拟线性双曲型方程组的精确能观性
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摘要
本文利用一阶拟线性双曲型方程组混合初边值问题的半整体C~1解理论,通过直接构造性方法建立了非自治一阶拟线性双曲型方程组的精确边界能观性理论,对相应的观测时间给出了精确的估计,并揭示其与自治系统情形的差异性。与此同时,本文还建立了一阶拟线性双曲型方程组一类具特征边界的边值问题的半整体C~1解理论,并且以此为基础,利用直接的构造性方法建立了具零特征的一阶拟线性双曲型方程组的局部精确能观性理论,并对相应的观测时间给出了精确的估计。通过这两项工作完善了拟线性双曲型方程组的精确能观性理论。
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观性的定义和相关问题的研究历史与现状,以及本文的主要工作及其意义。
作为本文的研究基础,在第二章,作者列举了一阶拟线性双曲型方程组半整体C~1解理论中的主要结论;在第三章,对一类具零特征的一阶拟线性双曲型方程组建立了其一类具特征边界的边值问题的半整体C~1解理论。
在第四章,作者以第二章中所列的半整体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型方程组的局部精确能观性。当系统不具有零特征时,证明了只需通过在一侧或双侧边界上的观测即可实现局部精确能观性。
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观性问题。利用第二章中所列的一阶拟线性双曲型方程组的半整体C~1解结果,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观性。同时,作者指出对具有旋转不变性的n(n＞1)维拟线性波动方程在适当的假设下可以得到相应的局部精确边界能观性;对一维本质自治的拟线性波动方程,建立了相应的局部精确边界能观性。
在第六章,作者以第三章中建立起来的具零特征的一阶拟线性双曲型方程组边值问题的半整体C~1解理论为基础,结合第二章中所列的半整体C~1解理论、对具零特征的一阶拟线性双曲型方程组建立了局部精确能观性理论,指出除了通过边界上的观测,还需在求解区域内部某一时刻进行观测,才能实现精确能观。此外,通过例子说明了该理论中所要求的观测时间是不可改进的,观测量的个数也是不可减少的。
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.
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观性的定义和相关问题的研究历史与现状,以及本文的主要工作及其意义。
作为本文的研究基础,在第二章,作者列举了一阶拟线性双曲型方程组半整体C~1解理论中的主要结论;在第三章,对一类具零特征的一阶拟线性双曲型方程组建立了其一类具特征边界的边值问题的半整体C~1解理论。
在第四章,作者以第二章中所列的半整体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型方程组的局部精确能观性。当系统不具有零特征时,证明了只需通过在一侧或双侧边界上的观测即可实现局部精确能观性。
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观性问题。利用第二章中所列的一阶拟线性双曲型方程组的半整体C~1解结果,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观性。同时,作者指出对具有旋转不变性的n(n＞1)维拟线性波动方程在适当的假设下可以得到相应的局部精确边界能观性;对一维本质自治的拟线性波动方程,建立了相应的局部精确边界能观性。
在第六章,作者以第三章中建立起来的具零特征的一阶拟线性双曲型方程组边值问题的半整体C~1解理论为基础,结合第二章中所列的半整体C~1解理论、对具零特征的一阶拟线性双曲型方程组建立了局部精确能观性理论,指出除了通过边界上的观测,还需在求解区域内部某一时刻进行观测,才能实现精确能观。此外,通过例子说明了该理论中所要求的观测时间是不可改进的,观测量的个数也是不可减少的。
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.
引文
[1]Alabau F.,Komornik V.,Observabilite,controlabilite er stabilisation frontiere du systeme d'elasticite lineaire,C.R.Acad.Sci.Paris.Serie I,324(1997),519-524.
[2]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.
[3]Cavalcanti M.M.,Exact controllability of the wave equation with mixed boundary condition and time-dependent coefficients,Arch.Math.(Brno),35(1999),29-57.
[4]Cirine M.,Boundary controllability of nonlinear hyperbolic systems,SIAM J.Control Optim.,7(1969),198-212.
[5]Cirina M.,Nonlinear hyperbolic problems with solutions on preassigned sets,Michigan Math.J.,17(1970),193-209.
[6]Dolecki S.,Russell D.,A general theory of observation and control.SIAM J.Control 15(1977),185-220.
[7]Guo Lina,Boundary value problems for first order quasilinear hyperbolic systems with zero eigenvalues,to appear.
[8]Guo Lina,Wang Zhiqiang,Exact boundary observability for nonautonomous first order quasilinear hyperbolic systems,Math.Meth.Appl.Sci,31(2008),1956-1971.
[9]Guo Lina,Wang Zhiqiang,Exact boundary observability for nonautonomous quasilinear wave equations,to appear.
[10]P.Hariman,A.Wintner,On hyperbolic partial differential equations,American Journal of Mathematics,74(1952),834-864.
[11]Infante J.,Zuazua E.,Boundary observability for the space semi-discretizations of the 1-D wave equation,Rairo-Math.Model.Num.33(1999),407-438.
[12]Infante J.,Zuazua E.,Boundary observability for the space-discretization of the 1-D wave equation,C.R.Acad.Sci.Paris,Sere I,326,713-718,(1998).
