Studies On Algorithm Of Identification Of Water Depth And Water Velocity Potential In Coastal Region

In oceanic and coastal engineering,a challenging task is to recover the water depth and the velocity of water waves.In the dissertation,the motion of water waves is assumed to be described by linearized water wave equation.It has been shown that the linearized water wave equation is well-posed.Based on the classical Luenberger observer for finite-dimensional systems,an infinite-dimensional observer is proposed for linearized water wave equation.From this observer,we can estimate the velocity potential of water waves by measuring the surface wave elevation.And based on the classical gradient descent method,two algorithms of identification are designed to simultaneously recover the water depth and the velocity potential of water waves.The main research content of the dissertation can be summarized as follows:(1)The linearized water wave equation is developed from the continuity equation of hydrodynamics.It has been shown that the linearized water wave equation is a second order partial differential equation.By using the substitution of variables method,the linearized water wave equation can be written as a state equation coupled with a Laplace equation.(2)The linearized water wave equation is proved to be well-posed.It is meant that the linearized water wave equation has a unique solution.Furthermore,this solution continuously depends on the initial values.It has been shown that the state space of the linearized water wave equation is a Hilbert space.And if the initial values of linearized water wave equation are in this Hilbert space,then the solution of linearized water wave equation exists.Comparing with previous studies,the state space chosen for this dissertation is more suitable for the linearized water wave equation.(3)A Luenberger-like observer is designed for the linearized water wave equation.This type of observer was used for linear systems in finite-dimension.By using the measurement records of the surface wave elevation,the velocity of water waves in the water domain can be estimated.It has been shown that the estimation error is exponentially stable.And then,a semi-discretized finite difference scheme is introduced to compute the numerical solution of linearized water wave equation and the observer.The convergence of our designed observer is shown by using the Matlab software.(4)In some sense,the dynamical system described by the linearized water wave equation is similar to the system of masse-string.The eigenvalues of the system operator are located at the imaginary axis of the complex plane.Moreover,the system is exactly observable if the measurement output of dynamical system is the displacement or the velocity.Hence based on the different measurement output,two algorithms of identification are designed.The first algorithm is to estimate the water depth and the the velocity potential of water waves by using the measurement records of wave elevation.The second algorithm is to estimate the water depth,the velocity potential of water waves and the wave elevation by using the measurement records of velocity of water waves on the surface.Both two algorithms are designed based on the gradient descent method.By computing the differential of the cost functional,the descent direction of the cost functional can be found at any arbitrary point.Finally,our proposed algorithms are tested by using the Matlab software.It can be seen from the simulations that the algorithms are convergent.Moreover,the algorithms are robust.Finally,the main results of the dissertation are concluded and some issues for further research and exploration are proposed.