Typesetting math: 100%
Search Advanced

Networks and Heterogeneous Media

2016, Volume 11, Issue 4: 563-601. doi: 10.3934/nhm.2016010
Previous Article Next Article

Stability of non-autonomous difference equations with applications to transport and wave propagation on networks

  • 1.
    Laboratoire des systèmes et signaux, Université Paris-Sud, CNRS, Supélec, 91192, Gif-sur-Yvette
  • 2.
    Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay
  • 3.
    Inria, team GECO & CMAP, École Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau cedex
  • Received: 01 September 2015 Revised: 01 March 2016

  • 39A30, 39A60, 35R02, 35B35, 39A06.

  • In this paper, we address the stability of transport systems and wave propagation on networks with time-varying parameters. We do so by reformulating these systems as non-autonomous difference equations and by providing a suitable representation of their solutions in terms of their initial conditions and some time-dependent matrix coefficients. This enables us to characterize the asymptotic behavior of solutions in terms of such coefficients. In the case of difference equations with arbitrary switching, we obtain a delay-independent generalization of the well-known criterion for autonomous systems due to Hale and Silkowski. As a consequence, we show that exponential stability of transport systems and wave propagation on networks is robust with respect to variations of the lengths of the edges of the network preserving their rational dependence structure. This leads to our main result: the wave equation on a network with arbitrarily switching damping at external vertices is exponentially stable if and only if the network is a tree and the damping is bounded away from zero at all external vertices but at most one.

    Citation: Yacine Chitour, Guilherme Mazanti, Mario Sigalotti. Stability of non-autonomous difference equations with applications to transport and wave propagation on networks[J]. Networks and Heterogeneous Media, 2016, 11(4): 563-601. doi: 10.3934/nhm.2016010

    Related Papers:

    [1] Yacine Chitour, Guilherme Mazanti, Mario Sigalotti . Stability of non-autonomous difference equations with applications to transport and wave propagation on networks. Networks and Heterogeneous Media, 2016, 11(4): 563-601. doi: 10.3934/nhm.2016010
    [2] Yaru Xie, Genqi Xu . The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. Networks and Heterogeneous Media, 2016, 11(3): 527-543. doi: 10.3934/nhm.2016008
    [3] Zhong-Jie Han, Enrique Zuazua . Decay rates for heat-wave planar networks. Networks and Heterogeneous Media, 2016, 11(4): 655-692. doi: 10.3934/nhm.2016013
    [4] Tong Li . Qualitative analysis of some PDE models of traffic flow. Networks and Heterogeneous Media, 2013, 8(3): 773-781. doi: 10.3934/nhm.2013.8.773
    [5] Martin Gugat, Mario Sigalotti . Stars of vibrating strings: Switching boundary feedback stabilization. Networks and Heterogeneous Media, 2010, 5(2): 299-314. doi: 10.3934/nhm.2010.5.299
    [6] Serge Nicaise, Julie Valein . Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Networks and Heterogeneous Media, 2007, 2(3): 425-479. doi: 10.3934/nhm.2007.2.425
    [7] Hyeontae Jo, Hwijae Son, Hyung Ju Hwang, Eun Heui Kim . Deep neural network approach to forward-inverse problems. Networks and Heterogeneous Media, 2020, 15(2): 247-259. doi: 10.3934/nhm.2020011
    [8] Victor A. Eremeyev . Anti-plane interfacial waves in a square lattice. Networks and Heterogeneous Media, 2025, 20(1): 52-64. doi: 10.3934/nhm.2025004
    [9] Fatih Bayazit, Britta Dorn, Marjeta Kramar Fijavž . Asymptotic periodicity of flows in time-depending networks. Networks and Heterogeneous Media, 2013, 8(4): 843-855. doi: 10.3934/nhm.2013.8.843
    [10] Yinlin Ye, Hongtao Fan, Yajing Li, Ao Huang, Weiheng He . An artificial neural network approach for a class of time-fractional diffusion and diffusion-wave equations. Networks and Heterogeneous Media, 2023, 18(3): 1083-1104. doi: 10.3934/nhm.2023047
  • In this paper, we address the stability of transport systems and wave propagation on networks with time-varying parameters. We do so by reformulating these systems as non-autonomous difference equations and by providing a suitable representation of their solutions in terms of their initial conditions and some time-dependent matrix coefficients. This enables us to characterize the asymptotic behavior of solutions in terms of such coefficients. In the case of difference equations with arbitrary switching, we obtain a delay-independent generalization of the well-known criterion for autonomous systems due to Hale and Silkowski. As a consequence, we show that exponential stability of transport systems and wave propagation on networks is robust with respect to variations of the lengths of the edges of the network preserving their rational dependence structure. This leads to our main result: the wave equation on a network with arbitrarily switching damping at external vertices is exponentially stable if and only if the network is a tree and the damping is bounded away from zero at all external vertices but at most one.


