Stability against small noises in control problems with non-Lipschitz right-hand side of the dynamic equation

Control problems in systems with non-Lipschitz right-hand side are studied for the performance functional continuously dependent on the path. Are considered two variants of the optimization problem depending on the fact whether the ally controls the realized path from the set generated by a useful control. Relaxation of original optimization problems, namely, a sequence of perturbed problems with vanishing perturbations (the right-hand side of the equation and initial conditions) is proposed. An asymptotically optimal solution to the relaxation problem is obtained by N. N. Krasovskii and A. I. Subbotin's extreme shift method. As is shown, the value achieved at this can be considerably better than the optimal result of the original problem.