The averaging principle in control problems.

*(Russian)*Zbl 0703.49005Recent years have seen the appearance of a large number of papers in which the possibilities of applying different asymptotic methods developed in differential equation theory for constructing approximate solutions of optimal control problems are discussed. An especially large number have been devoted to two lines of research. The first is related to the investigation of optimal control problems of so-called singularly perturbed systems and is an extension to the case of controlled motion of the results of A. N. Tikhonov and of A. B. Vasil’eva and V. T. Butuzov. The other is represented by papers whose purpose is to justify the averaging method in control problems, and is based on the results of N. N. Bogolyubov and Yu. A. Mitropol’skij and N. N. Moiseev.

The present paper contains results that enable us, if not to combine both lines of research, at least to establish a close connection between them. In formal terms these results can be considered as an attempt at generalization of the results of V. M. Volosov [Russ. Math. Surv. 17, No.6, 1-126 (1962); translation from Usp. Mat. Nauk 17, No.6(108), 3- 126 (1962; Zbl 0119.075)] and V. M. Volosov and B. I. Morgunov [“Method of averaging in the theory of nonlinear oscillatory systems” (Russian) (1971; Zbl 0232.70021)] to the case of controlled motion.

The present paper contains results that enable us, if not to combine both lines of research, at least to establish a close connection between them. In formal terms these results can be considered as an attempt at generalization of the results of V. M. Volosov [Russ. Math. Surv. 17, No.6, 1-126 (1962); translation from Usp. Mat. Nauk 17, No.6(108), 3- 126 (1962; Zbl 0119.075)] and V. M. Volosov and B. I. Morgunov [“Method of averaging in the theory of nonlinear oscillatory systems” (Russian) (1971; Zbl 0232.70021)] to the case of controlled motion.