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EXPONENTIAL STABILISATION OF A TREE-SHAPED NETWORK OF STRINGS WITH VARIABLE COEFFICIENTS*

Published online by Cambridge University Press:  10 March 2011

YAN NI GUO and
GEN QI XU
YAN NI GUO
Affiliation:
Institute of Applied Mathematics, College of Science, Civil Aviation University of China, Tianjin 300300, P.R. China e-mail: gyann@126.com
GEN QI XU
Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, P.R. China e-mail: gqxu@tju.edu.cn
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Abstract

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In this paper, we deal with a tree-shaped network of strings with a fixed root node. By imposing velocity feedback controllers on all vertices except the root node, we show that the spectrum of the system operator consists of all isolated eigenvalues of finite multiplicity and is distributed in a strip parallel to the imaginary axis under certain conditions. Moreover, we prove that there exists a sequence of eigenvectors and generalised eigenvectors that forms a Riesz basis with parentheses, and that the imaginary axis is not an asymptote of the spectrum. Thereby, we deduce that the system is exponentially stable.

Keywords

93C2093D1535B3535P1047A70

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011, pp. 481 - 499
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Ammari, K. and Jellouli, M., Stabilization of star-shaped tree of elastic strings, Differ. Int. Equ. 17 (2004), 1395–1410.Google Scholar
2.Ammari, K., Jellouli, M. and Khenissi, M., Stabilization of generic trees of strings, J. Dyn. Control Syst. 11 (2) (2005), 177–193.CrossRefGoogle Scholar
3.Avdonin, S. A. and Ivanov, S. A., Families of exponentials. The method of moments in controllability problems for distributed parameter systems (Cambridge University Press, Cambridge, UK, 1995).Google Scholar
4.Bondy, J. A. and Murty, U. S. R., Graph theory with applications (Macmillan, London, 1976).CrossRefGoogle Scholar
5.Chen, G., Delfour, M. C., Krall, A. M. and Payre, G., Modeling, stabilization and control of serially connected beams, SIAM J. Control Optim. 25 (3) (1987), 526–546.CrossRefGoogle Scholar
6.Dunford, N. and Schwartz, J. T., Linear operators, part III, spectral operators (Wiley-interscience, New York, 1971).Google Scholar
7.Dager, R. and Zuazua, E., Wave propagation, observation and control in 1-d flexible multi-structures, Mathematiques & Applications, vol. 50 (Springer-Verlag, Berlin, New York, 2006).CrossRefGoogle Scholar
8.Lagnese, J. E., Leugering, G. and Schmit, E. J. P. G., Modeling, analysis and control of dynamic elastic multi-link structures (Birkhaüser, Basel, 1994).CrossRefGoogle Scholar
9.Leugering, G. and Zuazua, E., Exact controllability of generic trees, in Control of systems governed by partial differential equations (Nancy, France, March 1999). ESAIM Proceeding.Google Scholar
10.Liu, K. S., Huang, F. L. and Chen, G., Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage, SIAM J. Appl. Math. 49 (1989), 1694–1707.CrossRefGoogle Scholar
11.Naimark, M. A., Linear differential operators (Frederick Ungar, New York, 1967).Google Scholar
12.Pazy, A., Semigroups of linear operators and applications to partial differential equations (Springer-Verlag, New York, 1983).CrossRefGoogle Scholar
13.Markus, A. S., Introduction to the spectral theory of polynormial pencils, AMS Translation of Mathematical Monographs, vol. 71 (American Mathematical Society, Providence, RI, 1988), 25–27.Google Scholar
14.Keldysh, M. V., On the completeness of the eigenfunctions for certain classes of nonselfadjoint linear operatrors, Uspelhi Mta. Nauk, 27 (4) (1971), (160), 15–41; English transl. in Russian Math. Serveys 27 (1971).Google Scholar
15.Shkalikov, A. A., Boundry problems for ordinary differential equations with parameter in the boundary conditions, J. Math. Sci. 33 (6) (1986), 1311–1342.CrossRefGoogle Scholar
16.Gohberg, I. C. and Krein, M. G., Introduction to the theory of linear nonselfadjoint operators, AMS of Translation Mathematical Monographs, vol. 18 (American Mathematical Society, Providence, RI, 1969).Google Scholar
17.Shubov, M. A., Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equations of nonhomogeneous damped string, Integr. Equ. Oper. Theor. 25 (1996), 289–328.CrossRefGoogle Scholar
18.Wang, J. M. and Guo, B. Z., Rasis basis and stabilization for the flexible structure of a symmetric tree-shaped beam networks, Math. Meth. Appl. Sci. 31 (2008), 289–314.CrossRefGoogle Scholar
19.Xu, G. Q. and Guo, B. Z., Riesz basis property of evolution equations in Hilbert space and application to a coupled string equation, SIAM Control Optim. 42 (3) (2003), 966–984.CrossRefGoogle Scholar
20.Xu, G. Q., Han, Z. J. and Yung, P., Riesz basis property of serially connected Timoshenko beams, Int. J. Control 80 (3) (2007), 470–485.CrossRefGoogle Scholar
21.Xu, G. Q. and Yung, S. P., The expansion of semigroup and criterion of Riesz basis, J. Differ. Equ. 210 (2005), 1–24.CrossRefGoogle Scholar
22.Xu, G. Q., Liu, D. Y. and Liu, Y. Q., Abstract second order hyperbolic system and applications, SIAM J. Control Optim. 47 (4) (2008), 1762–1784.CrossRefGoogle Scholar
23.Xu, G. Q. and Yung, S. P., Stability and Riesz basis property of a star-shaped network of Euler-Berboulli beams with joint damping, Netw. Heterogeneous Media. 3 (4) (2008), 723–747.Google Scholar