# American Institute of Mathematical Sciences

December  2016, 11(4): 563-601. doi: 10.3934/nhm.2016010

## 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, France 3 Inria, team GECO & CMAP, École Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau cedex, France

Received  September 2015 Revised  March 2016 Published  October 2016

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. Networks & Heterogeneous Media, 2016, 11 (4) : 563-601. doi: 10.3934/nhm.2016010
##### References:
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Media, 2 (2007), 751-759. doi: 10.3934/nhm.2007.2.751.  Google Scholar [6] R. K. Brayton, Bifurcation of periodic solutions in a nonlinear difference-differential equations of neutral type, Quart. Appl. Math., 24 (1966), 215-224.  Google Scholar [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.  Google Scholar [8] Y. Chitour, G. Mazanti and M. Sigalotti, Persistently damped transport on a network of circles,, Trans. Amer. Math. Soc., ().   Google Scholar [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.  Google Scholar [10] J.-M. Coron, G. Bastin and B. d'Andréa Novel, Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems, SIAM J. 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Matrices, 6 (2012), 107-128. doi: 10.7153/oam-06-08.  Google Scholar [20] D. Liberzon, Switching in Systems and Control, 1st edition, Birkhäuser Boston, 2003. doi: 10.1007/978-1-4612-0017-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### References:
 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] R. K. Brayton, Bifurcation of periodic solutions in a nonlinear difference-differential equations of neutral type, Quart. Appl. Math., 24 (1966), 215-224.  Google Scholar [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.  Google Scholar [8] Y. Chitour, G. Mazanti and M. Sigalotti, Persistently damped transport on a network of circles,, Trans. Amer. Math. Soc., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] J. J. Green, Uniform Convergence to the Spectral Radius and Some Related Properties in Banach Algebras, PhD thesis, University of Sheffield, 1996. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] B. Klöss, The flow approach for waves in networks, Oper. Matrices, 6 (2012), 107-128. doi: 10.7153/oam-06-08.  Google Scholar [20] D. Liberzon, Switching in Systems and Control, 1st edition, Birkhäuser Boston, 2003. doi: 10.1007/978-1-4612-0017-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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