Averaging method in optimal control problems for systems of ordinary differential equations

The system of ordinary differential equations with small parameter in standard Bogolyubov's form and with control parameter, which is Lebeg integrable function and takes on a value in closed subset of Euclidean space, is considered. Averaging control problem is put in accordance to initial control problem. The theorem of existence of optimal control for precise and averaging problems is proved. It is shown that optimal control of averaging problem is ε-optimal for precise problem on asymptotically finite (order 1/ε) and on infinite time intervals. At the same time unlike earlier known approach the new and more convenient procedure of averaging is offered and this permits to solve the practical optimal control problems.