Exact boundary synchronization for a coupled system of 1-D wave equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 2, p. 339-361

Several kinds of exact synchronizations and the generalized exact synchronization are introduced for a coupled system of 1-D wave equations with various boundary conditions and we show that these synchronizations can be realized by means of some boundary controls.

DOI : https://doi.org/10.1051/cocv/2013066
Classification:  35B37,  93B05,  93B07
Keywords: exact null controllability, exact synchronization, exact synchronization by groups, exact null controllability and synchronization by groups, generalized exact synchronization
@article{COCV_2014__20_2_339_0,
author = {Li, Tatsien and Rao, Bopeng and Hu, Long},
title = {Exact boundary synchronization for a coupled system of 1-D wave equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {20},
number = {2},
year = {2014},
pages = {339-361},
doi = {10.1051/cocv/2013066},
mrnumber = {3264207},
language = {en},
url = {http://www.numdam.org/item/COCV_2014__20_2_339_0}
}

Li, Tatsien; Rao, Bopeng; Hu, Long. Exact boundary synchronization for a coupled system of 1-D wave equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 2, pp. 339-361. doi : 10.1051/cocv/2013066. http://www.numdam.org/item/COCV_2014__20_2_339_0/

[1] F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems. SIAM J. Control Optim. 41 (2002) 511-541. | MR 1920269 | Zbl 1031.35023

[2] F. Alabau-Boussouira, A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems. SIAM J. Control Optim. 42 (2003) 871-906. | MR 2002139 | Zbl 1125.93311

[3] F. Alabau-Boussouira, M. Léautaud, Indirect stabilization of locally coupled wave-type systems. ESAIM: COCV 18 (2012) 548-582. | Numdam | MR 2954638 | Zbl 1259.35034

[4] H. Fujisaka and T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems. Progress Theoret. Phys. 69 (1983) 32-47. | MR 699754 | Zbl 1171.70306

[5] M. Gugat, Optimal boundary control in flood management, Control of Coupled Partial Differential Equations, edited by K. Kunisch, J. Sprekels, G. Leugering and F. Tröltzsch, vol. 155 of Int. Ser. Numer. Math., Birkhäuser Verlag, Basel/Switzerland (2007) 69-94. | MR 2328602 | Zbl 1239.49030

[6] 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. B 34 (2013) 479-490. | MR 3072243 | Zbl 1278.35152

[7] Ch. Huygens, Œuvres Complètes, vol. 15, edited by S. and B.V. Zeitlinger, Amsterdam (1967).

[8] Tatsien Li and Yi Jin, Semi-global C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems. Chin. Ann. Math. B 22 (2001) 325-336. | MR 1845753 | Zbl 1005.35058

[9] Tatsien Li, Exact boundary observability for 1-D quasilinear wave equations. Math. Meth. Appl. Sci. 29 (2006) 1543-1553. | MR 2249577 | Zbl 1100.35017

[10] Tatsien Li, Controllability and Observability for Quasilinear Hyperbolic Systems, vol. 3 of AIMS Ser. Appl. Math. AIMS and Higher Education Press (2010). | MR 2655971 | Zbl 1198.93003

[11] Tatsien Li and Bopeng Rao, Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems. Chin. Annal. Math. B 31 (2010) 723-742. | MR 2726064 | Zbl 1206.35168

[12] Tatsien Li and Bopeng Rao, Asymptotic controllability for linear hyperbolic systems. Asymp. Anal. 72 (2011) 169-187. | MR 2920605 | Zbl 1244.35083

[13] Tatsien Li and Bopeng Rao, Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls. Chin. Annal. Math. B 34 (2013) 139-160. | MR 3011463 | Zbl 1262.35155

[14] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués, Vol. 1, Masson (1988). | MR 963060 | Zbl 0653.93002

[15] J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems. SIAM Review 30 (1988) 1-68. | MR 931277 | Zbl 0644.49028

[16] D.L. Russell, Controllability and stabilization theory for linear partial differential equations: Recent progress and open questions. SIAM Review 20 (1978) 639-739. | MR 508380 | Zbl 0397.93001

[17] S. Strogatz, SYNC: The Emerging Science of Spontaneous Order, THEIA, New York (2003).

[18] Ke Wang, Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems. Chin. Ann. Math. B 32 (2011) 803-822. | MR 2852304 | Zbl 1270.35301

[19] Chai Wah Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems. World Scientific (2007). | Zbl 1135.34002

[20] 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. | MR 2603496 | Zbl 1186.35109