Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Persistently damped transport on a network of circles


Authors: Yacine Chitour, Guilherme Mazanti and Mario Sigalotti
Journal: Trans. Amer. Math. Soc. 369 (2017), 3841-3881
MSC (2010): Primary 35R02, 35B35, 35C05, 35L40, 93C20
DOI: https://doi.org/10.1090/tran/6778
Published electronically: October 12, 2016
MathSciNet review: 3624395
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we address the exponential stability of a system of transport equations with intermittent damping on a network of $ N \geq 2$ circles intersecting at a single point $ O$. The $ N$ equations are coupled through a linear mixing of their values at $ O$, described by a matrix $ M$. The activity of the intermittent damping is determined by persistently exciting signals, all belonging to a fixed class. The main result is that, under suitable hypotheses on $ M$ and on the rationality of the ratios between the lengths of the circles, such a system is exponentially stable uniformly with respect to the persistently exciting signals. The proof relies on an explicit formula for the solutions of this system, which allows one to track down the effects of the intermittent damping.


References [Enhancements On Off] (What's this?)

  • [1] Fatiha Alabau-Boussouira and Matthieu Léautaud, Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl. (9) 99 (2013), no. 5, 544-576 (English, with English and French summaries). MR 3039207, https://doi.org/10.1016/j.matpur.2012.09.012
  • [2] Felix Ali Mehmeti, Joachim von Below, and Serge Nicaise (eds.), Partial differential equations on multistructures, Lecture Notes in Pure and Applied Mathematics, vol. 219, Marcel Dekker, Inc., New York, 2001. MR 1824576
  • [3] Brian D. O. Anderson, Exponential stability on linear equations arising in adaptive identification, IEEE Trans. Automatic Control AC-22 (1977), no. 1, 83-88. MR 0446667
  • [4] B. D. O. Anderson, R. R. Bitmead, C. R. Johnson Jr., P. V. Kokotović, R. L. Kosut, I. M. Y. Mareels, L. Praly, and B. D. Riedle, Stability of adaptive systems: Passivity and averaging systems, MIT Press Series in Signal Processing, Optimization, and Control, 8, MIT Press, Cambridge, MA, 1986. MR 846209
  • [5] S. Andersson and P. Krishnaprasad, Degenerate gradient flows: a comparison study of convergence rate estimates, Decision and Control, 2002, Proceedings of the 41st IEEE Conference, vol. 4, IEEE, 2002, pp. 4712-4717.
  • [6] 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), no. 4, 751-759. MR 2357767, https://doi.org/10.3934/nhm.2007.2.751
  • [7] Alberto Bressan, Sunčica Čanić, Mauro Garavello, Michael Herty, and Benedetto Piccoli, Flows on networks: recent results and perspectives, EMS Surv. Math. Sci. 1 (2014), no. 1, 47-111. MR 3200227, https://doi.org/10.4171/EMSS/2
  • [8] A. Chaillet, Y. Chitour, A. Loría, and M. Sigalotti, Towards uniform linear time-invariant stabilization of systems with persistency of excitation, Decision and Control, 2007 46th IEEE Conference, IEEE, 2007, pp. 6394-6399.
  • [9] Antoine Chaillet, Yacine Chitour, Antonio Loría, and Mario Sigalotti, Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases, Math. Control Signals Systems 20 (2008), no. 2, 135-156. MR 2411416, https://doi.org/10.1007/s00498-008-0024-1
  • [10] Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, Mathematical Surveys and Monographs, vol. 70, American Mathematical Society, Providence, RI, 1999. MR 1707332
  • [11] Yacine Chitour, Fritz Colonius, and Mario Sigalotti, Growth rates for persistently excited linear systems, Math. Control Signals Systems 26 (2014), no. 4, 589-616. MR 3273660, https://doi.org/10.1007/s00498-014-0131-0
  • [12] Y. Chitour, G. Mazanti, and M. Sigalotti, Stabilization of persistently excited linear systems, J. Daafouz, S. Tarbouriech, and M. Sigalotti, editors, Hybrid Systems with Constraints, chapter 4, Wiley-ISTE, London, 2013.
  • [13] Yacine Chitour and Mario Sigalotti, On the stabilization of persistently excited linear systems, SIAM J. Control Optim. 48 (2010), no. 6, 4032-4055. MR 2645472, https://doi.org/10.1137/080737812
  • [14] René Dáger and Enrique Zuazua, Wave propagation, observation and control in $ 1$$ \text {-}d$ flexible multi-structures, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 50, Springer-Verlag, Berlin, 2006. MR 2169126
  • [15] Martin Gugat, Optimal switching boundary control of a string to rest in finite time, ZAMM Z. Angew. Math. Mech. 88 (2008), no. 4, 283-305. MR 2389094, https://doi.org/10.1002/zamm.200700154
  • [16] Martin Gugat, Contamination source determination in water distribution networks, SIAM J. Appl. Math. 72 (2012), no. 6, 1772-1791. MR 3022286, https://doi.org/10.1137/110859269
  • [17] Martin Gugat and Mario Sigalotti, Stars of vibrating strings: switching boundary feedback stabilization, Netw. Heterog. Media 5 (2010), no. 2, 299-314. MR 2661409, https://doi.org/10.3934/nhm.2010.5.299
  • [18] Jack K. Hale, Dynamical systems and stability, J. Math. Anal. Appl. 26 (1969), 39-59. MR 0244582
