Stabilization of wave dynamics by moving boundary

A wave equation on a time-dependent domain is considered. The shape of the domain changes according to a prescribed space/time-dependent velocity field. On the moving boundary the solution satisfies zero Dirichlet condition. It is known that if the domain keeps expanding at a “subsonic” speed, then the associated finite energy decays uniformly. Here, the scenario of interest is when the domain remains bounded and undergoes phases of expansion and contraction. Although the energy identity in this case is not necessarily conservative, it is shown that the L2 space-time norm of the normal trace remains a priori bounded at small normal speeds of the boundary, analogously to the classical Dirichlet wave problem on a static domain. In addition, it is demonstrated that small normal velocity but very large acceleration of the boundary is compatible with the known existence theory, provided the magnitude of the deformations is relatively small. An “adaptive” boundary movement control is proposed and implemented numerically. The control action is dynamically computed from the normal trace data and dissipates the energy by means of small deformations of the domain only.