In this paper, we introduce a new method for feedback controller design for the complex distributed parameter networks governed by wave equations, which ensures the stability of the closed loop system. This method is based on the uniqueness theory of ordinary differential equations and cutting-edge approach in the graph theory, but it is not a simple extension. As a realization of this idea, we investigate a bush-type wave network. The well-posedness of the closed loop system is obtained via Lax-Milgram's lemma and semigroup theory. The validity of cutting-edge method is proved by spectral analysis approach. In particular, we give a detailed procedure of cutting-edge for the bush-type wave networks. The results show that if we impose feedback controllers, consisting of velocity and position terms, at all the boundary vertices and at most three velocity feedback controllers on the cycle, the system is asymptotically stabilized. Finally, some examples are given.
Mots-clés : Bush-type, wave network, controller design, asymptotic stability, cutting-edge
@article{COCV_2012__18_1_208_0,
author = {Zhang, Yaxuan and Xu, Genqi},
title = {Controller design for bush-type 1-d wave networks},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {208--228},
publisher = {EDP-Sciences},
volume = {18},
number = {1},
year = {2012},
doi = {10.1051/cocv/2010050},
mrnumber = {2887933},
zbl = {1243.35167},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2010050/}
}
TY - JOUR AU - Zhang, Yaxuan AU - Xu, Genqi TI - Controller design for bush-type 1-d wave networks JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 208 EP - 228 VL - 18 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010050/ DO - 10.1051/cocv/2010050 LA - en ID - COCV_2012__18_1_208_0 ER -
%0 Journal Article %A Zhang, Yaxuan %A Xu, Genqi %T Controller design for bush-type 1-d wave networks %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 208-228 %V 18 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010050/ %R 10.1051/cocv/2010050 %G en %F COCV_2012__18_1_208_0
Zhang, Yaxuan; Xu, Genqi. Controller design for bush-type 1-d wave networks. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 208-228. doi : 10.1051/cocv/2010050. http://www.numdam.org/articles/10.1051/cocv/2010050/
[1] and , Stabilization of star-shaped tree of elastic strings. Differential Integral Equations 17 (2004) 1395-1410. | MR | Zbl
[2] and , Remark on stabilization of tree-shaped networks of strings. Appl. Math. 52 (2007) 327-343. | MR | Zbl
[3] and , Polynomial and analytic stabilization of a wave equation coupled with a Euler-Bernoulli beam. Math. Methods Appl. Sci. 32 (2009) 556-576. | MR | Zbl
[4] , and , Stabilization of generic trees of strings. J. Dyn. Control Syst. 11 (2005) 177-193. | MR | Zbl
[5] and , Graph Theory, Graduate Texts in Mathematics Series. Springer-Verlag, New York (2008). | MR | Zbl
[6] , Observation and control of vibrations in tree-shaped networks of strings. SIAM J. Control Optim. 43 (2004) 590-623. | MR | Zbl
[7] and , Controllability of star-shaped networks of strings. C. R. Acad. Sci. Paris, Sér. I 332 (2001) 621-626. | MR | Zbl
[8] and , Controllability of tree-shaped networks of vibrating strings. C. R. Acad. Sci. Paris, Sér. I 332 (2001) 1087-1092. | MR | Zbl
[9] and , Wave propagation, observation and control in 1-d flexible multistructures, Mathématiques and Applications 50. Springer-Verlag, Berlin (2006). | Zbl
[10] , Boundary feedback stabilization by time delay for one-dimensional wave equations. IMA J. Math. Control Inform. 27 (2010) 189-204. | MR | Zbl
[11] and , On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback. Nonlinear Anal. 71 (2009) 5961-5978. | MR | Zbl
[12] , Graphs, Networks and Algorithms, Algorithms and Computation in Mathematics 5. Springer-Verlag, New York, third edition (2008). | MR | Zbl
[13] , and , Modeling, analysis and control of dynamic elastic multi-link structures - Systems and control : Foundations and applications. Birkhäuser-Basel (1994). | MR | Zbl
[14] and , On the control of networks of vibrating strings and beams. Proc. of the 28th IEEE Conference on Decision and Control 3 (1989) 2287-2290.
[15] and , On exact controllability of generic trees. ESAIM : Proc. 8 (2000) 95-105. | MR | Zbl
[16] and , Asymptotic stability of linear differential equations in Banach spaces. Studia Math. 88 (1988) 34-37. | MR | Zbl
[17] and , Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Netw. Heterog. Media 2 (2007) 425-479. | MR | Zbl
[18] , Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, Berlin (1983). | MR | Zbl
[19] and , Stabilization of the wave equation on 1-d networks. SIAM J. Control Optim. 48 (2009) 2771-2797. | MR | Zbl
[20] , and , Abstract second order hyperbolic system and applications to controlled network of strings. SIAM J. Control Optim. 47 (2008) 1762-1784. | MR | Zbl
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