Asymptotic methods in the theory of optimal control

The paper gives a survey of some works devoted to methods of the small parameter for the solution of optimal control problems. It is well known that these problems are difficult both for analytical and numerical approaches. Exact analytical solutions exist only for specific classes of optimal control problems. A numerical solution is possible for optimal program (open-loop) controls, but it becomes very difficult for obtaining synthesis if the order of the system is high. Therefore, approximate methods based on the small parameter techniques seem to be useful. By means of these methods, one can find approximate optimal control (both program and synthesis) for a number of optimal control problems which arise in applications. The paper presents some approaches and results in this direction obtained in the Computing Center and the Institute for Problems of Mechanics of the USSR Academy of Sciences in the past 10-15 years.