doi: 10.3934/eect.2022024
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Stability of a damped wave equation on an infinite star-shaped network

1. 

LR Analyse Non-Linéaire et Géométrie, LR21ES08, Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia

2. 

LR Analysis and Control of PDEs, LR22ES03, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Monastir, Tunisia

* Corresponding author: Ahmed Bchatnia

Received  November 2021 Revised  March 2022 Early access April 2022

In this paper, we study the stability of an infinite star-shaped network of a linear viscous damped wave equation. We prove that, under some conditions, the whole system is asymptotically stable. Moreover we give a decay rate of the energy of the solution. Our technique is based on a frequency domain method.

Citation: Ahmed Bchatnia, Amina Boukhatem. Stability of a damped wave equation on an infinite star-shaped network. Evolution Equations and Control Theory, doi: 10.3934/eect.2022024
References:
[1]

K. AmmariA. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptot. Anal., 28 (2001), 215-240. 

[2]

K. AmmariA. Henrot and M. Tucsnak, Optimal location of the actuator for the pointwise stabilization of a string, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 275-280.  doi: 10.1016/S0764-4442(00)00113-0.

[3]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential Integral Equations, 17 (2004), 1395-1410. 

[4]

K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory., 4 (2015), 1-19.  doi: 10.3934/eect.2015.4.1.

[5]

K. Ammari and S. Nicaise, Stabilization of Elastic Systems by Collocated Feedback, Lecture Notes in Mathematics, 2124, Springer, Cham, 2015. doi: 10.1007/978-3-319-10900-8.

[6]

K. Ammari and F. Shel, Stability of Elastic Multi-Link Structures, SpringerBriefs in Mathematics, Springer, Cham, 2022. doi: 10.1007/978-3-030-86351-7.

[7]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.

[8]

R. Assel and M. Ghazel, Energy decay for the damped wave equation on an unbounded network, Evol. Equ. Control Theory., 7 (2018), 335-351.  doi: 10.3934/eect.2018017.

[9]

J. K. BattyR. Chill and Y. Tomilov, Fine scales of decay of operator semigroups, J. Eur. Math. Soc. (JEMS), 81 (2016), 853-929.  doi: 10.4171/JEMS/605.

[10]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multistructures, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[11]

P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory: With Applications to Schrödinger Operators, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0741-2.

show all references

References:
[1]

K. AmmariA. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptot. Anal., 28 (2001), 215-240. 

[2]

K. AmmariA. Henrot and M. Tucsnak, Optimal location of the actuator for the pointwise stabilization of a string, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 275-280.  doi: 10.1016/S0764-4442(00)00113-0.

[3]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential Integral Equations, 17 (2004), 1395-1410. 

[4]

K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory., 4 (2015), 1-19.  doi: 10.3934/eect.2015.4.1.

[5]

K. Ammari and S. Nicaise, Stabilization of Elastic Systems by Collocated Feedback, Lecture Notes in Mathematics, 2124, Springer, Cham, 2015. doi: 10.1007/978-3-319-10900-8.

[6]

K. Ammari and F. Shel, Stability of Elastic Multi-Link Structures, SpringerBriefs in Mathematics, Springer, Cham, 2022. doi: 10.1007/978-3-030-86351-7.

[7]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.

[8]

R. Assel and M. Ghazel, Energy decay for the damped wave equation on an unbounded network, Evol. Equ. Control Theory., 7 (2018), 335-351.  doi: 10.3934/eect.2018017.

[9]

J. K. BattyR. Chill and Y. Tomilov, Fine scales of decay of operator semigroups, J. Eur. Math. Soc. (JEMS), 81 (2016), 853-929.  doi: 10.4171/JEMS/605.

[10]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multistructures, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[11]

P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory: With Applications to Schrödinger Operators, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0741-2.

Figure 1.  Infinite star-shaped network
[1]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks and Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723

[2]

F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier. Dispersive waves with multiple tunnel effect on a star-shaped network. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 783-791. doi: 10.3934/dcdss.2013.6.783

[3]

Farah Abou Shakra. Asymptotics of wave models for non star-shaped geometries. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 347-362. doi: 10.3934/dcdss.2014.7.347

[4]

Walid Boughamda. Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1103-1125. doi: 10.3934/dcdss.2021139

[5]

Jason Metcalfe, Christopher D. Sogge. Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1589-1601. doi: 10.3934/dcds.2010.28.1589

[6]

Zhong-Jie Han, Enrique Zuazua. Decay rates for elastic-thermoelastic star-shaped networks. Networks and Heterogeneous Media, 2017, 12 (3) : 461-488. doi: 10.3934/nhm.2017020

[7]

Takahiro Hashimoto. Nonexistence of global solutions of nonlinear Schrodinger equations in non star-shaped domains. Conference Publications, 2007, 2007 (Special) : 487-494. doi: 10.3934/proc.2007.2007.487

[8]

Helmut Harbrecht, Thorsten Hohage. A Newton method for reconstructing non star-shaped domains in electrical impedance tomography. Inverse Problems and Imaging, 2009, 3 (2) : 353-371. doi: 10.3934/ipi.2009.3.353

[9]

Byung-Soo Lee. Strong convergence theorems with three-step iteration in star-shaped metric spaces. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 371-379. doi: 10.3934/naco.2011.1.371

[10]

Giuseppe Maria Coclite, Carlotta Donadello. Vanishing viscosity on a star-shaped graph under general transmission conditions at the node. Networks and Heterogeneous Media, 2020, 15 (2) : 197-213. doi: 10.3934/nhm.2020009

[11]

Karim El Mufti, Rania Yahia. Polynomial stability in viscoelastic network of strings. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1421-1438. doi: 10.3934/dcdss.2022073

[12]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861

[13]

Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441

[14]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021

[15]

Chunyan Zhao, Chengkui Zhong, Zhijun Tang. Asymptotic behavior of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022025

[16]

Shuichi Jimbo, Yoshihisa Morita. Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4013-4039. doi: 10.3934/dcds.2021026

[17]

Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network. Evolution Equations and Control Theory, 2020, 9 (3) : 673-692. doi: 10.3934/eect.2020028

[18]

Florian Monteghetti, Ghislain Haine, Denis Matignon. Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions. Mathematical Control and Related Fields, 2019, 9 (4) : 759-791. doi: 10.3934/mcrf.2019049

[19]

Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations and Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017

[20]

Yan Cui, Zhiqiang Wang. Asymptotic stability of wave equations coupled by velocities. Mathematical Control and Related Fields, 2016, 6 (3) : 429-446. doi: 10.3934/mcrf.2016010

2021 Impact Factor: 1.169

Article outline

Figures and Tables

[Back to Top]