American Institute of Mathematical Sciences

Advanced
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 - A, 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, (2003).   Google Scholar

[2]

R. A. Horn and C. R. Johnson, Matrix Analysis,, 2nd Edition, (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.  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).   Google Scholar

[5]

Tatsien Li, Controllability and Observability for Quasilinear Hyperbolic Systems,, AIMS Series on Applied Mathematics, (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.  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.  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, (1988).   Google Scholar

[11]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Review, 30 (1988), 1.  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.  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, (2003).   Google Scholar

[2]

R. A. Horn and C. R. Johnson, Matrix Analysis,, 2nd Edition, (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.  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).   Google Scholar

[5]

Tatsien Li, Controllability and Observability for Quasilinear Hyperbolic Systems,, AIMS Series on Applied Mathematics, (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.  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.  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, (1988).   Google Scholar

[11]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Review, 30 (1988), 1.  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.  doi: 10.1137/1020095.  Google Scholar

[1]

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
[2]

Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665
[3]

Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020
[4]

Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014
[5]

Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367
[6]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014
[7]

Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control & Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019
[8]

Ning-An Lai, Jinglei Zhao. Potential well and exact boundary controllability for radial semilinear wave equations on Schwarzschild spacetime. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1317-1325. doi: 10.3934/cpaa.2014.13.1317
[9]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325
[10]

Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419
[11]

Scott W. Hansen, Oleg Yu Imanuvilov. Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions. Mathematical Control & Related Fields, 2011, 1 (2) : 189-230. doi: 10.3934/mcrf.2011.1.189
[12]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. A note on the one-side exact boundary controllability for quasilinear hyperbolic systems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 405-418. doi: 10.3934/cpaa.2009.8.405
[13]

S. S. Krigman. Exact boundary controllability of Maxwell's equations with weak conductivity in the heterogeneous medium inside a general domain. Conference Publications, 2007, 2007 (Special) : 590-601. doi: 10.3934/proc.2007.2007.590
[14]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243
[15]

Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743
[16]

Enrique Fernández-Cara, Manuel González-Burgos, Luz de Teresa. Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 639-658. doi: 10.3934/cpaa.2006.5.639
[17]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020
[18]

Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020052
[19]

Abdelaziz Bennour, Farid Ammar Khodja, Djamel Teniou. Exact and approximate controllability of coupled one-dimensional hyperbolic equations. Evolution Equations & Control Theory, 2017, 6 (4) : 487-516. doi: 10.3934/eect.2017025
[20]

Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences