Constructive control of the motion of oscillating systems with discrete and distributed parameters

Abstract

The problem of controlling the motion of mechanical systems over an asymptotically large but fixed time interval is considered. The controlled objects may include elements with discrete parameters (material points, rigid bodies, weightless springs, etc.) and oscillating components with distributed parameters (strings, rods, elastic beams and shafts, membranes, plates, cavities with stratified liquid, etc.), whose frequency spectrum is denumerably infinite. The controlling actions, whether kinematic or dynamic in nature, are assumed to be concentrated with respect to the space variables. They may be movable, applied to the absolutely rigid parts of the system and / or fixed at the boundaries of the distributed elements (boundary control). This kind of control is of value in applications. On the assumption that techniques of mathematical physics /1, 2/ or the method of moments /3, 4/ yield an infinite-dimensional control problem for the Fourier coefficients of the solution relative to a set of basis functions for the boundary-value problem, an asymptotic approach is proposed for constructing approximate controls and the resulting scheme for the approximate solution of the problem is shown to be legitimate. A specific problem is examined as an illustration - rotation of an elastic rod in the plane by a torque applied at its end.