    [1] F. Alabau-Boussouira, V. Perrollaz and L. Rosier, Finite-time stabilization of a network of strings, Math. Control Relat. Fields, 5 (2015), 721-742. doi: 10.3934/mcrf.2015.5.721
    [2] F. A. Mehmeti, J. von Below and S. Nicaise (eds.), Partial Differential Equations on Multistructures, vol. 219 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker Inc., New York, 2001. doi: 10.1201/9780203902196
    [3] S. Amin, F. M. Hante and A. M. Bayen, Exponential stability of switched linear hyperbolic initial-boundary value problems, IEEE Trans. Automat. Control, 57 (2012), 291-301. doi: 10.1109/TAC.2011.2158171
    [4] C. E. Avellar and J. K. Hale, On the zeros of exponential polynomials, J. Math. Anal. Appl., 73 (1980), 434-452. doi: 10.1016/0022-247X(80)90289-9
    [5] G. Bastin, B. Haut, J.-M. Coron and B. D'Andréa-Novel, Lyapunov stability analysis of networks of scalar conservation laws, Netw. Heterog. Media, 2 (2007), 751-759. doi: 10.3934/nhm.2007.2.751
    [6] R. K. Brayton, Bifurcation of periodic solutions in a nonlinear difference-differential equations of neutral type, Quart. Appl. Math., 24 (1966), 215-224.
    [7] A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli, Flows on networks: recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111. doi: 10.4171/EMSS/2
    [8] Trans. Amer. Math. Soc., to appear, arXiv:1406.0731.
    [9] K. L. Cooke and D. W. Krumme, Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations, J. Math. Anal. Appl., 24 (1968), 372-387. doi: 10.1016/0022-247X(68)90038-3
    [10] J.-M. Coron, G. Bastin and B. d'Andréa Novel, Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems, SIAM J. Control Optim., 47 (2008), 1460-1498. doi: 10.1137/070706847
    [11] R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1−d Flexible Multi-structures, vol. 50 of Mathématiques & Applications, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3
    [12] E. Fridman, S. Mondié and B. Saldivar, Bounds on the response of a drilling pipe model, IMA J. Math. Control Inform., 27 (2010), 513-526. doi: 10.1093/imamci/dnq024
    [13] J. J. Green, Uniform Convergence to the Spectral Radius and Some Related Properties in Banach Algebras, PhD thesis, University of Sheffield, 1996.
    [14] M. Gugat, M. Herty, A. Klar, G. Leugering and V. Schleper, Well-posedness of networked hyperbolic systems of balance laws, in Constrained optimization and optimal control for partial differential equations, vol. 160 of Internat. Ser. Numer. Math., Birkhäuser/Springer Basel AG, Basel, 2012, 123-146. doi: 10.1007/978-3-0348-0133-1_7
