Lipschitz Continuous Solutions For First Order Quasilinear Hyperbolic Systems, And Their Exact Controllability And Exact Observability

In this Ph.D. thesis, we study systematically Lipschitz continuous solutions to the Cauchy problem and the mixed initial-boundary value problem for first order quasilinear hyperbolic systems. The local existence and uniqueness of Lipschitz continuous solutions to the Cauchy problem, the local existence and uniqueness of Lipschitz continuous solutions to the mixed initial-boundary value problem, the existence and uniqueness of semi-global Lipschitz continuous solutions to the mixed initial-boundary value problem, the corresponding exact boundary controllability and exact boundary observability for quasilinear hyperbolic systems are discussed respectively.The structure of the thesis is as follows.First of all, a brief introduction of the background and the present situation on the study of C1classical solutions as well as of Lipschitz continuous solutions for first order quasilinear hyperbolic systems is given in Chapter1.In Chapter2, we list some preliminaries, including some results on Lipschitz continuous functions, and the continuous dependency on parameters of solutions to ordinary differential equations.The C1classical solution to first order quasilinear hyperbolic systems has been investigated rather completely. The corresponding results in the framework of Lip-schitz continuous solutions will be under research. By a method of approximation, Wang&Wu proved that the existence of Lipschitz continuous solutions to the Cauchy problem such that the Lipschitz continuous solution satisfies differential equations almost everywhere, however, the Lipschitz continuous solution obtained by them depends on the choice of subsequences and there was no uniqueness result. In Chapter3, we prove the uniqueness of Lipschitz continuous solution and show the Lipschitz continuous solution to the Cauchy problem can be also defined by a system of integral equations.In Chapter4, we consider the mixed initial-boundary value problem for quasilin-ear hyperbolic systems with general nonlinear boundary conditions. By means of the corresponding system of integral equations, we prove the existence and uniqueness of Lipschitz continuous solutions. Moreover, in order to study the controllability and observability for Lipschitz continuous solutions to the mixed initial-boundary value problem, we extend the concept of semi-global classical solutions to Lipschitz con-tinuous solutions, and prove the existence and uniqueness of semi-global Lipschitz continuous solutions to the mixed initial-boundary value problem for quasilinear hyperbolic systems.Based on results of Chapter4, Chapter5deals with the exact boundary con-trollability for Lipschitz continuous solutions to first order quasilinear hyperbolic systems. In Chapter6, we consider the exact boundary observability for Lipschitz continuous solutions to first order quasilinear hyperbolic systems.