树状网络上的拟线性双曲组的精确边界能控性与能观性
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摘要
本文研究在具有一般拓扑形状的树状网络上,拟线性双曲型方程组的精确边界能控性与能观性。以双曲型方程组混合初边值问题的半整体C~1解的存在性与唯一性理论为基础,使用构造性的方法,实现了指定区域上的能控性与能观性。
本文在两个实际的物理模型下讨论了能控能观性,它们分别是河渠非定常流问题(Saint-venant方程组)和弦振动问题(拟线性波动方程组)。对于河渠非定常流问题,在分别得到了能控性和能观性的结果之后,还通过比较分析,得到了能控性与能观性间的一些对偶关系。而对于树状网络上的弦振动问题,对带有各种边界条件的拟线性波动方程组得到了类似的结论,特别,已有在线性得到具Dirichlet边界条件的树状网络上的能控性结果被推广到拟线性及具一般非线性边界条件的情况。
本文具体组织如下:
第一章介绍问题的背景及研究的现状,并对全文的内容作了一个简要的概述。
在第二章与第三章中,在亚临界与超临界这两种状态下,给出了树状河渠网络上非定常流问题的精确边界能控性。
在第四章中,通过把二阶拟线性波动方程化为一阶拟线性双曲组的手段,建立了树状网络上拟线性弦振动问题的能控性。
第五章与第六章在亚临界与超临界这两种状态研究了河渠非定常流问题的精确边界能观性,并通过与第二章、第三章的对比,找出了能控性与能观性间的对偶性质。
而第七章则与第四章相呼应,建立了树状网络上拟线性弦振动问题的能观性。
最后,在第八章里,对树状网络中拟线性双曲系统在连接点处的条件进行了讨论与分析。给出了对潜在物理模型的能控性结果中得以减少控制量个数而加以进行改进的可能性。
In this Ph.D thesis, based on the theory of the semi-global C~1 solution to the mixed initial-boundary value problem for first order quasilinear hyperbolic systems, by means of a constructive method, we deal with the exact boundary controllability and the exact boundary observability for quasilinear hyperbolic systems on a tree-like network with general topology.
Two physical models are considered in this thesis, one is the unsteady flows in a tree-like network of open canals (Saint-Venant systems), and another one is a tree-like network of vibrating strings (quasilinear wave equations). For the first one, we get the exact boundary controllability and the exact boundary observability in subcritical and supercritical situations, respectively, and we show some duality properties between controllability and observability. For the second one, we get similar results for quasilinear wave equations with various boundary conditions, in particular, the previous results on the exact boundary controllability for linear wave equations with Dirichlet boundary conditions on a tree-like network are extended to the quasilinear case with various boundary conditions.
The arrangement of the thesis is as follows:
First of all, in Chapter 1, a brief introduction is given for the background and present situation on the study and for the results in this thesis.
In Chapter 2 and Chapter 3, the exact boundary controllability of unsteady flows in a tree-like network of open canals is obtained for the subcritical and supercritical situations, respectively.
In Chapter 4, by transforming a second order quasilinear wave equation to a first order quasilinear system, the exact boundary controllability on a tree-like network of vibrating strings is realized.
In Chapter 5 and Chapter 6, the exact boundary observability of unsteady flows in a tree-like network of open canals is given for the subcritical and supercritical situations, respectively. Then, some duality properties between controllability and observability are obtained by comparing with the results mentioned in Chapter 2 and Chapter 3.
Corresponding to Chapter 4, in Chapter 7, the exact boundary observability on a tree-like network of vibrating strings is constructed.
At last, Chapter 8, deals with some discussions about the interface conditions. We show that it is possible to further reduce the number of controls under certain additional hypotheses, and it gives the possibility to improve the result on the controllability for some physical models.
