A Solution Method for Boundary Control of One-Dimensional Multispan Vibrating Systems

Abstract

Many flexible structures consist of a large number of components coupled end to end in the form of a chain. In this study, the authors consider a type of such structures formed by N one-dimensional coupled structures, with (N — 1) controllers at the coupled points. A method of solution for such a special class of optimal control problems is suggested by using an eigenfunction expansion and the maximum principle. This solution involves reducing the original problem to a system of ordinary differential equations. For comparative studies, the special problem is also solved by the variational approach, which leads to a system of integral equations as a necessary and sufficient condition of optimality The effectiveness of these approaches is demonstrated by means of a numerical solution for controlling the vibrations of two strings that are coupled at the connecting point.