Exact boundary synchronization for a coupled system of 1-D quasilinear wave equations

Hu, Long; Li, Tatsien; Qu, Peng

doi:10.1051/cocv/2016035

Hu, Long ^1,^2,³ ; Li, Tatsien ⁴ ; Qu, Peng ²

¹ School of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China.
² School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China.
³ Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
⁴ School of Mathematical Sciences, Fudan University; Shanghai Key Laboratory for Contemporary Applied Mathematics; Nonlinear Mathematical Modeling and Methods Laboratory, Shanghai 200433, P.R. China.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1163-1183.

Résumé

Based on the theory of semi-global classical solutions for quasilinear hyperbolic systems, under suitable hypotheses, an iteration procedure given by a unified constructive method is presented to establish the exact boundary synchronization for a coupled system of 1-D quasilinear wave equations with boundary conditions of various types.

Reçu le : 2016-06-06
Accepté le : 2016-06-07

MR Zbl

DOI : 10.1051/cocv/2016035

Classification : 93B05, 35L04
Mots clés : Exact boundary synchronization, coupled system of quasilinear wave equations

Affiliations des auteurs :

Hu, Long ^{1, 2, 3} ; Li, Tatsien ⁴ ; Qu, Peng ²

@article{COCV_2016__22_4_1163_0,
     author = {Hu, Long and Li, Tatsien and Qu, Peng},
     title = {Exact boundary synchronization for a coupled system of {1-D} quasilinear wave equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1163--1183},
     publisher = {EDP-Sciences},
     volume = {22},
     number = {4},
     year = {2016},
     doi = {10.1051/cocv/2016035},
     zbl = {1350.93040},
     mrnumber = {3570498},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2016035/}
}

TY  - JOUR
AU  - Hu, Long
AU  - Li, Tatsien
AU  - Qu, Peng
TI  - Exact boundary synchronization for a coupled system of 1-D quasilinear wave equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2016
SP  - 1163
EP  - 1183
VL  - 22
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2016035/
DO  - 10.1051/cocv/2016035
LA  - en
ID  - COCV_2016__22_4_1163_0
ER  -

%0 Journal Article
%A Hu, Long
%A Li, Tatsien
%A Qu, Peng
%T Exact boundary synchronization for a coupled system of 1-D quasilinear wave equations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2016
%P 1163-1183
%V 22
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2016035/
%R 10.1051/cocv/2016035
%G en
%F COCV_2016__22_4_1163_0

Hu, Long; Li, Tatsien; Qu, Peng. Exact boundary synchronization for a coupled system of 1-D quasilinear wave equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1163-1183. doi : 10.1051/cocv/2016035. http://www.numdam.org/articles/10.1051/cocv/2016035/

Bibliographie
Cité par

Hirokazu Fujisaka and Tomoji Yamada, Stability theory of synchronized motion in coupled-oscillator systems. Prog. Theor. Phys. 69 (1983) 32–47. | DOI | Zbl

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. 34 (2013) 379–390. | Zbl

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. Commun. Pure Appl. Anal. 13 (2014) 881–901. | Zbl

Ch. Huygens, Œuvres Complètes, Vol. 15. Swets Zeitlinger B.V., Amsterdam (1967).

Tatsien Li, Exact boundary controllability for quasilinear wave equations. J. Comput. Appl. Math. 190 (2006) 127–135. | DOI | Zbl

Tatsien Li, Controllability and Observability for Quasilinear Hyperbolic Systems. Vol. 3 of AIMS Series on Applied Mathematics. AIMS & Higher Education Press (2010). | Zbl

Tatsien Li, Yi Jin, Semi-global $C^{1}$ solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems. Chin. Ann. Math. 22 (2001) 325–336. | DOI | Zbl

Tatsien Li, Bopeng Rao, Local exact boundary controllability for a class of quasilinear hyperbolic systems. Chin. Ann. Math. 23 (2002) 209–218. | DOI | Zbl

Tatsien Li, Bopeng Rao, Exact boundary controllability for quasilinear hyperbolic systems. SIAM J. Control. Optim. 41 (2003) 1748–1755. | DOI | MR | Zbl

Tatsien Li, Bopeng Rao, Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems. Chin. Ann. Math. 31 (2010) 723–742. | DOI | MR | Zbl

Tatsien Li, Bopeng Rao, Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls. Chin. Ann. Math. 34 (2013) 139–160. | DOI | MR | Zbl

Tatsien. Li, Wenci. Yu, Boundary Value Problems for Quasilinear Hyperbolic systems. Duke University Mathematics Series V (1985). | MR | Zbl

Tatsien Li, Lixin Yu, Exact boundary controllability for 1-D quasilinear wave equations. SIAM J. Control. Optim. 45 (2006) 1074–1083. | DOI | MR | Zbl

Tatsien Li, Bopeng Rao, Long Hu, Exact boundary synchronization for a coupled system of 1-D wave equations. ESAIM: COCV 20 (2014) 339–361. | Numdam | MR | Zbl

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués. Vol. I. Masson (1988). | MR | Zbl

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30 (1988) 1–68. | DOI | MR | Zbl

L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems. Phys. Rev. Lett. 64 (1990) 821–824. | DOI | Zbl

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

Zhiqiang Wang, Exact controllability for nonautonomous quasilinear hyperbolic systems. Chin. Ann. Math. 27 (2006) 643–656. | DOI | Zbl

Zhiqiang Wang, Exact boundary controllability for nonautonomous quasilinear wave equations. Math. Methods Appl. Sci. 30 (2007) 1311–1327. | DOI | Zbl

Chai Wah Wu, Synchronizaton in complex networks of nonlinear dynamical systems. World scientific (2007). | Zbl

Cité par Sources :