Boundary Control and Stability of Linear Water Waves

This paper is the first of two parts which study controllability and stabilizability properties of small amplitude waves on a fluid surface. We first derive an evolution equation describing these waves, and then discuss a suitable form of boundary control.We show that, for a simple domain geometry, the system is null controllable (can be steered to the zero state) only on an infinite time interval. (The second part of this paper extends the result to finite, irregular domains.) We actually construct the Laplace transform of the open loop null control for infinite time, and show that no null control exists for finite time. We give sufficient conditions for convergence of the series describing the control.