Controllability and mixing for acoustic wave motions

This paper concerns the dynamical behaviors of acoustic wave motion driven by a force acting through the boundary. If the boundary force is a suitable control, we show that the dynamical system associated to the acoustic wave motion is exactly controllable. Furthermore, when it is random perturbation of white noise type, we prove that the corresponding stochastic system is strong mixing. The bridge between these two problems is the observability inequality, which will be established in this work.