Control of the evolution of a dynamical system under high-frequency excitations

A controlled dynamical system subjected to high-frequency excitations is investigated. A standard controlled system is constructed using a change of variables, which generalizes the Bogolyubov change of variables in the problem of a pendulum with a vibrating suspension point. An effective procedure is developed for the approximate solution of the problem of optimal control over an asymptotically large range of variation of the argument. The property of closeness of the approximate solution to the exact solution with respect to the slow variable and functional is established. A generalization of the algorithm for solving the problem to dynamical systems with variable parameters is given. The effectiveness of the approach is illustrated by investigating the problems of control of mechanical systems: the oscillations and rotations of a rigid body with a vibrating axis, and the motion of a “microparticle” in a force field, modelled by travelling and standing waves.