles with a non-single-point initial set is considered. On the reachable set at the moment $T>0$, the problem of maximization of a given continuous function is studied. This problem can be interpreted as a problem on the spread of trajectories of the controlled object. The corresponding maximum depends on the chosen admissible control $u(\cdot )$. In the article, the existence of a minimum on the set of admissible controls from this maximum is substantiated.