Advertisement

A Formal Proof in Coq of LaSalle’s Invariance Principle

  • Cyril CohenEmail author
  • Damien RouhlingEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10499)

Abstract

Stability analysis of dynamical systems plays an important role in the study of control techniques. LaSalle’s invariance principle is a result about the asymptotic stability of the solutions to a nonlinear system of differential equations and several extensions of this principle have been designed to fit different particular kinds of system. In this paper we present a formalization, in the Coq proof assistant, of a slightly improved version of the original principle. This is a step towards a formal verification of dynamical systems.

Keywords

Formal proofs Coq Dynamical systems Stability 

Notes

Acknowledgements

We thank the anonymous reviewers for their useful feedback.

References

  1. 1.
    Barkana, I.: Defending the beauty of the Invariance Principle. Int. J. Control 87(1), 186–206 (2014). http://dx.doi.org/10.1080/00207179.2013.826385 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.): ITP 2013. LNCS, vol. 7998. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39634-2 zbMATHGoogle Scholar
  3. 3.
    Boldo, S., Lelay, C., Melquiond, G.: Coquelicot: A User-Friendly Library of Real Analysis for Coq. Math. Comput. Sci. 9(1), 41–62 (2015). http://dx.doi.org/10.1007/s11786-014-0181-1 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cano, G.: Interaction entre algèbre linéaire et analyse en formalisation des mathématiques. (Interaction between linear algebra and analysis in formal mathematics). Ph.D. thesis, University of Nice Sophia Antipolis, France (2014). https://tel.archives-ouvertes.fr/tel-00986283
  5. 5.
    Chan, M., Ricketts, D., Lerner, S., Malecha, G.: Formal Verification of Stability Properties of Cyber-Physical Systems, January 2016Google Scholar
  6. 6.
    Chellaboina, V., Leonessa, A., Haddad, W.M.: Generalized Lyapunov and invariant set theorems for nonlinear dynamical systems. Syst. Control Lett. 38(4–5), 289–295 (1999). http://www.sciencedirect.com/science/article/pii/S0167691199000766 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Darmochwał, A.: Compact Spaces. Formaliz. Math. 1(2), 383–386 (1990). http://fm.mizar.org/1990-1/pdf1-2/compts_1.pdf Google Scholar
  8. 8.
    Fischer, N.R., Kamalapurkar, R., Dixon, W.E.: LaSalle-Yoshizawa Corollaries for Nonsmooth Systems. IEEE Trans. Automat. Control 58(9), 2333–2338 (2013). http://dx.doi.org/10.1109/TAC.2013.2246900 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gonthier, G.: Formal Proof - The Four-Color Theorem. Notices AMS 55(11), 1382–1393 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gonthier, G., et al.: A Machine-Checked Proof of the Odd Order Theorem. In: Blazy et al. [2], pp. 163–179 (2013). doi: 10.1007/978-3-642-39634-2_14
  11. 11.
    Gonthier, G., Mahboubi, A., Tassi, E.: A Small Scale Reflection Extension for the Coq system. Research Report RR-6455, Inria Saclay Ile de France (2015). https://hal.inria.fr/inria-00258384
  12. 12.
    Herencia-Zapana, H., Jobredeaux, R., Owre, S., Garoche, P.-L., Feron, E., Perez, G., Ascariz, P.: PVS linear algebra libraries for verification of control software algorithms in C/ACSL. In: Goodloe, A.E., Person, S. (eds.) NFM 2012. LNCS, vol. 7226, pp. 147–161. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-28891-3_15 CrossRefGoogle Scholar
  13. 13.
    Hölzl, J., Immler, F., Huffman, B.: Type Classes and Filters for Mathematical Analysis in Isabelle/HOL. In: Blazy et al. [2], pp. 279–294 (2013). doi: 10.1007/978-3-642-39634-2_21
  14. 14.
    Khalil, H.: Nonlinear Systems. Pearson Education, Prentice Hall (2002). https://books.google.fr/books?id=t_d1QgAACAAJ
  15. 15.
    LaSalle, J.: Some Extensions of Liapunov’s Second Method. IRE Trans. Circ. Theory 7(4), 520–527 (1960)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lester, D.R.: Topology in PVS: Continuous Mathematics with Applications. In: Proceedings of the Second Workshop on Automated Formal Methods, AFM 2007, pp. 11–20. ACM, New York (2007). http://doi.acm.org/10.1145/1345169.1345171
  17. 17.
    Liapounoff, A.: Problème général de la stabilité du mouvement. In: Annales de la Faculté des sciences de Toulouse: Mathématiques, vol. 9, pp. 203–474 (1907). http://eudml.org/doc/72801
  18. 18.
    Lozano, R., Fantoni, I., Block, D.: Stabilization of the inverted pendulum around its homoclinic orbit. Syst. Control Lett. 40(3), 197–204 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mancilla-Aguilar, J.L., García, R.A.: An extension of LaSalle’s invariance principle for switched systems. Syst. Control Lett. 55(5), 376–384 (2006). http://dx.doi.org/10.1016/j.sysconle.2005.07.009 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mayero, M.: Formalisation et automatisation de preuves en analyses réelle et numérique. Ph.D. thesis, Université Paris VI (décembre 2001)Google Scholar
  21. 21.
    Mitra, S., Chandy, K.M.: A Formalized Theory for Verifying Stability and Convergence of Automata in PVS. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 230–245. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-71067-7_20 CrossRefGoogle Scholar
  22. 22.
    Padlewska, B., Darmochwał, A.: Topological Spaces and Continuous Functions. Formaliz. Math. 1(1), 223–230 (1990). http://fm.mizar.org/1990-1/pdf1-1/pre_topc.pdf Google Scholar
  23. 23.
    The Coq Development Team: The Coq proof assistant reference manual, version 8.6. (2016). http://coq.inria.fr
  24. 24.
    Wilansky, A.: Topology for Analysis. Dover Books on Mathematics. Dover Publications, New York (2008). http://cds.cern.ch/record/2222525

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Université Côte d’Azur, InriaSophia AntipolisFrance

Personalised recommendations