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State Estimation of a Linear Water Wave System

In this article, we study the observation problem of linear water wave systems. The linear water wave system that we consider is a distributed parameter system governed by an infinite dimensional evolution equation coupled with a Laplace equation. We assume that the water wave elevation is measurable. Based on the measurement an observer is designed capable to estimate the water wave velocity potential. Then the water wave velocity field can be extracted from the gradient of the estimated velocity potential. First we prove that the water wave system is well-posed, then we show that the water wave state is observable by measuring the surface wave elevation only. Based on these facts, an infinite-dimensional observer is designed to estimate the velocity potential. Moreover we prove that the designed observer is exponentially convergent. The observer's performance is tested by numerical simulations.