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July  2014, 34(7): 2893-2905. doi: 10.3934/dcds.2014.34.2893

Generalized exact boundary synchronization for a coupled system of wave equations

Tatsien Li 1, , Bopeng Rao 2, and  Yimin Wei 3,

1. 

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

Institut de Recherche Mathématique Avancée, Université de Strasbourg, 67084 Strasbourg
3. 

School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai 200433

Received  August 2013 Revised  November 2013 Published  December 2013

By means of Moore-Penrose generalized inverse, a general framework is presented to treat the generalized exact boundary synchronization for a coupled systems of wave equations.
Keywords: Moore-Penrose generalized inverse., Coupled system of wave equations, exact boundary synchronization by groups, exact boundary null controllability and synchronization by groups, exact boundary synchronization, exact boundary null controllability, generalized exact boundary synchronization.
Mathematics Subject Classification: 93B05, 93B0.
Citation: Tatsien Li, Bopeng Rao, Yimin Wei. Generalized exact boundary synchronization for a coupled system of wave equations. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2893-2905. doi: 10.3934/dcds.2014.34.2893
References:
[1]

A. Ben-Israel and T. N. E. Greville, Generalized Inverses. Theory and Applications, 2nd Edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 15, Springer-Verlag, New York, 2003.  Google Scholar

[2]

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd Edition, Cambridge University Press, Cambridge, 2013.  Google Scholar

[3]

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. Ser. B, 34 (2013), 479-490. doi: 10.1007/s11401-013-0785-9.  Google Scholar

[4]

Long Hu, Tatsien Li and 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, 13 (2014). DOI: 10.3934/cpaa.2014.13 Google Scholar

[5]

Tatsien Li, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Series on Applied Mathematics, 3, American Institute of Mathematical Sciences (AIMS), Springfield, MO; Higher Education Press, Beijing, 2010.  Google Scholar

[6]

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

[7]

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

[8]

Tatsien Li and Bopeng Rao, A note on the exact synchronization by groups for a coupled system of wave equations,, to appear in Math. Meth. Appl. Sci., ().   Google Scholar

[9]

Tatsien Li, Bopeng Rao and Long Hu, Exact boundary synchronization for a coupled system of 1-D wave equations,, to appear in ESAIM: COCV. DOI: 10.1051/COCV/2013066, ().   Google Scholar

[10]

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

[11]

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

[12]

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

show all references

References:
[1]

A. Ben-Israel and T. N. E. Greville, Generalized Inverses. Theory and Applications, 2nd Edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 15, Springer-Verlag, New York, 2003.  Google Scholar

[2]

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd Edition, Cambridge University Press, Cambridge, 2013.  Google Scholar

[3]

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. Ser. B, 34 (2013), 479-490. doi: 10.1007/s11401-013-0785-9.  Google Scholar

[4]

Long Hu, Tatsien Li and 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, 13 (2014). DOI: 10.3934/cpaa.2014.13 Google Scholar

[5]

Tatsien Li, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Series on Applied Mathematics, 3, American Institute of Mathematical Sciences (AIMS), Springfield, MO; Higher Education Press, Beijing, 2010.  Google Scholar

[6]

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

[7]

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

[8]

Tatsien Li and Bopeng Rao, A note on the exact synchronization by groups for a coupled system of wave equations,, to appear in Math. Meth. Appl. Sci., ().   Google Scholar

[9]

Tatsien Li, Bopeng Rao and Long Hu, Exact boundary synchronization for a coupled system of 1-D wave equations,, to appear in ESAIM: COCV. DOI: 10.1051/COCV/2013066, ().   Google Scholar

[10]

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

[11]

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

[12]

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

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