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

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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

Abstract
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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 and 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.
[2]	Tatsien Li, "Controllability and Observability for Quasilinear Hyperbolic Systems," AIMS Series on Applied Mathematies, Vol. 3, AIMS & Higher Education Press, 2010.
[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.
[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.
[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., ().
[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.
[7]	J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués," Vol. 1, Masson, 1988.
[8]	J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.
[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.
[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.
[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.

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.
[2]	Tatsien Li, "Controllability and Observability for Quasilinear Hyperbolic Systems," AIMS Series on Applied Mathematies, Vol. 3, AIMS & Higher Education Press, 2010.
[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.
[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.
[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., ().
[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.
[7]	J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués," Vol. 1, Masson, 1988.
[8]	J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.
[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.
[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.
[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.

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