Uniform Approximation For Two Kinds Of Wave Equation Boundary Control Systems

It is by now well-known that a semi-discrete system obtained by discretizing in space is not uniformly controllable as the discretization parameter h goes to zero.Therefore,semi-discretized approximation of boundary control system for two kinds of wave equation are studied in this paper.Firstly,this thesis established the semi-discrete difference scheme for the following one-dimensional wave equation with Robin boundary damping.The semi-discretized scheme on equidistant grids is proposed,which doesn't need the nu-merical viscosity.And the uniformly exponential stability of semi-discretized scheme is demonstrated by introducing a Lyapunov function.A numerical experiment verififies the theoretical results.Next,we also established a semi-discrete finite difference scheme for the one-dimensional wave equation with Neumann boundary.In order to get uniformly controllability of the discrete systems,On one hand,We proved the uniformly observable inequality by spectral analysis of the eigenvalues;on the other hand,we introduce functional Jh to construct the discrete controls which is uniformly bounded.