【作者】 王志强；

【导师】 李大潜；

【作者基本信息】 复旦大学 ， 应用数学， 2006， 博士

【摘要】 本文将一阶拟线性双曲组混合初-边值问题的半整体C~1解理论推广到更为一般的形式，并且以此为基础，利用直接的构造性方法建立了非自治一阶拟线性双曲组的局部精确能控性理论，揭示其与自治系统情形的差异性，并对相应的控制时间给出了精确的估计．作为应用，本文解决了一维非自治拟线性波动方程和一维绝热流方程组的局部精确边界能控性问题．最后，本文对齐次的一阶拟线性对角型双曲组实现了整体精确能控性，并在一维等熵流方程组得到了应用．本文的具体安排如下：首先在第一章，作者简单介绍了精确能控性的定义以及相关问题的研究历史与现状．在第二章，作者结合一些例子说明了非自治双曲系统的精确能控性存在多种不同的可能性．通过与自治系统的比较，揭示出非自治双曲系统的精确能控性的一般特点，并指出对其研究的困难和意义．作为下一步研究的基础，在第三章中，作者对带非线性边界条件的一般形式的一阶拟线性双曲组建立了其混合初-边值问题的半整体C~1解理论．在第四章，作者以第三章建立起来的半整体C~1解理论为基础，采用一个与自治系统情况相似的直接构造方法得到了非自治一阶拟线性双曲组的局部精确能控性．当系统不具有零特征时，证明了只需通过作用在一侧或双侧边界上的控制即可实现局部精确能控性；而在系统具有零特征的情形，虽仍然可以得到的相应的局部精确能控性，但此时除了边界控制，还需要对相应于零特征的方程加入内部控制．在第五章，作者致力于研究一维非自治拟线性波动方程的局部精确能控性问题．利用第三章得到的一阶拟线性双曲组的半整体C~1解结果，作者可以用统一的方式处理各种不同类型的边界条件，得到一维非自治拟线性波动方程混合问题的半整体C~2解理论，并进而实现相应的双侧和单侧局部精确边界能控性．作为一个直接的应用，作者对具有旋转不变性的n(n＞1)维拟线性波动方程建立了相应的局部精确边界能控性．在第六章，作为已有结果的应用，作者研究了Lagrange坐标下的一维绝热流方程组的精确边界能控性问题，并通过对边界上速度与(或)压强的控制实现了其局部精确边界能控性．最后，在第七章，本文对齐次的一阶拟线性对角型双曲组实现了整体精确能控性，并在一维等熵流方程组得到了应用．更多还原

【Abstract】 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.更多还原