A solution method for boundary control of one-dimensional multispan vibrating systems

Many flexible structures consist of a large number of components coupled en d 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 sp ecial class of optimal control problems is suggested by using an eigenfunct ion expansion and the:maximum principle. This solution involves reducing th e original problem to a system of ordinary differential equations. For comp arative studies, the special problem is also solved by the variational appr oach, which leads to a system of integral equations as a necessary and suff icient condition of optimality. The effectiveness of these approaches is de monstrated by means of a numerical solution for controlling the vibrations of two strings that are coupled at the connecting point.