[13]Lasiecka I.,Triggiani R.,Yao Pengfei,Exact controllability for second-order hyperbolic equations with variable coefficient principal part and first-order terms,Nonlinear Anal.,TMA,30(1997),111-122.
[14]Li Tatsien,Controllability and Observability for Quasilinear Hyperbolic Systems,to print.
[15]Li Tatsien,Exact boundary observability for 1-D quasilinear wave equations,Math.Meth.Appl.Sci.,29(2006),1543-1553.
[16]Li Tatsien,Exact boundary observability for quasilinear hyperbolic systems,ESAIM:Control,Optimisation and Calculus of Variations,14(2008),759-766.
[17]Li Tatsien,Global Classical Solutions for Quasilinear Hyperbolic Systems,Masson,J.Wiley,1994.
[18]Li Tatsien,Observabilite exacte frontiere pour des systemes hyperboliques quasilineaires,C.R.Acad.Sci.Paris,Set.I,342(2006),937-942.
[19]Li Tatsien,Guo Lina,Exact observability for first order quasilinear hyperbolic systems with zero eigenvalues,to appear.
[20]Li Tatsien,Jin Yi,Semi-global C~1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems,Chinese Ann.Math.Ser.B,22(2001),325-336.
[21]Li Tatsien,Rao Bopeng,Local exact boundary controllability for a class of quasilinear hyperbolic systems,Chin.Ann.Math.Ser.B,23(2002),209-218.
[22]Li Tatsien,Rao Bopeng,Jin Yi,Semi-global C~1 solution and exact boundary controllability for reducible quasilinear hyperbolic systems,MSAN,34(2000),399-408.
[23]Li Tatsien,Wang Zhiqiang,A note on the exact controllability for nonautonomous hyperbolic systems,Commun.Pure Appl.Anal.,6(2007),229-235.
[24]Li Tatsien,Xu Yulan,Local exact boundary controllability for nonlinear vibrating string equations,International Journal of Modern Physics B,17(2003),4062-4071.
[25]Li Tatsien,Yu Lixin,Controlabilite exacte frontiere pour les equations des ondes quasi lineaires unidimensionnelles,C.R.Acad.Sci.Paris,Serie I,337(2003),271-276.
[26]Li Tatsien,Yu Lixin,Exact boundary controllability for 1-D quasilinear wave equations,SIAM J.Control Optim.,45,1074-1083,(2006).
[27]Li Tatsien,Yu Lixin,Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues,Chin.Ann.Math.Ser.B,24(2003),415-422.
[28]Li Tatsien,Yu Wenci,Boundary Value Problems for Quasilinear Hyperbolic Systems,Duke University Mathematics Series V,1985.
[29]李大潜,俞文(鱼此),一阶拟线性双曲型方程组的柯西问题,数学进展,2(1964),152-171.
[30]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A辑,2(1981),65-90.
[31]Lions J.L.,Exact controllability,stabilization and perturbations for distributed systems,SIAM Rev.,30(1988),1-68.
[32]Lions J.L.,Controlabilite Exacte,Perturbations et Stabilization de Systemes Distribues,Vol.Ⅰ,Masson,1988.
[33]Russell D.L.,Controllability and stabilizability theory for linear partial differential equations,Recent progress and open questions,SIAM Rev.,20(1978),639-739.
[34]Trooshin I.,Yamamoto M.,Identification problem for a one-dimensional vibrating system,Math.Meth.Appl.Sci.,28(2005),2037-2059.
[35]Wang Zhiqiang,Exact boundary controllability for nonautonomous quasilinear wave equations,Math.Meth.Appl.Sci.,30(2007),1311-1327.
[36]Wang Zhiqiang,Exact Controllability for nonautonomous first order quasilinear hyperbolic systems,Chin.Ann.Math.,27B(2006),643-656.
[37]王志强,非自治拟线性双曲型方程组的精确能控性,博士毕业论文.
[38]Yao Pengfei,On the observability inequalities for exact controllability of wave equations with variable coefficients,SIAM J.Control Optim.,37(1999),1568-1599.
[39]叶伟文,具零特征的一阶拟线性双曲组的精确能观性,硕士毕业论文.
[40]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A辑,25A:5(2004),549-560.
[41]Zuazua E.,Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square,J.Math.Pures et Appl.,78,523-563,(1999).
[2]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.
[3]Cavalcanti M.M.,Exact controllability of the wave equation with mixed boundary condition and time-dependent coefficients,Arch.Math.(Brno),35(1999),29-57.
[4]Cirine M.,Boundary controllability of nonlinear hyperbolic systems,SIAM J.Control Optim.,7(1969),198-212.
[5]Cirina M.,Nonlinear hyperbolic problems with solutions on preassigned sets,Michigan Math.J.,17(1970),193-209.
[6]Dolecki S.,Russell D.,A general theory of observation and control.SIAM J.Control 15(1977),185-220.
[7]Guo Lina,Boundary value problems for first order quasilinear hyperbolic systems with zero eigenvalues,to appear.
