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March  2014, 13(2): 881-901. doi: 10.3934/cpaa.2014.13.881

Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type

Long Hu 1, , Tatsien Li 1, and  Bopeng Rao 2,

1. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2. 

Institut de Recherche Mathématique Avancée, Université de Strasbourg, 67084 Strasbourg

Received  June 2013 Revised  September 2013 Published  October 2013

Several kinds of exact synchronizations are introduced for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type and these synchronizations can be realized by means of some boundary controls.
Keywords: Coupled boundary conditions of dissipative type, exact synchronization by groups, generalized exact synchronization., exact null controllability, exact null controllability and synchronization by groups, exact synchronization.
Mathematics Subject Classification: 35B37, 93B05, 93B0.
Citation: Long Hu, Tatsien Li, Bopeng Rao. Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type. Communications on Pure & Applied Analysis, 2014, 13 (2) : 881-901. doi: 10.3934/cpaa.2014.13.881
References:
[1]

Long Hu, Fanqiong Ji and Ke Wang, Exact boundary controllability and exact boundary observability for a coupled system of quasilinear wave equations, Chin. Ann. Math., 34B (2013), 479-490. doi: 10.1007/s11401-013-0785-9.  Google Scholar

[2]

Tatsien Li, "Controllability and Observability for Quasilinear Hyperbolic Systems," AIMS Series on Applied Mathematies, Vol. 3, AIMS & Higher Education Press, 2010.  Google Scholar

[3]

Tatsien Li and Bopeng Rao, Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems, Chin. Ann. Math., 31B (2010), 723-742. doi: 10.1007/s11401-010-0600-9.  Google Scholar

[4]

Tatsien Li and Bopeng Rao, Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls, Chin. Ann. Math., 34B (2013), 139-160. doi: 10.1007/s11401-012-0754-8.  Google Scholar

[5]

Tatsien Li, Bopeng Rao and Long Hu, Exact boundary synchronization for a coupled system of 1-D wave equations,, To appear in ESAIM:COCV., ().   Google Scholar

[6]

Tatsien Li and Lixin Yu, Exact boundary controllability for 1-D quasilinear wave equations, SIAM J. Control. Optim, 45 (2006), 1074-1083. doi: 10.1137/S0363012903427300.  Google Scholar

[7]

J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués," Vol. 1, Masson, 1988. Google Scholar

[8]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar

[9]

D. L. Russell, Controllability and stabilization theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.  Google Scholar

[10]

Ke Wang, Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems, Chin. Ann. Math., 32B (2011), 803-822. doi: 10.1007/s11401-011-0683-y.  Google Scholar

[11]

Lixin Yu, Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems and its applications, Math. Meth. Appl. Sci., 33 (2010), 273-286. doi: 10.1002/mma.1167.  Google Scholar

show all references

References:
[1]

Long Hu, Fanqiong Ji and Ke Wang, Exact boundary controllability and exact boundary observability for a coupled system of quasilinear wave equations, Chin. Ann. Math., 34B (2013), 479-490. doi: 10.1007/s11401-013-0785-9.  Google Scholar

[2]

Tatsien Li, "Controllability and Observability for Quasilinear Hyperbolic Systems," AIMS Series on Applied Mathematies, Vol. 3, AIMS & Higher Education Press, 2010.  Google Scholar

[3]

Tatsien Li and Bopeng Rao, Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems, Chin. Ann. Math., 31B (2010), 723-742. doi: 10.1007/s11401-010-0600-9.  Google Scholar

[4]

Tatsien Li and Bopeng Rao, Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls, Chin. Ann. Math., 34B (2013), 139-160. doi: 10.1007/s11401-012-0754-8.  Google Scholar

[5]

Tatsien Li, Bopeng Rao and Long Hu, Exact boundary synchronization for a coupled system of 1-D wave equations,, To appear in ESAIM:COCV., ().   Google Scholar

[6]

Tatsien Li and Lixin Yu, Exact boundary controllability for 1-D quasilinear wave equations, SIAM J. Control. Optim, 45 (2006), 1074-1083. doi: 10.1137/S0363012903427300.  Google Scholar

[7]

J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués," Vol. 1, Masson, 1988. Google Scholar

[8]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar

[9]

D. L. Russell, Controllability and stabilization theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.  Google Scholar

[10]

Ke Wang, Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems, Chin. Ann. Math., 32B (2011), 803-822. doi: 10.1007/s11401-011-0683-y.  Google Scholar

[11]

Lixin Yu, Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems and its applications, Math. Meth. Appl. Sci., 33 (2010), 273-286. doi: 10.1002/mma.1167.  Google Scholar

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