非自治拟线性双曲型方程组的精确能控性
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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．
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 there
is 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.
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．
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 there
is 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.
引文
[1] Bardos C., Lebeau G., Rauch J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30(1992), 1024-1065.
[2] Cavalcanti M. M., Exact controllability of the wave equation with mixed boundary condition and time-dependent coefficients, Arch. Math. (Brno), 35(1999), 29-57.
[3] Cirinà M., Boundary controllability of nonlinear hyperbolic systems, SIAM J. Control Optim., 7(1969), 198-212.
[4] Cirinà M., Nonlinear hyperbolic problems with solutions on preassigned sets, Michigan Math. J., 17(1970), 193-209.
[5] Emanuilov O. Yu., Boundary control by semilinear evolution equations, Russian Math. Surveys, Vol. 44, No.3 (1989), 183-184.
[6] Gugat M., Leugering G., Global boundary controllability of the de St. Venant equations between steady states, Ann. I. H. Poincare-AN 20, 1, (2003), 1-11.
[7] 金逸，拟线性双曲组混合初边值问题的半整体经典解，硕士论文．
[8] John F., Formation of singularities in one-dimensional nonlinear wave propagations, Comm. Pure Appl. Math., 27(1974), 377-405.
[9] Lasiecka I., Triggiani R., Exact controllability of semilinear abstract systems with applications to waves and plates boundary control problems, Appl. Math. Optim., 23 (1991), 109-154.
[10] Li Tatsien, Exact boundary controllability of unsteady flows in a network of open canals, Math. Nachr. 278:3(2005), 278-289.
[11] Li Tatsien, Exact controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals., Math. Methods Appl. Sci., 27(2004), 1089-1114.
[12] 李大潜，河渠非定常流的精确能控性，南通工学院学报(自然科学版)1：2(2002)，1-5．
[13] Li Tatsien, Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson, J. Wiley, 1994.
[14] 李大潜， 拟线性双曲型方程(组)的精确能控性，高校应用数学学报 A辑，20：2(2005)，127-146．
[15] Li Tatsien, Jin Yi, Semi-global C~1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems, Chin. Ann. Math., 22B (2001), 325-336.
[16] Li Tatsien, Peng Yuejun, Global C~1 solution to the initial-boundary value problem for diagonal hyperbolic systems with linearly degenerate characteristics, J. Partial Diff. Eqs., 16(2003), 8-17.
[17] Li Tatsien, Peng Yuejun, The mixed initial-bounadry value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics, Nonlinear Analysis, 52(2003), 573-583.
[18] Li Tatsien, Rao Bopeng, Exact boundary controllability for quasilinear hyperbolic system, SIAM J. Control Optim., 41(2003), 1748-1755.
[19] Li Tatsien, Rao Bopeng, Local exact boundary controllability for a class of quasilinear hyperbolic systems, Chin. Ann. Math., 23B(2002), 209-218.
[20] Li Tatsien, Rao Bopeng, Jin Yi, Semi-global C~1 solution and exact boundary controllability for reducible quasilinear hyperbolic systems, M2AN, 34(2000), 399-408.
[21] Li Tatsien, Rao Bopeng, Jin Yi, Solution C~1 semi-globale et controlabilité exacte frontière de systèmes hyperboliques quasi linéaires,, C. R. Acad. Sci. Paris, t.333, Série I (2001), 219-224.
[22] Li Tatsien, Wang Zhiqiang, A note on the exact controllability for nonautonomous hyperbolic systems, to appear in Commun. Pure Appl. Anal..
[23] Li Tatsien, Xu Yulan, Local exact boundary controllability for nonlinear vibrating string equations, International Journal of Modern Physics B, 17(2003), 4062-4071.
[24] Li Tatsien, Yu Lixin, ContrSlabilité exacte frontière pour les équations des ondes quasi linéaires unidimensionnelles, C. R. Acad. Sci. Paris, Série I, 337(2003), 271-276.
[25] Li Tatsien, Yu Lixin, Exact boundary controllability for 1-D quasilinear wave equations, to appear in SIAM J. Control. Optim..
[26] Li Tatsien, Yu Lixin, Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues, Chin. Ann. Math., 24B(2003), 415-422.
[27] Li Tatsien, Yu Wenci, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, 1985.
[28] 李大潜，俞文(鱼此)，沈玮熙，拟线性双曲-抛物耦合方程组的第二边值问题，数学年刊，2(1981)，65-90．
[29] Li Tatsien, Zhang Bingyu, Global exact controllability of a class of quasilinear hyperbolic systems, J. Math. Anal. Appl., 225(1998), 289-311.