    [15] M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization, Netw. Heterog. Media, 5 (2010), 299-314. doi: 10.3934/nhm.2010.5.299
    [16] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7
    [17] F. M. Hante, G. Leugering and T. I. Seidman, Modeling and analysis of modal switching in networked transport systems, Appl. Math. Optim., 59 (2009), 275-292. doi: 10.1007/s00245-008-9057-6
    [18] R. Jungers, The Joint Spectral Radius. Theory and Applications, vol. 385 of Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-95980-9
    [19] B. Klöss, The flow approach for waves in networks, Oper. Matrices, 6 (2012), 107-128. doi: 10.7153/oam-06-08
    [20] D. Liberzon, Switching in Systems and Control, 1st edition, Birkhäuser Boston, 2003. doi: 10.1007/978-1-4612-0017-8
    [21] W. Michiels, T. Vyhlídal, P. Zítek, H. Nijmeijer and D. Henrion, Strong stability of neutral equations with an arbitrary delay dependency structure, SIAM J. Control Optim., 48 (2009), 763-786. doi: 10.1137/080724940
    [22] W. L. Miranker, Periodic solutions of the wave equation with a nonlinear interface condition, IBM J. Res. Develop., 5 (1961), 2-24. doi: 10.1147/rd.51.0002
    [23] P. H. A. Ngoc and N. D. Huy, Exponential stability of linear delay difference equations with continuous time, Vietnam Journal of Mathematics, 43 (2015), 195-205. doi: 10.1007/s10013-014-0082-2
    [24] C. Prieur, A. Girard and E. Witrant, Stability of switched linear hyperbolic systems by Lyapunov techniques, IEEE Trans. Automat. Control, 59 (2014), 2196-2202. doi: 10.1109/TAC.2013.2297191
    [25] W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Book Co., New York, 1987.
    [26] E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings, SIAM J. Control Optim., 30 (1992), 229-245. doi: 10.1137/0330015
    [27] M. Slemrod, Nonexistence of oscillations in a nonlinear distributed network, J. Math. Anal. Appl., 36 (1971), 22-40. doi: 10.1016/0022-247X(71)90016-3
    [28] Z. Sun and S. S. Ge, Stability Theory of Switched Dynamical Systems, Communications and Control Engineering Series, Springer, London, 2011. doi: 10.1007/978-0-85729-256-8
    [29] J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim., 48 (2009), 2771-2797. doi: 10.1137/080733590
    [30] E. Zuazua, Control and stabilization of waves on 1-d networks, in Modelling and Optimisation of Flows on Networks, Springer, 2062 (2013), 463-493. doi: 10.1007/978-3-642-32160-3_9