本文在两个实际的物理模型下讨论了能控能观性,它们分别是河渠非定常流问题(Saint-venant方程组)和弦振动问题(拟线性波动方程组)。对于河渠非定常流问题,在分别得到了能控性和能观性的结果之后,还通过比较分析,得到了能控性与能观性间的一些对偶关系。而对于树状网络上的弦振动问题,对带有各种边界条件的拟线性波动方程组得到了类似的结论,特别,已有在线性得到具Dirichlet边界条件的树状网络上的能控性结果被推广到拟线性及具一般非线性边界条件的情况。
本文具体组织如下:
第一章介绍问题的背景及研究的现状,并对全文的内容作了一个简要的概述。
在第二章与第三章中,在亚临界与超临界这两种状态下,给出了树状河渠网络上非定常流问题的精确边界能控性。
在第四章中,通过把二阶拟线性波动方程化为一阶拟线性双曲组的手段,建立了树状网络上拟线性弦振动问题的能控性。
第五章与第六章在亚临界与超临界这两种状态研究了河渠非定常流问题的精确边界能观性,并通过与第二章、第三章的对比,找出了能控性与能观性间的对偶性质。
而第七章则与第四章相呼应,建立了树状网络上拟线性弦振动问题的能观性。
最后,在第八章里,对树状网络中拟线性双曲系统在连接点处的条件进行了讨论与分析。给出了对潜在物理模型的能控性结果中得以减少控制量个数而加以进行改进的可能性。
In this Ph.D thesis, based on the theory of the semi-global C~1 solution to the mixed initial-boundary value problem for first order quasilinear hyperbolic systems, by means of a constructive method, we deal with the exact boundary controllability and the exact boundary observability for quasilinear hyperbolic systems on a tree-like network with general topology.
Two physical models are considered in this thesis, one is the unsteady flows in a tree-like network of open canals (Saint-Venant systems), and another one is a tree-like network of vibrating strings (quasilinear wave equations). For the first one, we get the exact boundary controllability and the exact boundary observability in subcritical and supercritical situations, respectively, and we show some duality properties between controllability and observability. For the second one, we get similar results for quasilinear wave equations with various boundary conditions, in particular, the previous results on the exact boundary controllability for linear wave equations with Dirichlet boundary conditions on a tree-like network are extended to the quasilinear case with various boundary conditions.
The arrangement of the thesis is as follows:
First of all, in Chapter 1, a brief introduction is given for the background and present situation on the study and for the results in this thesis.
In Chapter 2 and Chapter 3, the exact boundary controllability of unsteady flows in a tree-like network of open canals is obtained for the subcritical and supercritical situations, respectively.
In Chapter 4, by transforming a second order quasilinear wave equation to a first order quasilinear system, the exact boundary controllability on a tree-like network of vibrating strings is realized.
In Chapter 5 and Chapter 6, the exact boundary observability of unsteady flows in a tree-like network of open canals is given for the subcritical and supercritical situations, respectively. Then, some duality properties between controllability and observability are obtained by comparing with the results mentioned in Chapter 2 and Chapter 3.
Corresponding to Chapter 4, in Chapter 7, the exact boundary observability on a tree-like network of vibrating strings is constructed.
At last, Chapter 8, deals with some discussions about the interface conditions. We show that it is possible to further reduce the number of controls under certain additional hypotheses, and it gives the possibility to improve the result on the controllability for some physical models.
引文
[1]K.Ammari and M.Jellouli,Stabilization of star-shaped networks of strings,Differential Integral Equations,17(2004),1395-1410.
[2]K.Ammari and M.Jellouli,Remark on stabilization of tree-shaped networks of strings,Applications of Mathematics,52(2007),327-343.
[3]K.Ammari,M.Jellouli,and M.Κhenissi,Stabilization of generic trees of strings,J.Dyn.Control Syst.,11(2005),177-193.
[4]C.Bardos,G.Lebeau,J.Rauch,Sharp sufficient conditions for the observation,control and stabilization of waves from the boundary,SIAM J.Control Optim.,30(1992),1024-1065.
[5]M.M.Cavalcanti,Exact controllability of the wave equation with mixed boundary condition and time-dependent coefficients,Arch.Math.(Brno),35(1999),29-57.
[6]M.Cirina,Boundary controllability of nonlinear hyperbolic systems,SIAM J.Control Optim.,7(1969),198-212.
[7]M.Cirina,Nonlinear hyperbolic problems with solutions on preassigned sets,Michigan Math.J.,17(1970),193-209.
[8]J.-M.Coron,B.d'Andrea-Novel and G.Bastin,A lyapunov approach to control irrigation canals modeled by Saint-Venant equations,in CD-Rom Proceedings,Paper F1008-5,ECC99,Karlsruhe,Germany,1999.
[9]R.Dager and E.Zuazua,Wave Propagation,Observation and Control in 1-D Flexible Multistructures,Mathematiques and Applications 50,2000.