[30] Lions J. L., Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30(1988), 1-68.
[31] Lions J. L., Controlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués, Vol. I, Masson, 1988.
[32] Liu Fagui, Global smooth resolvability for one dimensional nonisentropic gas dynamic systems, Nonlinear Analysis, 36(1999), 25-34.
[33] Liu Fagui, Zhang Yuanzhang, Initial boundary value problem for compressible fluids, J. Zhengzhou Univ., 36:1(2004), 1-6.
[34] Majda A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, New York, Springer-Verlag, 1984.
[35] Russell D. L., Controllability and stabilizability theory for linear partial differential equations, Recent progress and open questions, SIAM Rev., 20(1978), 639-739.
[36] Shubov M. A., Exact boundary and distributed Controllability of radial damped wave equation, J. Math. Pures et Appl., 77(1998), 415-437.
[37] Tataru D., Boundary controllability for conservative PDEs, Appl. Math. Optim., 31:3(1995), 257-295.
[38] Wang Zhiqiang, Exact boundary controllability]or nonautonomous quasilinear wave equations, submitted to Nonlinear Analysis.
[39] Wang Zhiqiang, Exact Controllability for nonautonomous first order quasilinear hyperbolic systems, to appear in Chin. Ann. Math..
[40] 王志强，于立新，一维绝热流方程组的精确边界能控性，将发表于高校应用数学学报．
[41] Yao Pengfei, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim., 37(1999), 1568-1599.
[42] 于立新，拟线性双曲型方程组的精确能控性，博士毕业论文．
[43] 于立新，一类拟线性双曲型方程组混合初-边值问题的半整体C~1解，数学年刊，25A：5(2004)，549-560．
[44] 于勇，河渠非定常流的精确边界能控性，高校应用数学学报，19(2004)，379-393．
[45] 于勇，河渠非定常流的精确边界能控性，硕士毕业论文．
[46] Zuazua E., Controllability of partial differential equations and its semi-discrete approximation, Discrete and Continuous Dynamical Systems, 8(2002), 469-513.
[47] Zuazua E., Exact controllability for the semilinear wave equation, J. Math. Pures et Appl., 69 (1990), 1-31.
[48] Zuazua E., Exact controllability for semilinear wave equations, Annales de l'Institut Henri Poincaré, 10(1993), 109-129.
[2] Cavalcanti M. M., Exact controllability of the wave equation with mixed boundary condition and time-dependent coefficients, Arch. Math. (Brno), 35(1999), 29-57.
[3] Cirinà M., Boundary controllability of nonlinear hyperbolic systems, SIAM J. Control Optim., 7(1969), 198-212.
[4] Cirinà M., Nonlinear hyperbolic problems with solutions on preassigned sets, Michigan Math. J., 17(1970), 193-209.
[5] Emanuilov O. Yu., Boundary control by semilinear evolution equations, Russian Math. Surveys, Vol. 44, No.3 (1989), 183-184.
[6] Gugat M., Leugering G., Global boundary controllability of the de St. Venant equations between steady states, Ann. I. H. Poincare-AN 20, 1, (2003), 1-11.
[7] 金逸，拟线性双曲组混合初边值问题的半整体经典解，硕士论文．
[8] John F., Formation of singularities in one-dimensional nonlinear wave propagations, Comm. Pure Appl. Math., 27(1974), 377-405.
[9] Lasiecka I., Triggiani R., Exact controllability of semilinear abstract systems with applications to waves and plates boundary control problems, Appl. Math. Optim., 23 (1991), 109-154.
[10] Li Tatsien, Exact boundary controllability of unsteady flows in a network of open canals, Math. Nachr. 278:3(2005), 278-289.
[11] Li Tatsien, Exact controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals., Math. Methods Appl. Sci., 27(2004), 1089-1114.
[12] 李大潜，河渠非定常流的精确能控性，南通工学院学报(自然科学版)1：2(2002)，1-5．
[13] Li Tatsien, Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson, J. Wiley, 1994.
[14] 李大潜， 拟线性双曲型方程(组)的精确能控性，高校应用数学学报 A辑，20：2(2005)，127-146．
[15] Li Tatsien, Jin Yi, Semi-global C~1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems, Chin. Ann. Math., 22B (2001), 325-336.
[16] Li Tatsien, Peng Yuejun, Global C~1 solution to the initial-boundary value problem for diagonal hyperbolic systems with linearly degenerate characteristics, J. Partial Diff. Eqs., 16(2003), 8-17.
