Control and stabilization for the wave equation with variable coefficients in domains with moving boundary

The study of stabilization and control for PDEs with variable coefficients involves higher level of complexity than the corresponding case of constant coefficients. Further compounding the complexity are the concern and effects of dynamic boundary motion. The problem in its general form is extremely challenging to treat, but under certain specific physical and geometric conditions, such as the time-likeness of the boundary and a limited speed of domain expansion, energy decay estimates can be established and the exact controllability can also be obtained by control-theoretic and Riemannian-geometric methods. Our approach here is based on the Bochner technique of differential geometry in terms of Riemannian metric and geometric multipliers, by generalizing an energy identity method used earlier in Bardos and Chen (1981). Concrete examples are also given to illustrate the geometric conditions and the theorems.