[10]B.De Saint-Venant,Theorie du mouvement non permanent des eaux,avec application aux crues des rivieres et l'introduction des marees dans leur lit,C.R.Acad.Sci.,73(1871),147-154,237-240.
[11]J.de Halleux,B.d'Andrea-Novel J.-M.,Coron and G.Bastin,A lyapunov approach for the control of multi reach channels medelled by Saint-Venant equations,in CD-Rom Proceedings,NOLCOS'01,St-Petersburg,Russia,June 2001.
[12]J.de Halleux and G.Bastin,Stabilization of Saint-Venant equations using Riemann invariants:Applidation to waterways with mobil apillways,in CD-Rom Proceedings,Barcelona,Spain,July 2002.
[13]J.de Halleux,C.Prieur,J.-M.Coron,B.d'Andrea-Novel and G.Bastin,Boundary feedback control in networks of open channels,Automatica,39(2003),1365-1376.
[14]O.Yu.Emanuilov,Boundary control by semilinear evolution equations,Russian Math.Surveys,44(1989),183-184.
[15]Qilong Gu,Exact boundary controllability of unsteady supercritical flows in a tree-like network of open canals,Math.Meth.Appl.Sci.,31(2008),1497-1508.
[16]Qilong Gu and Tatsien Li,Exact boundary observability of unsteady flows in a tree-like network of open canals,Math.Meth.Appl.Sci.,32(2009),395-418.
[17]Qilong Gu and Tatsien Li,Exact boundary controllability for quasilinear hyperbolic systems on a tree-like network and its applications,to appear in SIAM J.Control Optim..
[18]Qilong Gu and Tatsien Li,Exact boundary controllability for quasilinear wave equations in a planar tree-like network of strings,Ann.Inst.H.Poincare,Anal.Non Lineaire,26(2009),2373-2384.
[19]M.Gugat and G.Leugering,Global boundary controllability of the de St.Venant equations between steady states,Ann.I.H.Poincare-AN,20(2003),1-11.[20]M.Gugat,G.Leugering and EJP.Georg Schmidt,Global controllability between steady supereritical flows in channel networks,Math.Meth.Appl.Sci.,27(2004),781-802.
[21]Lina Guo and Zhiqiang Wang,Exact boundary observability for nonautonomous quasilinear wave equations,to appears.
[22]J.E.Lagnese,G.Leugering,and E.J.P.G.Schmidt,Modeling,analysis and control of multi-link structures,Systems and Control:Foundations and Applications,Birhauser-Basel,1994.
[23]I.Lasiecka and R.Triggiani,Exact controllability of semilinear abstract systems with applications to waves and plates boundary control problems,Appl.Math.Optim.,23(1991),109-154.
[24]I.Lasiecka,R.Triggiani and Pengfei Yao,Inverse/observability estimates for seconde-order hyperbolic equations with variable coefficients,J.Math.Anal.Appl.,235(1999),13-57.
[25]G.Leugering and E.G.Schmidt,On the modelling and stabilization of flows in networks of open canals,SIAM J.Control Optim.,41(2002),164-180.
[26]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.
[27]Tatsien Li,Exact boundary controllability of unsteady flows in a network of open canals,Math.Nachr.,278(2005),278-289.
[28]Tatsien Li,Exact controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals.,Math.Meth.Appl.Sci.,27(2004),1089-1114.
[29]Tatsien Li,Observabilite exacte frontiere pour des systemes hyperboliques quasi lineaires,C.R.Acad.Sci.Paris,Set.I,342(2006),937-942.
[30]Tatsien Li,Exact boundary observability for quasilinear hyperbolic systems,ESIAM:Control,Optimisation and Calculus of Variations(2008).
[31]Tatsien Li,Exact boundary observability for 1-D quasilinear wave equations,Math.Meth.Appl.Sci.,29(2006),1543-1553.
[32]Tatsien Li,Cont ollability and Observability for Quasilinear Hyperbolic Systems,AIMS Series on Applied Mathematics,Vol.3,AIMS & Higher Education Press,2010.
[33]Tatsien Li,Global Classical Solutions for Quasilinear Hyperbolic Systems,Masson,J.Wiley,1994.
[34]Tatsien Li and Yi Jin,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.
[35]Tatsien Li and Bopeng Rao,Exact boundary controllability for quasilinear hyperbolic systems,SIAM J.Control Optim.,41(2003),1748-1755.
