Numerical solution of the optimal boundary control of transverse vibrations of a beam

Boundary control is an effective means for suppressing excessive structural vibrations. By introducing a quadratic index of performance in terms of di splacement and velocity, as well as the control farce, and an adjoint probl em, it is possible to determine the optimal control. This optimal control i s expressed in terms of the adjoint variable by utilizing a maximum princip le. With the optimal control applied, the determination of the correspondin g displacement and velocity is reduced to solving a set of partial differen tial equations involving the state variable, as well as the adjoint variabl e, subject to boundary, initial, and terminal conditions. The set of equati ons may not be separable and analytical solutions may only be found in spec ial cases. Furthermore, the computational effort to determine an analytic s olution may also be excessive. Herein a numerical algorithm is presented, w hich easily solves the optimal boundary control problem in the spacetime do main. An example of a continuous system is analyzed. This is the case of th e vibrating cantilever beam. Using a finite element recurrence scheme, nume rical solutions are obtained, which compare the behavior of the controlled and uncontrolled systems, Also, the analytic solution to the problem is com pared with the results obtained using the numerical scheme presented. (C) 1 999 John Wiley & Sons, Inc.