[8]Guo Lina,Wang Zhiqiang,Exact boundary observability for nonautonomous first order quasilinear hyperbolic systems,Math.Meth.Appl.Sci,31(2008),1956-1971.
[9]Guo Lina,Wang Zhiqiang,Exact boundary observability for nonautonomous quasilinear wave equations,to appear.
[10]P.Hariman,A.Wintner,On hyperbolic partial differential equations,American Journal of Mathematics,74(1952),834-864.
[11]Infante J.,Zuazua E.,Boundary observability for the space semi-discretizations of the 1-D wave equation,Rairo-Math.Model.Num.33(1999),407-438.
[12]Infante J.,Zuazua E.,Boundary observability for the space-discretization of the 1-D wave equation,C.R.Acad.Sci.Paris,Sere I,326,713-718,(1998).
[13]Lasiecka I.,Triggiani R.,Yao Pengfei,Exact controllability for second-order hyperbolic equations with variable coefficient principal part and first-order terms,Nonlinear Anal.,TMA,30(1997),111-122.
[14]Li Tatsien,Controllability and Observability for Quasilinear Hyperbolic Systems,to print.
[15]Li Tatsien,Exact boundary observability for 1-D quasilinear wave equations,Math.Meth.Appl.Sci.,29(2006),1543-1553.
[16]Li Tatsien,Exact boundary observability for quasilinear hyperbolic systems,ESAIM:Control,Optimisation and Calculus of Variations,14(2008),759-766.
[17]Li Tatsien,Global Classical Solutions for Quasilinear Hyperbolic Systems,Masson,J.Wiley,1994.
[18]Li Tatsien,Observabilite exacte frontiere pour des systemes hyperboliques quasilineaires,C.R.Acad.Sci.Paris,Set.I,342(2006),937-942.
[19]Li Tatsien,Guo Lina,Exact observability for first order quasilinear hyperbolic systems with zero eigenvalues,to appear.
[20]Li Tatsien,Jin Yi,Semi-global C~1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems,Chinese Ann.Math.Ser.B,22(2001),325-336.
[21]Li Tatsien,Rao Bopeng,Local exact boundary controllability for a class of quasilinear hyperbolic systems,Chin.Ann.Math.Ser.B,23(2002),209-218.
[22]Li Tatsien,Rao Bopeng,Jin Yi,Semi-global C~1 solution and exact boundary controllability for reducible quasilinear hyperbolic systems,MSAN,34(2000),399-408.
[23]Li Tatsien,Wang Zhiqiang,A note on the exact controllability for nonautonomous hyperbolic systems,Commun.Pure Appl.Anal.,6(2007),229-235.
[24]Li Tatsien,Xu Yulan,Local exact boundary controllability for nonlinear vibrating string equations,International Journal of Modern Physics B,17(2003),4062-4071.
[25]Li Tatsien,Yu Lixin,Controlabilite exacte frontiere pour les equations des ondes quasi lineaires unidimensionnelles,C.R.Acad.Sci.Paris,Serie I,337(2003),271-276.
[26]Li Tatsien,Yu Lixin,Exact boundary controllability for 1-D quasilinear wave equations,SIAM J.Control Optim.,45,1074-1083,(2006).
[27]Li Tatsien,Yu Lixin,Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues,Chin.Ann.Math.Ser.B,24(2003),415-422.
[28]Li Tatsien,Yu Wenci,Boundary Value Problems for Quasilinear Hyperbolic Systems,Duke University Mathematics Series V,1985.
[29]李大潜,俞文(鱼此),一阶拟线性双曲型方程组的柯西问题,数学进展,2(1964),152-171.
[30]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A辑,2(1981),65-90.
[31]Lions J.L.,Exact controllability,stabilization and perturbations for distributed systems,SIAM Rev.,30(1988),1-68.
[32]Lions J.L.,Controlabilite Exacte,Perturbations et Stabilization de Systemes Distribues,Vol.Ⅰ,Masson,1988.
[33]Russell D.L.,Controllability and stabilizability theory for linear partial differential equations,Recent progress and open questions,SIAM Rev.,20(1978),639-739.
[34]Trooshin I.,Yamamoto M.,Identification problem for a one-dimensional vibrating system,Math.Meth.Appl.Sci.,28(2005),2037-2059.
[35]Wang Zhiqiang,Exact boundary controllability for nonautonomous quasilinear wave equations,Math.Meth.Appl.Sci.,30(2007),1311-1327.
[36]Wang Zhiqiang,Exact Controllability for nonautonomous first order quasilinear hyperbolic systems,Chin.Ann.Math.,27B(2006),643-656.
[37]王志强,非自治拟线性双曲型方程组的精确能控性,博士毕业论文.
[38]Yao Pengfei,On the observability inequalities for exact controllability of wave equations with variable coefficients,SIAM J.Control Optim.,37(1999),1568-1599.
[39]叶伟文,具零特征的一阶拟线性双曲组的精确能观性,硕士毕业论文.
[40]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A辑,25A:5(2004),549-560.
[41]Zuazua E.,Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square,J.Math.Pures et Appl.,78,523-563,(1999).