[36]Tatsien Li and Bopeng Rao,Local exact boundary controllability for a class of quasilinear hyperbolic systems,Chinese Ann.Math.Ser.B,23(2002),209-218.
[37]Tatsien Li and Bopeng Rao,Controlabilite exacte frontiere de l'ecoulement d'un fluide nonstationnaire dans un reseau du type d'arbre de canaux ouverts,C.R.Acad.Sci.Paris,Set.I,339(2004),867-872.
[38]Tatsien Li and Bopeng Rao,Exact boundary controllability of unsteady flows in a tree-like network of open canals,Methods and Applications of Analysis,11(2004),353-366.
[39]Tatsien Li,Bopeng Rao and Yi Jin,Semi-global C~1 solution and exact boundary controllability for reducible quasilinear hyperbolic systems,M2AN,34(2000),399-408.
[40]Tatsien Li,Bopeng Rao and Yi Jin,Solution C~1 semi-globale et controlabilite exacte frontiere de systemes hyperboliques quasi lineaires,,C.R.Acad.Sci.Paris,Ser.I,333(2001),219-224.
[41]Tatsien Li and Zhiqiang Wang,A note on the exact controllability for nonautonomous hyperbolic systems,Commun.Pure Appl.Anal.,6(2007),229-235.
[42]Tatsien Li and Lixin Yu,Controlabilite exacte frontiere pour les equations des ondes quasi lineaires unidimensionnelles,C.R.Acad.Sci.Paris,Ser.I,337(2003),271-276.
[43]Tatsien Li and Lixin Yu,Exact boundary controllability for 1-D quasilinear wave equations,SIAM J.Control Optim.,45(2006),1074-1083.
[44]Tatsien Li and Lixin Yu,Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues,Chinese Ann.Math.Ser.B,24(2003),415-422.
[45]Tatsien Li and Wenci Yu,Boundary Value Problems for Quasilinear Hyperbolic Systems,Duke University Mathematics Series V,1985.
[46]Tatsien Li and Bingyu Zhang,Global exact controllability of a class of quasilinear hyperbolic systems,J.Math.Anal.Appl.,225(1998),289-311.
[47]J.L.Lions,Controlabilite Exacte,Perturbations et Stabilization de Systemes Distribues,Vol.I,Masson,1988.
[48]J.L.Lions,Exact controllability,stabilization and perturbations for distributed systems,SIAM Rev.,30(1988),1-68.
[49]S.Nicaise and J.Valein,Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,Netw.Heterog.Media(electronic),2(2007),425-479.
[50]D.L.Russell,Controllability and stabilizability theory for linear partial differential equations,Recent progress and open questions,SIAM Rev.,20(1978),639-739.
[51]E.J.P.G.Schmidt,On the modeling and exact controllability of networks of vibrating strings,SIAM J.Control.Optim.,30(1992),229-245.
[52]Zhiqiang Wang,Exact boundary controllability for nonautonomous quasilinear wave equations,Math.Meth.Appl.Sci.,30(2007),1311-1327.
[53]Zhiqiang Wang,Exact Controllability for nonautonomous first order quasilinear hyperbolic systems,Chinese Ann.Math.Ser.B.,27(2006),643-656.
[54]Pengfei Yao,On the observability inequalities for exact controllability of wave equations with variable coefficients,SIAM J.Control Optim.,37(1999),1568-1599.
[55]E.Zuazua,Exact controllability for the semilinear wave equation,J.Math.Pures et Appl.,69(1990),1-31.
[56]E.Zuazua,Exact controllability for semilinear wave equations,Ann.I.H.Poincare-AN,10(1993),109-129.
[57]E.Zuazua,Controllability of partial differential equations and its semi-discrete approximation,Discrete Contin.Dyn.Syst.,8(2002),469-513.
[2]K.Ammari and M.Jellouli,Remark on stabilization of tree-shaped networks of strings,Applications of Mathematics,52(2007),327-343.
[3]K.Ammari,M.Jellouli,and M.Κhenissi,Stabilization of generic trees of strings,J.Dyn.Control Syst.,11(2005),177-193.
[4]C.Bardos,G.Lebeau,J.Rauch,Sharp sufficient conditions for the observation,control and stabilization of waves from the boundary,SIAM J.Control Optim.,30(1992),1024-1065.
[5]M.M.Cavalcanti,Exact controllability of the wave equation with mixed boundary condition and time-dependent coefficients,Arch.Math.(Brno),35(1999),29-57.
[6]M.Cirina,Boundary controllability of nonlinear hyperbolic systems,SIAM J.Control Optim.,7(1969),198-212.
[7]M.Cirina,Nonlinear hyperbolic problems with solutions on preassigned sets,Michigan Math.J.,17(1970),193-209.
[8]J.-M.Coron,B.d'Andrea-Novel and G.Bastin,A lyapunov approach to control irrigation canals modeled by Saint-Venant equations,in CD-Rom Proceedings,Paper F1008-5,ECC99,Karlsruhe,Germany,1999.
[9]R.Dager and E.Zuazua,Wave Propagation,Observation and Control in 1-D Flexible Multistructures,Mathematiques and Applications 50,2000.
[10]B.De Saint-Venant,Theorie du mouvement non permanent des eaux,avec application aux crues des rivieres et l'introduction des marees dans leur lit,C.R.Acad.Sci.,73(1871),147-154,237-240.
[11]J.de Halleux,B.d'Andrea-Novel J.-M.,Coron and G.Bastin,A lyapunov approach for the control of multi reach channels medelled by Saint-Venant equations,in CD-Rom Proceedings,NOLCOS'01,St-Petersburg,Russia,June 2001.
[12]J.de Halleux and G.Bastin,Stabilization of Saint-Venant equations using Riemann invariants:Applidation to waterways with mobil apillways,in CD-Rom Proceedings,Barcelona,Spain,July 2002.
[13]J.de Halleux,C.Prieur,J.-M.Coron,B.d'Andrea-Novel and G.Bastin,Boundary feedback control in networks of open channels,Automatica,39(2003),1365-1376.
[14]O.Yu.Emanuilov,Boundary control by semilinear evolution equations,Russian Math.Surveys,44(1989),183-184.
[15]Qilong Gu,Exact boundary controllability of unsteady supercritical flows in a tree-like network of open canals,Math.Meth.Appl.Sci.,31(2008),1497-1508.
[16]Qilong Gu and Tatsien Li,Exact boundary observability of unsteady flows in a tree-like network of open canals,Math.Meth.Appl.Sci.,32(2009),395-418.
[17]Qilong Gu and Tatsien Li,Exact boundary controllability for quasilinear hyperbolic systems on a tree-like network and its applications,to appear in SIAM J.Control Optim..
[18]Qilong Gu and Tatsien Li,Exact boundary controllability for quasilinear wave equations in a planar tree-like network of strings,Ann.Inst.H.Poincare,Anal.Non Lineaire,26(2009),2373-2384.
[19]M.Gugat and G.Leugering,Global boundary controllability of the de St.Venant equations between steady states,Ann.I.H.Poincare-AN,20(2003),1-11.[20]M.Gugat,G.Leugering and EJP.Georg Schmidt,Global controllability between steady supereritical flows in channel networks,Math.Meth.Appl.Sci.,27(2004),781-802.
[21]Lina Guo and Zhiqiang Wang,Exact boundary observability for nonautonomous quasilinear wave equations,to appears.
[22]J.E.Lagnese,G.Leugering,and E.J.P.G.Schmidt,Modeling,analysis and control of multi-link structures,Systems and Control:Foundations and Applications,Birhauser-Basel,1994.
[23]I.Lasiecka and R.Triggiani,Exact controllability of semilinear abstract systems with applications to waves and plates boundary control problems,Appl.Math.Optim.,23(1991),109-154.
[24]I.Lasiecka,R.Triggiani and Pengfei Yao,Inverse/observability estimates for seconde-order hyperbolic equations with variable coefficients,J.Math.Anal.Appl.,235(1999),13-57.
[25]G.Leugering and E.G.Schmidt,On the modelling and stabilization of flows in networks of open canals,SIAM J.Control Optim.,41(2002),164-180.
[26]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.
[27]Tatsien Li,Exact boundary controllability of unsteady flows in a network of open canals,Math.Nachr.,278(2005),278-289.
[28]Tatsien Li,Exact controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals.,Math.Meth.Appl.Sci.,27(2004),1089-1114.
