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

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.
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.  doi: 10.1007/s11401-013-0785-9.  Google Scholar

[2]

Tatsien Li, "Controllability and Observability for Quasilinear Hyperbolic Systems,", AIMS Series on Applied Mathematies, (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.  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.  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.  doi: 10.1137/S0363012903427300.  Google Scholar

[7]

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

[8]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1.  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.  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.  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.  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.  doi: 10.1007/s11401-013-0785-9.  Google Scholar

[2]

Tatsien Li, "Controllability and Observability for Quasilinear Hyperbolic Systems,", AIMS Series on Applied Mathematies, (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.  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.  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.  doi: 10.1137/S0363012903427300.  Google Scholar

[7]

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

[8]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1.  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.  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.  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.  doi: 10.1002/mma.1167.  Google Scholar

[1]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933

[2]

Gelasio Salaza, Edgardo Ugalde, Jesús Urías. Master--slave synchronization of affine cellular automaton pairs. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 491-502. doi: 10.3934/dcds.2005.13.491

[3]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020027

[4]

Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201

[5]

Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141

[6]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379

[7]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[8]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[9]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027

[10]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185

[11]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[12]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

[13]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[14]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363

[15]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623

[16]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91

[17]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[18]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[19]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[20]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]