Abstract
In this paper we study the stability problem of a tree of elastic strings with local Kelvin–Voigt damping on some of the edges. Under the compatibility condition of displacement and strain and continuity condition of damping coefficients at the vertices of the tree, exponential/polynomial stability are proved. Our results generalize the case of single elastic string with local Kelvin–Voigt damping in Liu and Rao (Z. Angew Math Phys 56:630–644, 2005), Liu and Liu (Z. Angew Math Phys 53:265–280, 2002).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alves, M., Revera, J.M., Sepúlveda, M., Villagrán, O.V., Gary, M.Z.: The asymptotic behavior of the linear transmission problem in viscoelasticity. Math. Nachr. 287, 483–497 (2014)
Ammari, K., Nicaise, S.: Stabilization of Elastic Systems by Collocated Feedback. Lecture Notes in Mathematics, vol. 2124. Springer, Cham (2015)
Ammari, K., Mercier, D.: Boundary feedback stabilization of a chain of serially connected strings. Evol. Equ. Control Theory 1, 1–19 (2015)
Ammari, K., Mercier, D., Régnier, V.: Spectral analysis of the Schrödinger operator on binary tree-shaped networks and applications. J. Differ. Equ. 259, 6923–6959 (2015)
Ammari, K., Mercier, D., Régnier, V., Valein, J.: Spectral analysis and stabilization of a chain of serially connected Euler–Bernoulli beams and strings. Commun. Pure Appl. Anal. 11, 785–807 (2012)
Ammari, K., Tucsnak, M.: Stabilization of Bernoulli–Euler beams by means of a pointwise feedback force. SIAM J. Control Optim. 39, 1160–1181 (2000)
Ammari, K., Henrot, A., Tucsnak, M.: Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string. Asymptot. Anal. 28, 215–240 (2001)
Ammari, K., Jellouli, M.: Remark in stabilization of tree-shaped networks of strings. Appl. Maths. 4, 327–343 (2007)
Ammari, K., Tucsnak, M.: Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM Control Optim. Calc. Var. 6, 361–386 (2001)
Ammari, K., Jellouli, M.: Stabilization of star-shaped networks of strings. Differ. Integr. Equ. 17, 1395–1410 (2004)
Ammari, K., Jellouli, M., Khenissi, M.: Stabilization of generic trees of strings. J. Dyn. Contin. Syst. 11, 177–193 (2005)
Arendt, W., Batty, C.J.K.: Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 305, 837–852 (1988)
Banks, H.T., Smith, R.C., Wang, Y.: Smart Materials Structures. Wiley, Hoboken (1996)
Borichev, A., Tomilov, Y.: Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347, 455–478 (2010)
Brezis, H.: Analyse Fonctionnelle. Théorie et Applications, Masson, Paris (1983)
Chen, S., Liu, K., Liu, Z.: Spectrum and stability for elastic systems with global or local Kelvin–Voigt damping. SIAM J. Appl. Math. 59, 651–668 (1999)
Dáger, R., Zuazua, E.: Wave propagation, observation and control in 1-\(d\) flexible multi-structures, volume 50 of Mathématiques & Applications. Springer, Berlin (2006)
Hassine, F.: Stability of elastic transmission systems with a local Kelvin–Voigt damping. Eur. J. Control 23, 84–93 (2015)
Huang, F.: Characteristic conditions for exponential stability of linear dynamical systems in Hilbert space. Ann. Differ. Equ. 1, 43–56 (1985)
Lagnese, J., Leugering, G., Schmidt, E.J.P.G.: Modeling, Analysis of Dynamic Elastic Multi-Link Structures. Birkhäuser, Boston (1994)
Liu, Z., Rao, B.: Frequency domain characterization of rational decay rate for solution of linear evolution equations. Z. Angew. Math. Phys. 56, 630–644 (2005)
Liu, Z., Zhang, Q.: Stability of a string with local Kelvin–Voigt damping and non-smooth coefficient at interface. ESAIM Control Optim. Calc. Var. 23, 443–454 (2017)
Liu, K., Liu, Z.: Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping. SIAM J. Control Optim. 36, 1086–1098 (1998)
Liu, K., Liu, Z.: Exponential decay of energy of vibrating strings with local viscoelasticity. Z. Angew. Math. Phys. 53, 265–280 (2002)
Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems. Chapman & Hall/CRC Research Notes in Mathematics, vol. 398. Chapman & Hall/CRC, Boca Raton (1999)
Liu, K., Liu, Z., Zhang, Q.: Eventual differentiability of a string with local Kelvin–Voigt damping. SIAM J. Control Optim. 54, 1859–1871 (2016)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Prüss, J.: On the spectrum of \(C_0\)-semigroups. Trans. Am. Math. Soc. 248, 847–857 (1984)
Tucsnak, M., Weiss, G.: Observation and Control for Operator Semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Abdelaziz Rhandi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ammari, K., Liu, Z. & Shel, F. Stability of the wave equations on a tree with local Kelvin–Voigt damping. Semigroup Forum 100, 364–382 (2020). https://doi.org/10.1007/s00233-019-10064-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-019-10064-7