[29]Tatsien Li,Observabilite exacte frontiere pour des systemes hyperboliques quasi lineaires,C.R.Acad.Sci.Paris,Set.I,342(2006),937-942.
[30]Tatsien Li,Exact boundary observability for quasilinear hyperbolic systems,ESIAM:Control,Optimisation and Calculus of Variations(2008).
[31]Tatsien Li,Exact boundary observability for 1-D quasilinear wave equations,Math.Meth.Appl.Sci.,29(2006),1543-1553.
[32]Tatsien Li,Cont ollability and Observability for Quasilinear Hyperbolic Systems,AIMS Series on Applied Mathematics,Vol.3,AIMS & Higher Education Press,2010.
[33]Tatsien Li,Global Classical Solutions for Quasilinear Hyperbolic Systems,Masson,J.Wiley,1994.
[34]Tatsien Li and Yi Jin,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.
[35]Tatsien Li and Bopeng Rao,Exact boundary controllability for quasilinear hyperbolic systems,SIAM J.Control Optim.,41(2003),1748-1755.
[36]Tatsien Li and Bopeng Rao,Local exact boundary controllability for a class of quasilinear hyperbolic systems,Chinese Ann.Math.Ser.B,23(2002),209-218.
[37]Tatsien Li and Bopeng Rao,Controlabilite exacte frontiere de l'ecoulement d'un fluide nonstationnaire dans un reseau du type d'arbre de canaux ouverts,C.R.Acad.Sci.Paris,Set.I,339(2004),867-872.
[38]Tatsien Li and Bopeng Rao,Exact boundary controllability of unsteady flows in a tree-like network of open canals,Methods and Applications of Analysis,11(2004),353-366.
[39]Tatsien Li,Bopeng Rao and Yi Jin,Semi-global C~1 solution and exact boundary controllability for reducible quasilinear hyperbolic systems,M2AN,34(2000),399-408.
[40]Tatsien Li,Bopeng Rao and Yi Jin,Solution C~1 semi-globale et controlabilite exacte frontiere de systemes hyperboliques quasi lineaires,,C.R.Acad.Sci.Paris,Ser.I,333(2001),219-224.
[41]Tatsien Li and Zhiqiang Wang,A note on the exact controllability for nonautonomous hyperbolic systems,Commun.Pure Appl.Anal.,6(2007),229-235.
[42]Tatsien Li and Lixin Yu,Controlabilite exacte frontiere pour les equations des ondes quasi lineaires unidimensionnelles,C.R.Acad.Sci.Paris,Ser.I,337(2003),271-276.
[43]Tatsien Li and Lixin Yu,Exact boundary controllability for 1-D quasilinear wave equations,SIAM J.Control Optim.,45(2006),1074-1083.
[44]Tatsien Li and Lixin Yu,Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues,Chinese Ann.Math.Ser.B,24(2003),415-422.
[45]Tatsien Li and Wenci Yu,Boundary Value Problems for Quasilinear Hyperbolic Systems,Duke University Mathematics Series V,1985.
[46]Tatsien Li and Bingyu Zhang,Global exact controllability of a class of quasilinear hyperbolic systems,J.Math.Anal.Appl.,225(1998),289-311.
[47]J.L.Lions,Controlabilite Exacte,Perturbations et Stabilization de Systemes Distribues,Vol.I,Masson,1988.
[48]J.L.Lions,Exact controllability,stabilization and perturbations for distributed systems,SIAM Rev.,30(1988),1-68.
[49]S.Nicaise and J.Valein,Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,Netw.Heterog.Media(electronic),2(2007),425-479.
[50]D.L.Russell,Controllability and stabilizability theory for linear partial differential equations,Recent progress and open questions,SIAM Rev.,20(1978),639-739.
[51]E.J.P.G.Schmidt,On the modeling and exact controllability of networks of vibrating strings,SIAM J.Control.Optim.,30(1992),229-245.
[52]Zhiqiang Wang,Exact boundary controllability for nonautonomous quasilinear wave equations,Math.Meth.Appl.Sci.,30(2007),1311-1327.
[53]Zhiqiang Wang,Exact Controllability for nonautonomous first order quasilinear hyperbolic systems,Chinese Ann.Math.Ser.B.,27(2006),643-656.
[54]Pengfei Yao,On the observability inequalities for exact controllability of wave equations with variable coefficients,SIAM J.Control Optim.,37(1999),1568-1599.
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