Semi-discretized Approximation Of Uniformly Exponential Stability For Two Kinds Of Wave Equation With Damped Boundary

The stabilization is an important topic in distributed parameter control theory.Howev-er,in the process of numerical discretization,the decay rate is not uniform with respect to the net-spacing size?.Therefore,in this theis semi-discretized approximation of uniformly exponential stability for two kinds of wave equation with damped boundary are studied.Firstly,this thesis established the semi-discrete difference scheme for the following one-dimensional wave equation with Neumann boundary damping.(?)A semi-discretized finite difference scheme on equidistant grids is proposed for 1D wave equation with Neumann damped boundary by introducing an average operator in time di-rection.The discretized multiplier method is adopted to prove the uniformly observable inequality of semi-discretized scheme for conjugate system.The uniformly exponential sta-blility of semi-discretized scheme for original system is demonstrated further.Secondly,a novel semi-discretized finite difference scheme on equidistant grids is pro-posed for 1D wave equation with damped boundary,which holds the uniformly exponential stability of original system while without the need of numerical viscosity.The uniformly exponential stability of semi-discretized scheme is demonstrated by introducing a Lyapunov function.We proof the convergence result of the semi-discretized scheme by introducing the extension operators.A numerical experiment verifies the theoretical results.Finally,we also studied the semi-difference scheme for the one-dimensional wave e-quation with Robin boundary damping.(?)We know that after semi-discretization on spatial variable,the semi-dicrete system is not uniformly decaying with respect to the step size.So we focus on to add a suitable vanishing numerical viscosity term in the wave equation with Robin boundary,and then we proof that the energy of solution will decay exponentially to zero as time goes infinity.