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Deductive Stability Proofs for Ordinary Differential Equations

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12652)

Abstract

Stability is required for real world controlled systems as it ensures that those systems can tolerate small, real world perturbations around their desired operating states. This paper shows how stability for continuous systems modeled by ordinary differential equations (ODEs) can be formally verified in differential dynamic logic (dL). The key insight is to specify ODE stability by suitably nesting the dynamic modalities of dL with first-order logic quantifiers. Elucidating the logical structure of stability properties in this way has three key benefits: i) it provides a flexible means of formally specifying various stability properties of interest, ii) it yields rigorous proofs of those stability properties from dL’s axioms with dL’s ODE safety and liveness proof principles, and iii) it enables formal analysis of the relationships between various stability properties which, in turn, inform proofs of those properties. These benefits are put into practice through an implementation of stability proofs for several examples in KeYmaera X, a hybrid systems theorem prover based on dL.

Keywords

  • differential equations
  • stability
  • differential dynamic logic

This research was sponsored by the AFOSR under grant number FA9550-16-1-0288.

The first author was supported by A*STAR, Singapore.

References

  1. Ahmed, D., Peruffo, A., Abate, A.: Automated and sound synthesis of Lyapunov functions with SMT solvers. In: Biere, A., Parker, D. (eds.) TACAS. LNCS, vol. 12078, pp. 97–114. Springer (2020). https://doi.org/10.1007/978-3-030-45190-5_6

  2. Alur, R.: Principles of Cyber-Physical Systems. MIT Press (2015)

    Google Scholar 

  3. Ashbaugh, M.S., Chicone, C.C., Cushman, R.H.: The twisting tennis racket. Journal of Dynamics and Differential Equations 3, 67–85 (1991). https://doi.org/10.1007/BF01049489

  4. Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry. Springer, Heidelberg (1998). https://doi.org/10.1007/978-3-662-03718-8

  5. Branicky, M.S.: Introduction to hybrid systems. In: Hristu-Varsakelis, D., Levine, W.S. (eds.) Handbook of Networked and Embedded Control Systems, pp. 91–116. Birkhäuser (2005). https://doi.org/10.1007/0-8176-4404-0_5

  6. Chicone, C.: Ordinary Differential Equations with Applications. Springer, New York, second edn. (2006). https://doi.org/10.1007/0-387-35794-7

  7. Cohen, C., Rouhling, D.: A formal proof in Coq of LaSalle’s invariance principle. In: Ayala-Rincón, M., Muñoz, C.A. (eds.) ITP. LNCS, vol. 10499, pp. 148–163. Springer (2017). https://doi.org/10.1007/978-3-319-66107-0_10

  8. Doyen, L., Frehse, G., Pappas, G.J., Platzer, A.: Verification of hybrid systems. In: Clarke, E.M., Henzinger, T.A., Veith, H., Bloem, R. (eds.) Handbook of Model Checking, pp. 1047–1110. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-10575-8_30

  9. Forsman, K.: Construction of Lyapunov functions using Gröbner bases. In: CDC. vol. 1, pp. 798–799. IEEE (1991). https://doi.org/10.1109/CDC.1991.261424

  10. Fulton, N., Mitsch, S., Bohrer, B., Platzer, A.: Bellerophon: Tactical theorem proving for hybrid systems. In: Ayala-Rincón, M., Muñoz, C.A. (eds.) ITP. LNCS, vol. 10499, pp. 207–224. Springer (2017). https://doi.org/10.1007/978-3-319-66107-0_14

  11. Fulton, N., Mitsch, S., Quesel, J., Völp, M., Platzer, A.: KeYmaera X: an axiomatic tactical theorem prover for hybrid systems. In: Felty, A.P., Middeldorp, A. (eds.) CADE. LNCS, vol. 9195, pp. 527–538. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21401-6_36

  12. Gao, S., Kapinski, J., Deshmukh, J.V., Roohi, N., Solar-Lezama, A., Aréchiga, N., Kong, S.: Numerically-robust inductive proof rules for continuous dynamical systems. In: Dillig, I., Tasiran, S. (eds.) CAV. LNCS, vol. 11562, pp. 137–154. Springer (2019). https://doi.org/10.1007/978-3-030-25543-5_9

  13. Goebel, R., Sanfelice, R.G., Teel, A.R.: Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton University Press (2012)

    Google Scholar 

  14. Haddad, W.M., Chellaboina, V.: Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach. Princeton University Press (2008)

    Google Scholar 

  15. Hirsch, M.W.: The dynamical systems approach to differential equations. Bull. Amer. Math. Soc. (N.S.) 11(1), 1–64 (07 1984)

