Optimal piezoelectric control of a plate subject to time-dependent boundary moments and forcing function for vibration damping

Active vibration control problem for a rectangular plate subject to moment boundary conditions and forcing function is solved by means of a maximum principle. The control is exercised by a patch actuator and the solution is formulated using an adjoint variable leading to a coupled boundary–initial-terminal value problem. The application of the maximum principle yields the optimal control expression as well as the explicit solution for a simply supported plate. The objective functional to be minimized is defined as a quadratic functional of displacement and velocity and also includes a penalty in terms of control voltage applied to the piezoelectric patch actuator. The penalty term limits the amount of control energy spent during the control process. Numerical results are presented to assess the effect of the optimal control algorithm.