Verifying Safety and Persistence in Hybrid Systems Using Flowpipes and Continuous Invariants

  • Andrew SogokonEmail author
  • Paul B. Jackson
  • Taylor T. Johnson


We describe a method for verifying the temporal property of persistence in non-linear hybrid systems. Given some system and an initial set of states, the method establishes that system trajectories always eventually evolve into some specified target subset of the states of one of the discrete modes of the system, and always remain within this target region. The method also computes a time-bound within which the target region is always reached. The approach combines flowpipe computation with deductive reasoning about invariants and is more general than each technique alone. We illustrate the method with a case study showing that potentially destructive stick-slip oscillations of an oil-well drill eventually die away for a certain choice of drill control parameters. The case study demonstrates how just using flowpipes or just reasoning about invariants alone can be insufficient and shows the richness of systems that one can handle with the proposed method, since the systems features modes with non-polynomial ODEs. We also propose an alternative method for proving persistence that relies solely on flowpipe computation.


Persistence verification Safety verification Ordinary differential equations Hybrid systems Metric temporal logic Flowpipes Positively invariant sets 



The authors wish to thank the anonymous reviewers for their careful reading and valuable suggestions for improving this work and extend special thanks to Dr. E.M. Navarro-López for pointing out the highly relevant work on deadness [50] before it appeared in print.