  • [19] Falk M. Hante, Mario Sigalotti, and Marius Tucsnak, On conditions for asymptotic stability of dissipative infinite-dimensional systems with intermittent damping, J. Differential Equations 252 (2012), no. 10, 5569-5593. MR 2902127, https://doi.org/10.1016/j.jde.2012.01.037
  • [20] Falk M. Hante, Günter Leugering, and Thomas I. Seidman, Modeling and analysis of modal switching in networked transport systems, Appl. Math. Optim. 59 (2009), no. 2, 275-292. MR 2480783, https://doi.org/10.1007/s00245-008-9057-6
  • [21] Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
  • [22] Tosio Kato, Linear evolution equations of ``hyperbolic'' type. II, J. Math. Soc. Japan 25 (1973), 648-666. MR 0326483
  • [23] J. E. Lagnese, Günter Leugering, and E. J. P. G. Schmidt, Modeling, analysis and control of dynamic elastic multi-link structures, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1279380
  • [24] Daniel Liberzon, Switching in systems and control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 1987806
  • [25] A. Loría, A. Chaillet, G. Besançon, and Y. Chitour, On the PE stabilization of time-varying systems: Open questions and preliminary answers, Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC'05. 44th IEEE Conference, IEEE, 2005, pp. 6847-6852.
  • [26] G. Lumer, Connecting of local operators and evolution equations on networks, Potential theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), Lecture Notes in Math., vol. 787, Springer, Berlin, 1980, pp. 219-234. MR 587842
  • [27] Michael Margaliot, Stability analysis of switched systems using variational principles: an introduction, Automatica J. IFAC 42 (2006), no. 12, 2059-2077. MR 2259150, https://doi.org/10.1016/j.automatica.2006.06.020
  • [28] Guilherme Mazanti, Stabilization of persistently excited linear systems by delayed feedback laws, Systems Control Lett. 68 (2014), 57-67. MR 3199878, https://doi.org/10.1016/j.sysconle.2014.03.006
  • [29] Guilherme Mazanti, Yacine Chitour, and Mario Sigalotti, Stabilization of two-dimensional persistently excited linear control systems with arbitrary rate of convergence, SIAM J. Control Optim. 51 (2013), no. 2, 801-823. MR 3032896, https://doi.org/10.1137/110848153
  • [30] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
  • [31] R. S. Phillips, Perturbation theory for semi-groups of linear operators, Trans. Amer. Math. Soc. 74 (1953), 199-221. MR 0054167
  • [32] Robert Shorten, Fabian Wirth, Oliver Mason, Kai Wulff, and Christopher King, Stability criteria for switched and hybrid systems, SIAM Rev. 49 (2007), no. 4, 545-592. MR 2375524, https://doi.org/10.1137/05063516X
  • [33] M. Slemrod, The LaSalle invariance principle in infinite-dimensional Hilbert space, Dynamical systems approaches to nonlinear problems in systems and circuits (Henniker, NH, 1986) SIAM, Philadelphia, PA, 1988, pp. 53-59. MR 970041
  • [34] Masayasu Suzuki, Jun-ichi Imura, and Kazuyuki Aihara, Analysis and stabilization for networked linear hyperbolic systems of rationally dependent conservation laws, Automatica J. IFAC 49 (2013), no. 11, 3210-3221. MR 3115791, https://doi.org/10.1016/j.automatica.2013.08.016
  • [35] Julie Valein and Enrique Zuazua, Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim. 48 (2009), no. 4, 2771-2797. MR 2558320, https://doi.org/10.1137/080733590
  • [36] Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
  • [37] Enrique Zuazua, Control and stabilization of waves on 1-d networks, Modelling and optimisation of flows on networks, Lecture Notes in Math., vol. 2062, Springer, Heidelberg, 2013, pp. 463-493. MR 3050288, https://doi.org/10.1007/978-3-642-32160-3_9

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35R02, 35B35, 35C05, 35L40, 93C20

Retrieve articles in all journals with MSC (2010): 35R02, 35B35, 35C05, 35L40, 93C20


Additional Information

Yacine Chitour
Affiliation: Laboratoire des Signaux et Systèmes, Université Paris-Sud, CNRS, CentraleSupélec, 91192 Gif-sur-Yvette, France
Email: yacine.chitour@lss.supelec.fr

Guilherme Mazanti
Affiliation: CMAP & Inria, team GECO, École Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau cedex, France
Address at time of publication: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
Email: guilherme.mazanti@polytechnique.edu

Mario Sigalotti
Affiliation: Inria, team GECO & CMAP, École Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau cedex, France
Email: mario.sigalotti@inria.fr

DOI: https://doi.org/10.1090/tran/6778
Keywords: Transport equation, persistent excitation, exponential stability, multistructures
Received by editor(s): June 3, 2014
Received by editor(s) in revised form: May 22, 2015
Published electronically: October 12, 2016
Additional Notes: This research was partially supported by the iCODE Institute, research project of the IDEX Paris-Saclay, and by the Hadamard Mathematics LabEx (LMH) through grant number ANR-11-LABX-0056-LMH in the “Programme des Investissements d’Avenir”
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society