Observability And Stability Of 1-D Wave Equations With Moving Boundary Feedback At Endpoints And Internal Points

In this thesis,we are concerned with a wave equation on a time-dependent domain with a Dirichlet boundary condition at the endpoint x=0 and a boundary feedback at the moving endpoint x=kt.We discuss stability and exact boundary observability and internal point’s observability of the 1-dimensional wave equation with moving boundary feedback.By using generalized Fourier series and Parseval’s equality in weighted L2-spaces,we derive a precise polynomial asymptotic stability for the energy function of the solution.Moreover,the exact boundary observability and internal point’s observability of the solution are established in minimal time.The observability constants are explicitly given for each endpoints and internal points.The thesis consists of three sections.In Chapter 1,preface.In Chapter 2,we first study the following the 1-dimensional wave equation with moving boundary feedback The subscripts t and x stand for the derivatives with respect to time and space respectively.(w0,w1)∈HL1(0,ktp)×L2(0,kt0)is any given initial data,and c∈R is the boundary feedback coefficient.In this thesis,we first give the exact solution of the above-mentioned wave equation,and by using the exact expression of the solution,we obtain that the energy function of the above problem is polynomial asymptotically stable(the stability depends on the boundary feedback coefficient c and the boundary moving speed k).Then we give the direct inequality and inverse inequality at each endpoint,and give the optimal time of observability at each endpoint.In Chapter 3,In this thesis,we discuss the observability of the problem at the interior point.The observability at the interior point is divided into the observability at the fixed interior point and the observability at the moving interior point,But in this paper,we only discuss the observability at the moving interior point.Similar to the previous chapter,by using the generalized Fourier series and the Parseval equation of weighted L2-space,we obtain the direct inequality and inverse inequality at the interior point,and obtain the optimal observable time at the moving interior point.