Exact Controllability For Nonautonomous Quailinear Hyperbolic Systems

The present Ph.D. thesis deals with the exact controllability for nonautonomous quailinear hyperbolic systems. As a basis of the exact controllability, the author proves the existence and uniqueness of semiglobal C~1 solution to the mixed initial-boundary value problem for general first order quasilinear hyperbolic systems in two variables with general nonlinear boundary conditions. Then by means of a constructive method, the author realizes the local exact controllability for nonautonomous first order quasilinear hyperbolic systems and presents sharp estimates on the exact controllability time. Moreover, the author reveals the essential difference between the nonautonomous hyperbolic case and the autonomous case. As applications, the author gets the local exact boundary controllability for one-dimensional nonautonomous quasilinear wave equations and the one-dimensional adiabatic flow system. At the end, the author establishes the global exact boundary controllability for first order quasilinear hyperbolic systems of diagonal form, taking the one-dimensional isentropic flow system as an example.The arrangement of the thesis is as follows:First of all in Chapter 1, the author gives a brief introduction on the exact controllability.In Chapter 2, by choosing suitable examples, the author shows that, quite different from the autonomous hyperbolic case, the exact boundary controllability for nonautonomous hyperbolic systems possesses various possibilities. And then the author points out the difficulty and significance of studying the exact controllability for nonautonomous hyperbolic systems.As a basis of further study, in Chapter 3, the author proves the existence and uniqueness of semiglobal C~1 solution to the mixed initial-boundary value problem for general first order quasilinear hyperbolic systems with general nonlinear boundary conditions.In Chapter 4, by means of the result obtained in Chapter 2 and as in the autonomous case, the author adopts a direct constructive method and obtains the local exact controllability for general nonautonomous first order quasilinear hyperbolic systems. When thereis no zero eigenvalue, the author proves that the exact controllability can be realized with boundary controls acting on one end or on two ends. While in the case that there are some zero eigenvalues, in order to realize the corresponding exact controllability, one should use not only boundary controls but also some suitable internal controls in those equations corresponding to zero eigenvalues.Chapter 5 is devoted to the local exact boundary controllability for one-dimensional nonautonomous quasilinear wave equations. By the results of semiglobal C~1 solution to first order quasilinear hyperbolic systems obtained in Chapter 3, 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 controllability for the one-dimensional nonautonomous quasilinear wave equation in both cases of two-sides and one-side control. As an application, the corresponding results on the exact boundary controllability for n-dimensional quasilinear wave equation with rotation invariance are obtained.In Chapter 6, the author studies the controllability for the system of one-dimensional adiabatic flow in Lagrangian representation. By controlling the velocity and/or the pressure on the boundary, the local exact boundary controllability is obtained.At last, in Chapter 7, the author establishes the theory on global exact boundary controllability for first order quasilinear hyperbolic system of diagonal form and applies it to one-dimensional isentropic flow system.