# 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 and Heterogeneous Media, 2016, 11 (4) : 563-601. doi: 10.3934/nhm.2016010
##### 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. [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] Y. Chitour, G. Mazanti and M. Sigalotti, Persistently damped transport on a network of circles, 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.

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##### 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. [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] Y. Chitour, G. Mazanti and M. Sigalotti, Persistently damped transport on a network of circles, 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.
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