  • This article has been cited by:

    1. Ya-Xuan Zhang, Zhong-Jie Han, Gen-Qi Xu, Stability and Spectral Properties of General Tree-Shaped Wave Networks with Variable Coefficients, 2019, 164, 0167-8019, 219, 10.1007/s10440-018-00236-y
    2. Cristian M. Cazacu, Liviu I. Ignat, Ademir F. Pazoto, Null-Controllability of the Linear Kuramoto--Sivashinsky Equation on Star-Shaped Trees, 2018, 56, 0363-0129, 2921, 10.1137/16M1103348
    3. Yacine Chitour, Guilherme Mazanti, Mario Sigalotti, Approximate and exact controllability of linear difference equations, 2019, 7, 2270-518X, 93, 10.5802/jep.112
    4. Yacine Chitour, Swann Marx, Guilherme Mazanti, G. Buttazzo, E. Casas, L. de Teresa, R. Glowinski, G. Leugering, E. Trélat, X. Zhang, One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates, 2021, 27, 1292-8119, 84, 10.1051/cocv/2021067
    5. Georges Bastin, Jean-Michel Coron, Amaury Hayat, Diffusion and robustness of boundary feedback stabilization of hyperbolic systems, 2023, 35, 0932-4194, 159, 10.1007/s00498-022-00335-0
    6. Laurent Baratchart, Sébastien Fueyo, Jean-Baptiste Pomet, Exponential stability of periodic difference delay systems and 1-D hyperbolic PDEs of conservation laws, 2022, 55, 24058963, 228, 10.1016/j.ifacol.2022.11.362
    7. L. Baratchart, S. Fueyo, G. Lebeau, J.-B. Pomet, Sufficient Stability Conditions for Time-varying Networks of Telegrapher's Equations or Difference-Delay Equations, 2021, 53, 0036-1410, 1831, 10.1137/19M1301795
    8. Guilherme Mazanti, Relative Controllability of Linear Difference Equations, 2017, 55, 0363-0129, 3132, 10.1137/16M1073157
    9. Irina Kmit, Natalya Lyul'ko, Finite Time Stabilization of Nonautonomous First-Order Hyperbolic Systems, 2021, 59, 0363-0129, 3179, 10.1137/20M1343610
    10. SWITCHING SYNCHRONIZED CHAOTIC SYSTEMS APPLIED TO SECURE COMMUNICATION, 2018, 8, 2156-907X, 413, 10.11948/2018.413
    11. Ke Wang, Zhiqiang Wang, Wancong Yao, Boundary feedback stabilization of quasilinear hyperbolic systems with partially dissipative structure, 2020, 146, 01676911, 104815, 10.1016/j.sysconle.2020.104815
    12. Nguyen Khoa Son, Le Trung Hieu, Cao Thanh Tinh, Do Duc Thuan, New criteria for exponential stability of a class of nonlinear continuous-time difference systems with delays, 2022, 0020-7179, 1, 10.1080/00207179.2022.2063191
    13. Mouataz Billah Mesmouli, Abdelouaheb Ardjouni, Hicham Saber, Asymptotic Behavior of Solutions in Nonlinear Neutral System with Two Volterra Terms, 2023, 11, 2227-7390, 2676, 10.3390/math11122676
    14. Yacine Chitour, Sébastien Fueyo, Guilherme Mazanti, Mario Sigalotti, Approximate and exact controllability criteria for linear one-dimensional hyperbolic systems, 2024, 190, 01676911, 105834, 10.1016/j.sysconle.2024.105834
    15. Gonzalo Arias, Swann Marx, Guilherme Mazanti, 2023, Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE, 979-8-3503-0124-3, 1949, 10.1109/CDC49753.2023.10383213
    16. Sébastien Fueyo, Yacine Chitour, A corona theorem for an algebra of Radon measures with an application to exact controllability for linear controlled delayed difference equations, 2024, 362, 1778-3569, 851, 10.5802/crmath.604
    17. Mouataz Billah Mesmouli, Cemil Tunç, Taher S. Hassan, Hasan Nihal Zaidi, Adel A. Attiya, Asymptotic behavior of Levin-Nohel nonlinear difference system with several delays, 2023, 9, 2473-6988, 1831, 10.3934/math.2024089
    18. Laurent Baratchart, Sébastien Fueyo, Jean-Baptiste Pomet, INTEGRAL REPRESENTATION FORMULA FOR LINEAR NONAUTONOMOUS DIFFERENCE-DELAY EQUATIONS, 2024, 36, 0897-3962, 10.1216/jie.2024.36.407
    19. Sébastien Fueyo, Left-coprimeness condition for the reachability in finite time of pseudo-rational systems of order zero with an application to difference delay systems, 2025, 198, 01676911, 106051, 10.1016/j.sysconle.2025.106051
    20. Ismaïla Balogoun, Jean Auriol, Islam Boussaada, Guilherme Mazanti, A novel necessary and sufficient condition for the stability of 2×2 first-order linear hyperbolic systems, 2025, 199, 01676911, 106066, 10.1016/j.sysconle.2025.106066
  • Reader Comments
  • © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Networks and Heterogeneous Media
1.2 1.8

Metrics

Article views(4832) PDF downloads(61) Cited by(20)

Article has an altmetric score of 1
Article has an altmetric score of 1
Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog