Exact Observability For Quasilinear Hyperbolic Systems

The present Ph.D. thesis deals with the exact observability for quasilinear hyperbolic systems. First of all, the exact boundary observability is shown for nonautonomous quasilinear hyperbolic systems, by means of a direct and constructive method based on the theory on semiglobal C~1 solution. Moreover, the author presents sharp estimates on the exact observability time and reveals the essential difference between the nonautonomous hyperbolic case and the autonomous case. Secondly, as a basis of the exact observability with zero eigenvalues, a theory on the semiglobal C~1 solution to a kind of boundary value problem with characteristic boundary for first order quasilinear hyperbolic systems is established. Then, the author realizes the local exact observability for first order quasilinear hyperbolic systems with zero eigenvalues and presents sharp estimates on the exact observability time. These two studies above complement the theory on the exact observability for quasilinear hyperbolic systems.The arrangement of the thesis is as follows:In Chapter 1, the author gives a brief introduction to the exact observability.As the basis of further study, the author cites the theory on semiglobal C~1 solution to first order quasilinear hyperbolic systems in Chapter 2, and then in Chapter 3, proves the existence and uniqueness of the semiglobal C~1 solution to a kind of boundary value problem with characteristic boundary for first order quasilinear hyperbolic systems.In Chapter 4, by means of the result cited in Chapter 2 and as in the autonomous case, the author adopts a direct constructive method and obtains the local exact observability for general nonautonomous first order quasilinear hyperbolic systems. When there is no zero eigenvalue, the author proves that the exact observability can be realized with boundary observation on one side or on two sides.Chapter 5 is devoted to the local exact boundary observability for one-dimensional nonautonomous quasilinear wave equations. By the results of semiglobal C~1 solution to first order quasilinear hyperbolic systems cited in Chapter 2, the author deals with various types of boundary conditions in a unified way and establishes the semiglobal C~2 solution to the mixed initial-boundary value problem for one-dimensional nonautonomous quasilinear wave equation. Then the author gets the local exact boundary observability for the one-dimensional nonautonomous quasilinear wave equation in both cases of two-sided and one-sided observation. As a special case, the corresponding results on the exact boundary observability for one-dimensional essential autonomous quasilinear wave equation are obtained.At last, in Chapter 6, by means of the result obtained in Chapter 3, the author establishes the theory on local exact observability for first order quasilinear hyperbolic systems with zero eigenvalues and reveals that the observation should be given not only on the boundary but also in the domain under construction. Moreover, certain examples are illustrated to show the sharpness of both the exact observability time and the number of the observed values.