[17] Li Tatsien, Peng Yuejun, The mixed initial-bounadry value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics, Nonlinear Analysis, 52(2003), 573-583.
[18] Li Tatsien, Rao Bopeng, Exact boundary controllability for quasilinear hyperbolic system, SIAM J. Control Optim., 41(2003), 1748-1755.
[19] Li Tatsien, Rao Bopeng, Local exact boundary controllability for a class of quasilinear hyperbolic systems, Chin. Ann. Math., 23B(2002), 209-218.
[20] Li Tatsien, Rao Bopeng, Jin Yi, Semi-global C~1 solution and exact boundary controllability for reducible quasilinear hyperbolic systems, M2AN, 34(2000), 399-408.
[21] Li Tatsien, Rao Bopeng, Jin Yi, Solution C~1 semi-globale et controlabilité exacte frontière de systèmes hyperboliques quasi linéaires,, C. R. Acad. Sci. Paris, t.333, Série I (2001), 219-224.
[22] Li Tatsien, Wang Zhiqiang, A note on the exact controllability for nonautonomous hyperbolic systems, to appear in Commun. Pure Appl. Anal..
[23] Li Tatsien, Xu Yulan, Local exact boundary controllability for nonlinear vibrating string equations, International Journal of Modern Physics B, 17(2003), 4062-4071.
[24] Li Tatsien, Yu Lixin, ContrSlabilité exacte frontière pour les équations des ondes quasi linéaires unidimensionnelles, C. R. Acad. Sci. Paris, Série I, 337(2003), 271-276.
[25] Li Tatsien, Yu Lixin, Exact boundary controllability for 1-D quasilinear wave equations, to appear in SIAM J. Control. Optim..
[26] Li Tatsien, Yu Lixin, Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues, Chin. Ann. Math., 24B(2003), 415-422.
[27] Li Tatsien, Yu Wenci, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, 1985.
[28] 李大潜，俞文(鱼此)，沈玮熙，拟线性双曲-抛物耦合方程组的第二边值问题，数学年刊，2(1981)，65-90．
[29] Li Tatsien, Zhang Bingyu, Global exact controllability of a class of quasilinear hyperbolic systems, J. Math. Anal. Appl., 225(1998), 289-311.
[30] Lions J. L., Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30(1988), 1-68.
[31] Lions J. L., Controlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués, Vol. I, Masson, 1988.
[32] Liu Fagui, Global smooth resolvability for one dimensional nonisentropic gas dynamic systems, Nonlinear Analysis, 36(1999), 25-34.
[33] Liu Fagui, Zhang Yuanzhang, Initial boundary value problem for compressible fluids, J. Zhengzhou Univ., 36:1(2004), 1-6.
[34] Majda A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, New York, Springer-Verlag, 1984.
[35] Russell D. L., Controllability and stabilizability theory for linear partial differential equations, Recent progress and open questions, SIAM Rev., 20(1978), 639-739.
[36] Shubov M. A., Exact boundary and distributed Controllability of radial damped wave equation, J. Math. Pures et Appl., 77(1998), 415-437.
[37] Tataru D., Boundary controllability for conservative PDEs, Appl. Math. Optim., 31:3(1995), 257-295.
[38] Wang Zhiqiang, Exact boundary controllability]or nonautonomous quasilinear wave equations, submitted to Nonlinear Analysis.
[39] Wang Zhiqiang, Exact Controllability for nonautonomous first order quasilinear hyperbolic systems, to appear in Chin. Ann. Math..
[40] 王志强，于立新，一维绝热流方程组的精确边界能控性，将发表于高校应用数学学报．
[41] Yao Pengfei, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim., 37(1999), 1568-1599.
[42] 于立新，拟线性双曲型方程组的精确能控性，博士毕业论文．
[43] 于立新，一类拟线性双曲型方程组混合初-边值问题的半整体C~1解，数学年刊，25A：5(2004)，549-560．
[44] 于勇，河渠非定常流的精确边界能控性，高校应用数学学报，19(2004)，379-393．
[45] 于勇，河渠非定常流的精确边界能控性，硕士毕业论文．
[46] Zuazua E., Controllability of partial differential equations and its semi-discrete approximation, Discrete and Continuous Dynamical Systems, 8(2002), 469-513.
[47] Zuazua E., Exact controllability for the semilinear wave equation, J. Math. Pures et Appl., 69 (1990), 1-31.
[48] Zuazua E., Exact controllability for semilinear wave equations, Annales de l'Institut Henri Poincaré, 10(1993), 109-129.