手机体育投注平台

Advertisement

Complete Lyapunov Functions: Determination of the Chain-Recurrent Set Using the Gradient

  • Carlos ArgáezEmail author
  • Peter Giesl
  • Sigurdur Freyr Hafstein
Conference paper
  • 35 Downloads
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1260)

Abstract

手机体育投注平台Complete Lyapunov functions (CLF) are scalar-valued functions, which are non-increasing along solutions of a given autonomous ordinary differential equation. They separate the phase-space into the chain-recurrent set, where the CLF is constant along solutions, and the set where the flow is gradient-like and the CLF is strictly decreases along solutions. Moreover, one can deduce the stability of connected components of the chain-recurrent set from the CLF.

While the existence of CLFs was shown about 50 years ago, in recent years algorithms to construct CLFs have been designed to determine the chain-recurrent set using the orbital derivative. These algorithms require iterative methods that constructed better and better approximations to a CLF, based on previous iterations. A drawback of these methods is the overestimation of the chain-recurrent set, which has been addressed by different methods.

In this paper, we construct a CLF using the previous method, but in contrast to previous work we will use the norm of the gradient of the computed CLF, rather than its orbital derivative, to determine the chain-recurrent set. We will show in this paper that this new approach determines the chain-recurrent set very well without the need of iterations or further methods to reduce the overestimation.

Keywords

Complete Lyapunov functions Chain-recurrent set Dynamical systems 

References

  1. 1.
    Anderson, J., Papachristodoulou, A.: Advances in computational Lyapunov analysis using sum-of-squares programming. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2361–2381 (2015)
  2. 2.
    Argáez, C., Giesl, P., Hafstein, S.: Analysing dynamical systems towards computing complete Lyapunov functions. In: Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, SIMULTECH 2017, Madrid, Spain (2017)
  3. 3.
    Argáez, C., Giesl, P., Hafstein, S.: Computation of complete Lyapunov functions for three-dimensional systems. In: Proceedings of the 57rd IEEE Conference on Decision and Control (CDC), Miami Beach, FL, USA, 2018, pp. 4059-4064 (2018)
  4. 4.
    Argáez, C., Giesl, P., Hafstein, S.: Computational approach for complete Lyapunov functions. In: Dynamical Systems in Theoretical Perspective, Springer Proceedings in Mathematics and Statistics, vol. 248, pp. 1–11. Springer (2018)
  5. 5.
    Argáez, C., Giesl, P., Hafstein, S.: Iterative construction of complete Lyapunov functions. In: Proceedings of 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2018), pp. 211–222 (2018).  . ISBN 978-989-758-323-0
  6. 6.
    Argáez, C., Giesl, P., Hafstein, S.: Construction of a complete Lyapunov function using quadratic programming. In: Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO), Porto, Portugal, 2018, pp. 560–568 (2018)
  7. 7.
    Argáez, C., Giesl, P., Hafstein, S.: Middle point reduction of the chain-recurrent set. In: Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH), Prague, Czech Republic, 2019, pp. 141–152 (2019)
  8. 8.
    Argáez, C., Giesl, P., Hafstein, S.: Complete Lyapunov functions: computation and applications. In: Simulation and Modeling Methodologies, Technologies and Applications Series: Advances in Intelligent Systems and Computing, vol. 873, pp. 200–221. Springer (2019)
  9. 9.
    Argáez, C., Giesl, P., Hafstein, S.: Clustering algorithm for generalized recurrences using complete Lyapunov functions. In: Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO), Prague, Czech Republic, 2019, pp. 138–146 (2019)
  10. 10.
    Argáez, C., Giesl, P., Hafstein, S.: Improved estimation of the chain-recurrent set. In: Proceedings of the 18th European Control Conference (ECC), Napoli, Italy, 2019, pp. 1622–1627 (2019)
  11. 11.
    Argáez, C., Berthet, J.-C., Björnsson, H., Giesl, P., Hafstein, S.: LyapXool - a program to compute complete Lyapunov functions. SoftwareX 10, 100325 (2019). ISSN 2352-7110
  12. 12.
    Argáez, C., Giesl, P., Hafstein, S.: Critical tolerance evolution: classification of the chain-recurrent set. In: Proceedings of the 15th International Conference on Dynamical Systems: Theory and Applications (DSTA), Volume: Mathematical and Numerical Aspects of Dynamical System Analysis, Lodz, Poland, pp. 21–32 (2019)
  13. 13.
    Buhmann, M.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)
  14. 14.
    Conley, C.: Isolated invariant sets and the morse index. In: CBMS Regional Conference Series no. 38. American Mathematical Society (1978)
  15. 15.
    Conley, C.: The gradient structure of a flow I. Ergodic Theory Dynam. Syst. 8, 11–26 (1988)
  16. 16.
    Dellnitz, M., Junge, O.: Set oriented numerical methods for dynamical systems. Handbook of dynamical systems, vol. 2, pp. 221–264. North-Holland, Amsterdam (2002)
  17. 17.
    Giesl, P.: Construction of Global Lyapunov Functions Using Radial Basis Functions. Lecture Notes in Math, vol. 1904. Springer, Heidelberg (2007)
  18. 18.
    Giesl, P., Hafstein, S.: Review on computational methods for Lyapunov functions. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2291–2331 (2015)
  19. 19.
    Hsu, C.S.: Cell-to-Cell Mapping. Applied Mathematical Sciences, vol. 64. Springer, New York (1987)
  20. 20.
    Hurley, M.: Chain recurrence, semiflows, and gradients. J. Dyn. Diff. Equat. 7(3), 437–456 (1995)
  21. 21.
    Hurley, M.: Lyapunov functions and attractors in arbitrary metric spaces. Proc. Amer. Math. Soc. 126, 245–256 (1998)
  22. 22.
    Iske, A.: Perfect centre placement for radial basis function methods. Technical report TUM-M9809, TU Munich, Germany (1998)
  23. 23.
    Krauskopf, B., Osinga, H., Doedel, E.J., Henderson, M., Guckenheimer, J., Vladimirsky, A., Dellnitz, M., Junge, O.: A survey of methods for computing (un)stable manifolds of vector fields. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15(3), 763–791 (2005)
  24. 24.
    Lyapunov, A.M.: The general problem of the stability of motion. Internat. J. Control 55(3), 521–790 (1992). Translated by A. T. Fuller from Édouard Davaux’s French translation (1907) of the 1892 Russian original
  25. 25.
    Wendland, H.: Error estimates for interpolation by compactly supported Radial Basis Functions of minimal degree. J. Approx. Theory 93, 258–272 (1998)
  26. 26.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)

Copyright information

© Springer Nature Switzerland AG 2021

Authors and Affiliations

  • Carlos Argáez
    • 1
    Email author
  • Peter Giesl
    • 2
  • Sigurdur Freyr Hafstein
    • 1
  1. 1.Science InstituteUniversity of IcelandReykjavíkIceland
  2. 2.University of SussexFalmerUK

Personalised recommendations