Optimality conditions for the control of a double-membrane complex system by a modal decomposition technique

The optimal of damping out the oscillations of an elastically rectangular double-membrane system by means of point-wise actuators is solved analytically. The membrane is clamped along the boundaries. The motion of the system is initiated by given initial displacement and velocity conditions. The basic control problem is to minimize the deflection and the velocity of displacements at a specified time with the minimum expenditure of actuation energy. A quadratic performance functional is chosen as the cost functional which comprises the functionals of the deflection, velocity and the point-wise actuators. Necessary and sufficient conditions of optimality are investigated. The necessary conditions of optimality are obtained from a variational approach and formulated in the form of degenerate integrals which lead to explicit optimal control laws for the actuators. Numerical results are given for various problem parameters and the efficiency of the control mechanism is investigated.