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Controller design for bush-type 1-d wave networks

Published online by Cambridge University Press:  02 December 2010

Yaxuan Zhang
Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, P.R. China. bunnyxuan@tju.edu.cn, gqxu@tju.edu.cn
Genqi Xu
Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, P.R. China. bunnyxuan@tju.edu.cn, gqxu@tju.edu.cn
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Abstract

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2010

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References

Ammari, K. and Jellouli, M., Stabilization of star-shaped tree of elastic strings. Differential Integral Equations 17 (2004) 13951410. Google Scholar
Ammari, K. and Jellouli, M., Remark on stabilization of tree-shaped networks of strings. Appl. Math. 52 (2007) 327343. Google Scholar
Ammari, K. and Nicaise, S., Polynomial and analytic stabilization of a wave equation coupled with a Euler-Bernoulli beam. Math. Methods Appl. Sci. 32 (2009) 556576. Google Scholar
Ammari, K., Jellouli, M. and Khenissi, M., Stabilization of generic trees of strings. J. Dyn. Control Syst. 11 (2005) 177193. Google Scholar
J.A. Bondy and U.S.R. Murty, Graph Theory, Graduate Texts in Mathematics Series. Springer-Verlag, New York (2008).
Dáger, R., Observation and control of vibrations in tree-shaped networks of strings. SIAM J. Control Optim. 43 (2004) 590623. Google Scholar
Dáger, R. and Zuazua, E., Controllability of star-shaped networks of strings. C. R. Acad. Sci. Paris, Sér. I 332 (2001) 621626. Google Scholar
Dáger, R. and Zuazua, E., Controllability of tree-shaped networks of vibrating strings. C. R. Acad. Sci. Paris, Sér. I 332 (2001) 10871092. Google Scholar
R. Dáger and E. Zuazua, Wave propagation, observation and control in 1-d flexible multistructures, Mathématiques and Applications 50. Springer-Verlag, Berlin (2006).
Gugat, M., Boundary feedback stabilization by time delay for one-dimensional wave equations. IMA J. Math. Control Inform. 27 (2010) 189204. Google Scholar
Guo, B.Z. and Shao, Z.C., On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback. Nonlinear Anal. 71 (2009) 59615978. Google Scholar
D. Jungnickel, Graphs, Networks and Algorithms, Algorithms and Computation in Mathematics 5. Springer-Verlag, New York, third edition (2008).
J.E. Lagnese, G. Leugering and E.J.P.G. Schmidt, Modeling, analysis and control of dynamic elastic multi-link structures – Systems and control : Foundations and applications. Birkhäuser-Basel (1994).
Leugering, G. and Schmidt, E.J.P.G., On the control of networks of vibrating strings and beams. Proc. of the 28th IEEE Conference on Decision and Control 3 (1989) 22872290. Google Scholar
Leugering, G. and Zuazua, E., On exact controllability of generic trees. ESAIM : Proc. 8 (2000) 95105. Google Scholar
Lyubich, Yu.I. and Phóng, V.Q., Asymptotic stability of linear differential equations in Banach spaces. Studia Math. 88 (1988) 3437. Google Scholar
Nicaise, S. and Valein, J., Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Netw. Heterog. Media 2 (2007) 425-479. Google Scholar
A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, Berlin (1983).
Valein, J. and Zuazua, E., Stabilization of the wave equation on 1-d networks. SIAM J. Control Optim. 48 (2009) 27712797. Google Scholar
Xu, G.Q., Liu, D.Y. and Liu, Y.Q., Abstract second order hyperbolic system and applications to controlled network of strings. SIAM J. Control Optim. 47 (2008) 17621784. Google Scholar

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