A finite element approximations of a structural acoustic control problem with a Timoshenko beam interface

In this work we introduce and analyze a finite element approximation of an active control problem: the reduction of the vibration in a structural acoustic chamber where one of the walls is an elastic beam. The coupled fluid–solid system is subjected to an external harmonic excitation force, the control variable corresponds to the amplitude of the forces applied on the beam, and a distributed sensor is placed in a region of the fluid domain. The problem is considered in the framework of the mathematical theory of optimal control. The beam is modeled by a Timoshenko equation and the acoustic fluid is described by the pressure variable. The optimal order of convergence is proved for a finite element approximations using low order piecewise polynomials. Numerical test are included to demonstrate the good performance of the methods considered.