The stability of general tree-shaped wave networks with variable coefficients under boundary feedback controls is considered. Making full use of the tree-shaped structures, we present a detailed asymptotic spectral analysis of the networks. By proposing the from-root-to-leaf calculating technique, we deduce an explicit recursive expression for the asymptotic characteristic equation and the spectral properties are further obtained. We show that the spectrum-determined-growth (SDG) condition holds. Thus the stability analysis of the closed-loop system can be completely converted to the infimum estimation of the asymptotic characteristic equation. Especially, we further show that the infimum is positive so as to obtain the exponential stability by estimating the recursive expression in from-leaf-to-root order. Some numerical simulations are presented to illustrate and support the theoretical results.
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Lagnese, J., Leugering, G., Schmidt, E.J.P.G.: Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Systems and Control: Foundations and Applications. Birkhäuser, Boston (1994)
Huang, F.L.: Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1, 43–56 (1985)
Dager, R., Zuazua, E.: Wave Propagation, Observation and Control in 1-D Flexible Multi-Structures. Springer, Berlin (2006)
Komornik, V.: Exact Controllability and Stabilization: The Multiplier Method. Wiley/Masson, Chichester/Paris (1994)
Valein, J., Zuazua, E.: Stabilization of the wave equation on 1-d networks. SIAM J. Control Optim. 48(4), 2771–2797 (2009)
Yao, P.F.: Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation. Chin. Ann. Math., Ser. B 31(1), 59–70 (2010)
Guo, B.Z., Xie, Y.: A sufficient condition on Riesz basis with parentheses of non-self-adjoint operator and application to a serially connected string system under joint feedbacks. SIAM J. Control Optim. 43(4), 1234–1252 (2004)
Xu, G.Q., Han, Z.J., Yung, S.P.: Riesz basis property of serially connected Timoshenko beams. Int. J. Control 80(3), 470–485 (2007)
Dáger, R., Zuazua, E.: Controllability of star-shaped networks of strings. C. R. Acad. Sci., Sér. 1 Math. 332(7), 621–626 (2001)
Dáger, R., Zuazua, E.: Controllability of tree-shaped networks of vibrating strings. C. R. Acad. Sci., Sér. 1 Math. 332(12), 1087–1092 (2001)
Dáger, R., Zuazua, E.: Spectral boundary controllability of networks of strings. C. R. Math. Acad. Sci. Paris 334(7), 545–550 (2002)
Dager, R.: Observation and control of vibrations in tree-shaped networks of strings. SIAM J. Control Optim. 43(2), 590–623 (2004)
Ammari, K., Jellouli, M.: Stabilization of star-shaped networks of strings. Differ. Integral Equ. 17(11), 1395–1410 (2004)
Ammari, K., Jellouli, M., Khenissi, M.: Stabilization of generic trees of strings. J. Dyn. Control Syst. 11(2), 177–193 (2005)
Ammari, K.: Asymptotic behavior of some elastic planar networks of Bernoulli-Euler beams. Appl. Anal. 86(12), 1529–1548 (2007)
Han, Z.J., Xu, G.Q.: Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks. Netw. Heterog. Media 5(2), 315–334 (2010)
Han, Z.J., Xu, G.Q.: Output feedback stabilisation of a tree-shaped network of vibrating strings with non-collocated observation. Int. J. Control 84(3), 458–475 (2011)
Han, Z.J., Wang, L.: Riesz basis property and stability of planar networks of controlled strings. Acta Appl. Math. 110, 511–533 (2010)
Guo, Y.N., Chen, Y.L., Xu, G.Q., Zhang, Y.X.: Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks. J. Math. Anal. Appl. 395, 727–746 (2012)
Xu, G.Q., Liu, D.Y., Liu, Y.Q.: Abstract second order hyperbolic system and applications to controlled network of strings. SIAM J. Control Optim. 47(4), 1762–1784 (2008)
Jellouli, M.: Spectral analysis for a degenerate tree and applications. Int. J. Control 88(8), 1647–1662 (2015)
Han, Z.J., Zuazua, E.: Decay rates for 1-d heat-wave planar networks. Netw. Heterog. Media 11(4), 655–692 (2016)
Alabau-Boussouira, F., Perrollaz, V., Rosier, L.: Finite-time stabilization of a network of strings. Math. Control Relat. Fields 5(4), 721–742 (2015)
Zhang, Y.X., Xu, G.Q.: Exponential and super stability of a wave network. Acta Appl. Math. 124(1), 19–41 (2013)
Nicaise, S., Valein, J.: Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. Netw. Heterog. Media 2, 425–479 (2007)
Han, Z.J., Xu, G.Q.: Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Netw. Heterog. Media 6, 297–327 (2011)
Mercier, D., Régnier, V.: Spectrum of a network of Euler-Bernoulli beams. J. Math. Anal. Appl. 337, 174–196 (2008)
Ammari, K., Mercier, D., Régnier, V.: Spectral analysis of the Schrodinger operator on binary tree-shaped networks and applications. J. Differ. Equ. 259(12), 6923–6959 (2015)
Chitour, Y., Mazanti, G., Sigalotti, M.: Stability of non-autonomous difference equations with applications to transport and wave propagation on networks. Netw. Heterog. Media 11(4), 563–601 (2016)
Wang, J.M., Guo, B.Z.: Riesz basis and stabilization for the flexible structure of a symmetric tree-shaped beam network. Math. Methods Appl. Sci. 31, 289–314 (2008)
von Below, J.: Classical solvability of linear parabolic equations on networks. J. Differ. Equ. 72, 316–337 (1988)
Naimark, M.A.: Linear Differential Operators. Ungar, New York (1967)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)
Dunford, N., Schwartz, J.T.: Linear Operators, Part III, Spectral Operators. Interscience, New York (1971)
Avdonin, S.A., Ivanov, S.A.: Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge University Press, Cambridge (1995)
Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980)
Lyubich, Y.L., Phóng, V.Q.: Asymptotic stability of linear differential equations in Banach spaces. Stud. Math. 88, 34–37 (1988)
Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)
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This research is supported by the Natural Science Foundation of China grant NSFC-61503385, 61573252 and 11705279.
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Zhang, Y., Han, Z. & Xu, G. Stability and Spectral Properties of General Tree-Shaped Wave Networks with Variable Coefficients. Acta Appl Math 164, 219–249 (2019). https://doi.org/10.1007/s10440-018-00236-y
- Wave network
- Variable coefficient
- Exponential stability
- Feedback control
- Recursive characteristic equation
- Spectrum-determined-growth (SDG) condition
Mathematics Subject Classification (2000)