  1. 1.
    Akbarpour, B., Paulson, L.C.: MetiTarski: an automatic theorem prover for real-valued special functions. J. Autom. Reason. 44(3), 175–205 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.: Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In: Grossman, R.L., Nerode, A., Ravn, A.P., Rischel, H. (eds.) Hybrid Systems, Volume 736 of LNCS, pp. 209–229. Springer, Berlin (1992)Google Scholar
  3. 3.
    Baier, C., Tinelli, C. (eds.): Tools and Algorithms for the Construction and Analysis of Systems—21st International Conference, TACAS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, London, UK, April 11–18, 2015. Proceedings, volume 9035 of LNCS. Springer (2015)Google Scholar
  4. 4.
    Bemporad, A., Bicchi, A., Buttazzo, G.C. (eds.): Hybrid Systems: Computation and Control, 10th International Workshop, HSCC 2007, Pisa, Italy, April 3–5, 2007, Proceedings, Volume 4416 of LNCS. Springer (2007)Google Scholar
  5. 5.
    Berz, M., Makino, K.: Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models. Reliab. Comput. 4(4), 361–369 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blanchini, F.: Set invariance in control. Automatica 35(11), 1747–1767 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carter, R.A:. Verification of liveness properties on hybrid dynamical systems. Ph.D. thesis, University of Manchester, School of Computer Science (2013)Google Scholar
  8. 8.
    Chen, X., Ábrahám, E., Sankaranarayanan, S.: Flow*: an analyzer for non-linear hybrid systems. In: Sharygina and Veith [69], pp. 258–263CrossRefGoogle Scholar
  9. 9.
    Clarke, E.M., Fehnker, A., Han, Z., Krogh, B.H., Ouaknine, J., Stursberg, O., Theobald, M.: Abstraction and counterexample-guided refinement in model checking of hybrid systems. Int. J. Found. Comput. Sci. 14(4), 583–604 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cohen, C., Mahboubi, A.: Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination. Log. Methods Comput. Sci. 8(1), 1–40 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Collins, G.E.: Hauptvortrag: quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Barkhage, H. (ed.) Automata Theory and Formal Languages, 2nd GI Conference, Kaiserslautern, May 20–23, 1975, Volume 33 of LNCS, pp. 134–183. Springer, Berlin (1975)Google Scholar
  12. 12.
    Davenport, J.H., England, M.: Recent advances in real geometric reasoning. In: Botana, F., Quaresma, P. (eds.) Automated Deduction in Geometry—10th International Workshop, ADG 2014, Coimbra, Portugal, July 9–11, 2014, Revised Selected Papers, Volume 9201 of LNCS, pp. 37–52. Springer (2014)Google Scholar
  13. 13.
    Davison, E., Kurak, E.: A computational method for determining quadratic Lyapunov functions for non-linear systems. Automatica 7(5), 627–636 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Donzé, A., Maler, O.: Systematic simulation using sensitivity analysis. In: Bemporad et al. [4], pp. 174–189Google Scholar
  15. 15.
    Duggirala, P.S., Mitra, S.: Abstraction refinement for stability. In: 2011 IEEE/ACM International Conference on Cyber-Physical Systems, ICCPS 2011, Chicago, Illinois, USA, 12–14 April, 2011, pp. 22–31. IEEE Computer Society (2011)Google Scholar
  16. 16.
    Duggirala, P.S., Mitra, S.: Lyapunov abstractions for inevitability of hybrid systems. In: Dang, T., Mitchell, I.M. (eds.) Hybrid Systems: Computation and Control (Part of CPS Week 2012), HSCC’12, Beijing, China, April 17–19, 2012, pp. 115–124. ACM (2012)Google Scholar
  17. 17.
    Eggers, A., Ramdani, N., Nedialkov, N.S., Fränzle, M.: Improving the SAT modulo ODE approach to hybrid systems analysis by combining different enclosure methods. Softw. Syst. Model. 14(1), 121–148 (2015)CrossRefzbMATHGoogle Scholar
  18. 18.
    Fan, C., Kapinski, J., Jin, X., Mitra, S.: Locally optimal reach set over-approximation for nonlinear systems. In: 2016 International Conference on Embedded Software, EMSOFT 2016, Pittsburgh, Pennsylvania, USA, October 1–7, 2016, pp. 6:1–6:10. ACM (2016)Google Scholar
  19. 19.
    Fan, C., Kapinski, J., Jin, X., Mitra, S.: Simulation-driven reachability using matrix measures. ACM Trans. Embed. Comput. Syst. 17(1):21:1–21, 28 (2018)Google Scholar
  20. 20.
    Forsman, K.: Construction of Lyapunov functions using Gröbner bases, Vol. 1, pp. 798–799. IEEE(1991)Google Scholar
  21. 21.
    Frehse, G., Guernic, C.L., Donzé, A., Cotton, S., Ray, R., Lebeltel, O., Ripado, R., Girard, A., Dang, T., Maler, O.: SpaceEx: scalable verification of hybrid systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) Computer Aided Verification—23rd International Conference, CAV 2011, Snowbird, UT, USA, July 14–20, 2011. Proceedings, Volume 6806 of LNCS, pp. 379–395. Springer (2011)CrossRefGoogle Scholar
  22. 22.
    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.) Automated Deduction—CADE-25—25th International Conference on Automated Deduction, Berlin, Germany, August 1–7, 2015, Proceedings, Volume 9195 of LNCS, pp. 527–538. Springer (2015)Google Scholar
  23. 23.
    Ghorbal, K., Platzer, A.: Characterizing algebraic invariants by differential radical invariants. In: Ábrahám, E., Havelund, K. (eds.) Tools and Algorithms for the Construction and Analysis of Systems—20th International Conference, TACAS 2014, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2014, Grenoble, France, April 5–13, 2014. Proceedings, Volume 8413 of LNCS, pp. 279–294. Springer (2014)Google Scholar