    Google Scholar 

  16. Hölzl, J., Immler, F., Huffman, B.: Type classes and filters for mathematical analysis in Isabelle/HOL. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP. LNCS, vol. 7998, pp. 279–294. Springer (2013). https://doi.org/10.1007/978-3-642-39634-2_21

  17. Kapinski, J., Deshmukh, J.V., Sankaranarayanan, S., Aréchiga, N.: Simulation-guided Lyapunov analysis for hybrid dynamical systems. In: Fränzle, M., Lygeros, J. (eds.) HSCC. pp. 133–142. ACM (2014). https://doi.org/10.1145/2562059.2562139

  18. Khalil, H.K.: Nonlinear systems. Macmillan Publishing Company, New York (1992)

    Google Scholar 

  19. Liapounoff, A.: Probléme général de la stabilité du mouvement. Annales de la Faculté des sciences de Toulouse : Mathématiques 9, 203–474 (1907)

    Google Scholar 

  20. Liberzon, D.: Switching in Systems and Control. Systems & Control: Foundations & Applications, Birkhäuser (2003). https://doi.org/10.1007/978-1-4612-0017-8

  21. Liu, J., Zhan, N., Zhao, H.: Automatically discovering relaxed Lyapunov functions for polynomial dynamical systems. Math. Comput. Sci. 6(4), 395–408 (2012). https://doi.org/10.1007/s11786-012-0133-6

  22. Nguyen, L.V., Kapinski, J., Jin, X., Deshmukh, J.V., Johnson, T.T.: Hyperproperties of real-valued signals. In: Talpin, J., Derler, P., Schneider, K. (eds.) MEMOCODE. pp. 104–113. ACM (2017). https://doi.org/10.1145/3127041.3127058

  23. Papachristodoulou, A., Prajna, S.: On the construction of Lyapunov functions using the sum of squares decomposition. In: CDC. vol. 3, pp. 3482–3487. IEEE (2002). https://doi.org/10.1109/CDC.2002.1184414

  24. Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. thesis, California Institute of Technology (2000)

    Google Scholar 

  25. Platzer, A.: The complete proof theory of hybrid systems. In: LICS. pp. 541–550. IEEE Computer Society (2012). https://doi.org/10.1109/LICS.2012.64

  26. Platzer, A.: A complete uniform substitution calculus for differential dynamic logic. J. Autom. Reasoning 59(2), 219–265 (2017). https://doi.org/10.1007/s10817-016-9385-1

  27. Platzer, A.: Logical Foundations of Cyber-Physical Systems. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-63588-0

  28. Platzer, A., Tan, Y.K.: Differential equation invariance axiomatization. J. ACM 67(1) (2020). https://doi.org/10.1145/3380825

  29. Podelski, A., Wagner, S.: Model checking of hybrid systems: From reachability towards stability. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC. LNCS, vol. 3927, pp. 507–521. Springer (2006). https://doi.org/10.1007/11730637_38

  30. Poincaré, H.: Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris (1892–1899)

    Google Scholar 

  31. Rouche, N., Habets, P., Laloy, M.: Stability Theory by Liapunov’s Direct Method. Springer, New York (1977). https://doi.org/10.1007/978-1-4684-9362-7

  32. Rouhling, D.: A formal proof in Coq of a control function for the inverted pendulum. In: Andronick, J., Felty, A.P. (eds.) CPP. pp. 28–41. ACM (2018). https://doi.org/10.1145/3167101

  33. Sankaranarayanan, S., Chen, X., Ábrahám, E.: Lyapunov function synthesis using Handelman representations. In: Tarbouriech, S., Krstic, M. (eds.) NOLCOS. pp. 576–581. IFAC (2013). https://doi.org/10.3182/20130904-3-FR-2041.00198

  34. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, Boulder, CO, second edn. (2015)

    Google Scholar 

  35. Tan, Y.K., Platzer, A.: Deductive stability proofs for ordinary differential equations. CoRR abs/2010.13096 (2020), https://arxiv.org/abs/2010.13096

  36. Tan, Y.K., Platzer, A.: An axiomatic approach to existence and liveness for differential equations. Formal Aspects Comput. (to appear). https://doi.org/10.1007/s00165-020-00525-0

  37. Topcu, U., Packard, A.K., Seiler, P.J.: Local stability analysis using simulations and sum-of-squares programming. Autom. 44(10), 2669–2675 (2008). https://doi.org/10.1016/j.automatica.2008.03.010

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Tan, Y.K., Platzer, A. (2021). Deductive Stability Proofs for Ordinary Differential Equations. In: Groote, J.F., Larsen, K.G. (eds) Tools and Algorithms for the Construction and Analysis of Systems. TACAS 2021. Lecture Notes in Computer Science(), vol 12652. Springer, Cham. https://doi.org/10.1007/978-3-030-72013-1_10

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  • DOI: https://doi.org/10.1007/978-3-030-72013-1_10

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