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Efficient computation of Lyapunov functions for nonlinear systems by integrating numerical solutions

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Abstract

A strict Lyapunov function for an equilibrium of a dynamical system asserts its asymptotic stability and gives a lower bound on its basin of attraction. For nonlinear systems, the explicit construction of a Lyapunov function taking the nonlinear dynamics into account remains a difficult problem and one often resorts to numerical methods. We improve and analyse a method that is based on a converse theorem in the Lyapunov stability theory and compare it to different methods in the literature. Our method is of low complexity, and its workload is perfectly parallel. Further, its free parameters allow it to be adapted to the problem at hand and we show that our method matches or gives a larger lower bound on the equilibrium’s basin of attraction than other approaches in the literature in most examples. Finally, we apply our method to a model of a genetic toggle switch in Escherichia coli and we demonstrate that our novel method delivers important information on the model’s dynamics for different parameters.

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References

  1. Björnsson, J., Giesl, P., Hafstein, S., Kellett, C., Li, H.: Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction. In: Proceedings of the CDC, 53rd IEEE Conference on Decision and Control, pp. 5506–5511. Los Angeles (2014)

  2. Björnsson, J., Hafstein, S.: Efficient Lyapunov function computation for systems with multiple exponentially stable equilibria. Procedia Comput. Sci. 108, 655–664 (2017). Proceedings of the International Conference on Computational Science (ICCS), Zurich, Switzerland, 2017

  3. Chesi, G.: Estimating the domain of attraction for non-polynomial systems via LMI optimizations. Automatica 45, 1536–1541 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chesi, G.: Domain of Attraction: Analysis and Control via SOS Programming. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  5. Doban, A.: Stability domains computation and stabilization of nonlinear systems: implications for biological systems. PhD thesis: Eindhoven University of Technology (2016)

  6. Doban, A., Lazar, M.: Computation of Lyapunov functions for nonlinear differential equations via a Yoshizawa-type construction. IFAC-PapersOnLine 49(18), 29–34 (2016)

    Article  Google Scholar 

  7. Doban, A., Lazar, M.: Computation of Lyapunov functions for nonlinear differential equations via a Massera-type construction. IEEE Trans. Autom. Control 63(5), 1259–1272 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gardner, T., Cantor, C., Collins, J.: Construction of a genetic toggle switch in Escherichia coli. Nature 403(6767), 339–342 (2000)

    Article  Google Scholar 

  9. Genesio, R., Tartaglia, M., Vicino, A.: On the estimation of asymptotic stability regions: state of the art and new proposals. IEEE Trans. Autom. Control 30(8), 747–755 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Giesl, P., Hafstein, S.: Revised CPA method to compute Lyapunov functions for nonlinear systems. J. Math. Anal. Appl. 410, 292–306 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Giesl, P., Hafstein, S.: Computation and verification of Lyapunov functions. SIAM J. Appl. Dyn. Syst. 14(4), 1663–1698 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Giesl, P., Hafstein, S.: Review of computational methods for Lyapunov functions. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2291–2331 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hachicho, O., Tibken, B.: Estimating domains of attraction of a class of nonlinear dynamical systems with LMI methods based on the theory of moments. In: Proceedings of the 41th IEEE Conference on Decision and Control (CDC), pp. 3150–3155. Los Angeles (2002)

  14. Hafstein, S.: A constructive converse Lyapunov theorem on exponential stability. Discrete Contin. Dyn. Syst. 10(3), 657–678 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hafstein, S., Kellett, C., Li, H.: Computing continuous and piecewise affine Lyapunov functions for nonlinear systems. J. Comput. Dyn. 2(2), 227–246 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hafstein, S., Valfells, A.: Study of dynamical systems by fast numerical computation of Lyapunov functions. In: Proceedings of the 14th International Conference on Dynamical Systems: Theory and Applications (DSTA), Mathematical and Numerical Aspects of Dynamical System Analysis, pp. 220–240 (2017)

  17. Kellett, C.: Converse theorems in Lyapunov’s second method. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2333–2360 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Khalil, H.: Nonlinear Systems, 3rd edn. Prentice-Hall, New Jersey (2002)

    MATH  Google Scholar 

  19. Lugagne, J., Carrillo, S., Kirch, M., Köhler, A., Batt, G., Hersen, P.: Balancing a genetic toggle switch by real-time feedback control and periodic forcing. Nat. Commun. 8, 1671 (2017)

    Article  Google Scholar 

  20. Massera, J.: Contributions to stability theory. Ann. Math. 64, 182–206 (1956). (Erratum. Annals of Mathematics, 68:202, 1958)

    Article  MathSciNet  MATH  Google Scholar 

  21. Matallana, L., Blanco, A., Bandoni, J.: Estimation of domains of attraction: a global optimization approach. Math. Comput. Model. 52(3–4), 574–585 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sastry, S.: Nonlinear Systems: Analysis, Stability, and Control. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  23. Vannelli, A., Vidyasagar, M.: Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems. Automatica 21(1), 69–80 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Vidyasagar, M.: Nonlinear System Analysis, Classics in Applied Mathematics, 2nd edn. SIAM, Philadelphia (2002)

    Book  Google Scholar 

  25. Wang, W., Ruan, S.: Bifurcations in an epidemic model with constant removal rate of infectives. J. Math. Anal. Appl. 291(1), 775–793 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yoshizawa, T.: Stability Theory by Liapunov’s Second Method. Publications of the Mathematical Society of Japan, No. 9. The Mathematical Society of Japan, Tokyo (1966)

    Google Scholar 

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Acknowledgements

This work was supported by the Icelandic Research Fund in the project Algorithms to compute Lyapunov functions (No. 130677-051). Additionally, the authors would like to thank the anonymous reviewers, whose suggestions improved this manuscript considerably.

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  1. Faculty of Physical Sciences, University of Iceland, Dunhagi 5, 107, Reykjavík, Iceland

    Sigurdur Freyr Hafstein & Asgeir Valfells

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  1. Sigurdur Freyr Hafstein
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  2. Asgeir Valfells
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Correspondence to Sigurdur Freyr Hafstein.

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Hafstein, S.F., Valfells, A. Efficient computation of Lyapunov functions for nonlinear systems by integrating numerical solutions. Nonlinear Dyn 97, 1895–1910 (2019). https://doi.org/10.1007/s11071-018-4729-5

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  • DOI: https://doi.org/10.1007/s11071-018-4729-5

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