On the problem of controlling oscillations of a flat membrane by distributed force actions
Abstract
On the problem of controlling oscillations of a flat membrane by distributed force actions
Incoming article date: 23.11.2022The paper considers the problem of bringing vibrations of a flat membrane to rest, controlled by forces applied to the entire area of the membrane and limited in absolute value. Sufficient conditions are given for the initial data of the deviation and velocity of the membrane, under which a complete stop of motion in a finite time is possible. The resting time is also evaluated. The theorem on estimating the eigenfunctions of the Dirichlet problem for the Laplace equation used in the work makes it possible to refine the mentioned sufficient condition in comparison with the work of F.L. Chernousko. In this work, a similar problem is considered, and the method of expanding the unknown control and the corresponding solution in terms of eigenfunctions of the Dirichlet problem for the Laplace equation is also applied. In this paper, the problem of bringing various elastic oscillatory systems with distributed parameters (membrane, rod, plate, etc.) to a state of rest in a finite time is reduced by means of expansion into a Fourier series in the corresponding system of eigenfunctions to the study of the problem of stopping a counting system of pendulums, connected with each other only through the values of external control actions, the sum of the values of which should not exceed in absolute value some given constant. In order to fulfill this limitation, it is necessary to use estimates for the absolute values of the eigenfunctions, normalized in the mean square. In this paper, we use some estimates for the absolute values of eigenfunctions, previously obtained by Eidus D.M. This allows us to refine the results of F.L. Chernousko for sufficient conditions on the initial functions of the oscillatory system, under which we must dampen the oscillations. These conditions consist in the fact that the initial functions belong to Sobolev spaces with certain indices and in the fulfillment of some additional boundary conditions on the boundary of the domain in which the system is defined.
Keywords: control, wave equation, limited distributed force, Fourier method, counting system of harmonic oscillators