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. 9 2006 Elsevier Ltd. All rights reserved.
1. FORMULATION OF THE PROBLEM
A controlled dynamical system, subjected to high-frequency periodic excitations, is considered [1]. It is assumed that the equations of motion (for example, in Lagrange form) can be represented in terms of dimensionless quantities as follows ii = Q(O, q, L 1, u, ~,), q(to) = q0 q(t0 ) = q0 (1.1)
Here a dot denotes a derivative with respect to time, (q, q) is a 2n-vector of the phase variables, 0 is the fast phase of the external periodic excitation, u is the r-vector of control, and U is a fixed set. The numerical parameter ~, can take asymptotically large values (~, ---) oo), i.e. e = ~-1 is a small parameter. It is assumed that the quantities I q I, I q I, l u I are of the order of unity with respect to the large parameter ~,. The quantities to, qO, qO are assumed to be given, and the controlled motion of the system (1.1) is considered in a fixed time interval to ---t ___ T (of the order of unity). The function Q must be 2n-periodic and piecewise-continuous in 0; it is assumed to be fairly smooth enough with respect to the remaining arguments. The structural characteristics and properties of smoothness of the functions Q and u will be refined below. They are needed to enable the problem of control and optimization for system (1.1) to be reduced to a standard form, allowing of the use of asymptotic methods [1-3].
We will formulate the optimal control problem. We will assume that the final conditions imposed on the variables q and 9 at a fixed instant of time t = T have the general form M(q, Ll)Jr = O, M = (M n ..... Mm), O<m<2n
(1.2)
In particular, conditions (1.2) may be absent (m = 0), or correspond to the two-point problem: q(T) = qT, 9(T) = 9 T, when qT, 9 ~ are known. The vector function M is assumed to be smooth enough in the region under consideration, and a possible regular dependence on %, for example a smooth dependence on e, I~1 -~0, is not indicated for brevity.