A generalization of the method of averaging for the optimal control of non-linear oscillations

The problem of the optimal control of oscillations using small control inputs is considered. It is assumed that in the first approximation of the averaging method there is no change in the slow phase vector. A second-order averaging scheme is developed, which enables the control problem to be solved over a time interval of length inversely proportional to the square of a small parameter, that is, over an “elongated time interval”. Error estimates are obtained with respect to the phase trajectory, boundary conditions, functional, and control. Results are presented for a special case - a linear-quadratic control problem with periodic coefficients. The control of the phase and amplitude of non-linear oscillating systems is considered in model examples.