# Controller design for bush-type 1-d wave networks∗

ESAIM: Control, Optimisation and Calculus of Variations (2012)

- Volume: 18, Issue: 1, page 208-228
- ISSN: 1292-8119

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topZhang, Yaxuan, and Xu, Genqi. "Controller design for bush-type 1-d wave networks∗." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 208-228. <http://eudml.org/doc/277823>.

@article{Zhang2012,

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. },

author = {Zhang, Yaxuan, Xu, Genqi},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Bush-type; wave network; controller design; asymptotic stability; cutting-edge; number and locations of the controllers},

language = {eng},

month = {2},

number = {1},

pages = {208-228},

publisher = {EDP Sciences},

title = {Controller design for bush-type 1-d wave networks∗},

url = {http://eudml.org/doc/277823},

volume = {18},

year = {2012},

}

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

DA - 2012/2//

PB - EDP Sciences

VL - 18

IS - 1

SP - 208

EP - 228

AB - 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.

LA - eng

KW - Bush-type; wave network; controller design; asymptotic stability; cutting-edge; number and locations of the controllers

UR - http://eudml.org/doc/277823

ER -

## References

top- K. Ammari and M. Jellouli, Stabilization of star-shaped tree of elastic strings. Differential Integral Equations17 (2004) 1395–1410.
- K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings. Appl. Math.52 (2007) 327–343.
- K. Ammari and S. Nicaise, Polynomial and analytic stabilization of a wave equation coupled with a Euler-Bernoulli beam. Math. Methods Appl. Sci.32 (2009) 556–576.
- K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings. J. Dyn. Control Syst.11 (2005) 177–193.
- J.A. Bondy and U.S.R. Murty, Graph Theory, Graduate Texts in Mathematics Series. Springer-Verlag, New York (2008).
- R. Dáger, Observation and control of vibrations in tree-shaped networks of strings. SIAM J. Control Optim.43 (2004) 590–623.
- R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings. C. R. Acad. Sci. Paris, Sér. I332 (2001) 621–626.
- R. Dáger and E. Zuazua, Controllability of tree-shaped networks of vibrating strings. C. R. Acad. Sci. Paris, Sér. I332 (2001) 1087–1092.
- R. Dáger and E. Zuazua, Wave propagation, observation and control in 1-d flexible multistructures, Mathématiques and Applications50. Springer-Verlag, Berlin (2006).
- M. Gugat, Boundary feedback stabilization by time delay for one-dimensional wave equations. IMA J. Math. Control Inform.27 (2010) 189–204.
- B.Z. Guo and Z.C. Shao, On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback. Nonlinear Anal.71 (2009) 5961–5978.
- D. Jungnickel, Graphs, Networks and Algorithms, Algorithms and Computation in Mathematics5. 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).
- G. Leugering and E.J.P.G. Schmidt, On the control of networks of vibrating strings and beams. Proc. of the 28th IEEE Conference on Decision and Control3 (1989) 2287–2290.
- G. Leugering and E. Zuazua, On exact controllability of generic trees. ESAIM : Proc.8 (2000) 95–105.
- Yu.I. Lyubich and V.Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces. Studia Math.88 (1988) 34–37.
- S. Nicaise and J. Valein, Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Netw. Heterog. Media2 (2007) 425-479.
- A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, Berlin (1983).
- J. Valein and E. Zuazua, Stabilization of the wave equation on 1-d networks. SIAM J. Control Optim.48 (2009) 2771–2797.
- G.Q. Xu, D.Y. Liu and Y.Q. Liu, Abstract second order hyperbolic system and applications to controlled network of strings. SIAM J. Control Optim.47 (2008) 1762–1784.

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