  24. 24.
    Ghorbal, K., Sogokon, A., Platzer, A.: A hierarchy of proof rules for checking positive invariance of algebraic and semi-algebraic sets. Comput. Lang. Syst. Struct. 47, 19–43 (2017)zbMATHGoogle Scholar
  25. 25.
    Goubault, E., Jourdan, J., Putot, S., Sankaranarayanan, S.: Finding non-polynomial positive invariants and Lyapunov functions for polynomial systems through Darboux polynomials. In: American Control Conference, ACC 2014, Portland, OR, USA, June 4–6, 2014, pp. 3571–3578. IEEE (2014)Google Scholar
  26. 26.
    Goubault, E., Putot, S.: Forward inner-approximated reachability of non-linear continuous systems. In: Frehse, G., Mitra, S. (eds.) Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control, HSCC 2017, Pittsburgh, PA, USA, April 18–20, 2017, pp. 1–10. ACM (2017)Google Scholar
  27. 27.
    Gulwani, S., Tiwari, A.: Constraint-based approach for analysis of hybrid systems. In: Gupta and Malik [28], pp. 190–203Google Scholar
  28. 28.
    Gupta, A., Malik, S. (eds.): Computer Aided Verification, 20th International Conference, CAV 2008, Princeton, NJ, USA, July 7–14, 2008, Proceedings, Volume 5123 of LNCS. Springer (2008)Google Scholar
  29. 29.
    Henzinger, T.A.: The theory of hybrid automata. In: Proceedings, 11th Annual IEEE Symposium on Logic in Computer Science, New Brunswick, New Jersey, USA, July 27–30, 1996, pp. 278–292. IEEE Computer Society (1996)Google Scholar
  30. 30.
    Immler, F.: Verified reachability analysis of continuous systems. In: Baier and Tinelli [3], pp. 37–51Google Scholar
  31. 31.
    Jirstrand, M.: Cylindrical algebraic decomposition—an introduction. Technical Report 1807, Linköping University, Automatic Control (1995)Google Scholar
  32. 32.
    Kapela, T., Mrozek, M., Pilarczyk, P., Wilczak, D., Zgliczyński, P.: CAPD—a rigorous toolbox for computer assisted proofs in dynamics. Technical report, Jagiellonian University, Krakow, Poland (2010). Accessed 20 Nov 2018
  33. 33.
    Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, Upper Saddle River (2002)zbMATHGoogle Scholar
  34. 34.
    Kong, S., Gao, S., Chen, W., Clarke, E.M.: dreach: \({\delta }\)-reachability analysis for hybrid systems. In: Baier and Tinelli [3], pp. 200–205Google Scholar
  35. 35.
    Koymans, R.: Specifying real-time properties with metric temporal logic. Real Time Syst. 2(4), 255–299 (1990)CrossRefGoogle Scholar
  36. 36.
    Lin, Y., Stadtherr, M.A.: Validated solutions of initial value problems for parametric ODEs. Appl. Numer. Math. 57(10), 1145–1162 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Liu, J., Lv, J., Quan, Z., Zhan, N., Zhao, H., Zhou, C., Zou, L.: A calculus for hybrid CSP. In: Ueda, K. (ed) Programming Languages and Systems—8th Asian Symposium, APLAS 2010, Shanghai, China, November 28–December 1, 2010. Proceedings, Volume 6461 of LNCS, pp. 1–15. Springer (2010)Google Scholar
  38. 38.
    Liu, J., Zhan, N., Zhao, H.: Computing semi-algebraic invariants for polynomial dynamical systems. In: Chakraborty, S., Jerraya, A., Baruah, S.K., Fischmeister, S. (eds.) Proceedings of the 11th International Conference on Embedded Software, EMSOFT 2011, Part of the Seventh Embedded Systems Week, ESWeek 2011, Taipei, Taiwan, October 9–14, 2011, pp. 97–106. ACM (2011)Google Scholar
  39. 39.
    Lygeros, J., Johansson, K.H., Simić, S.N., Zhang, J., Sastry, S.S.: Dynamical properties of hybrid automata. IEEE Trans. Autom. Control 48(1), 2–17 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Mahboubi, A.: Programming and certifying a CAD algorithm in the Coq system. In: Coquand, T., Lombardi, H., Roy, M. (eds.) Mathematics, Algorithms, Proofs, 9–14. January 2005, Volume 05021 of Dagstuhl Seminar Proceedings. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany (2005)Google Scholar
  41. 41.
    Maidens, J.N., Arcak, M.: Trajectory-based reachability analysis of switched nonlinear systems using matrix measures. In: 53rd IEEE Conference on Decision and Control, CDC 2014, Los Angeles, CA, USA, December 15–17, 2014, pp. 6358–6364. IEEE (2014)Google Scholar
  42. 42.
    Maidens, J.N., Arcak, M.: Reachability analysis of nonlinear systems using matrix measures. IEEE Trans. Autom. Control 60(1), 265–270 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Makino, K., Berz, M.: COSY INFINITY version 9. Nucl. Instrum. Methods Phys. Res. Sect. A 558(1), 346–350 (2006)CrossRefGoogle Scholar
  44. 44.
    Manna, Z., Pnueli, A.: A hierarchy of temporal properties. In: Dwork, C. (ed) Proceedings of the Ninth Annual ACM Symposium on Principles of Distributed Computing, Quebec City, Quebec, Canada, August 22–24, 1990, pp. 377–410. ACM (1990)Google Scholar
  45. 45.
    Martin-Dorel, É., Roux, P.: A reflexive tactic for polynomial positivity using numerical solvers and floating-point computations. In: Bertot, Y., Vafeiadis, V. (eds.) Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs, CPP 2017, Paris, France, January 16–17, 2017, pp. 90–99. ACM (2017)Google Scholar
  46. 46.
    Mitrohin, C., Podelski, A.: Composing stability proofs for hybrid systems. In: Fahrenberg, U., Tripakis, S. (eds.) Formal Modeling and Analysis of Timed Systems—9th International Conference, FORMATS 2011, Aalborg, Denmark, September 21–23, 2011. Proceedings, Volume 6919 of LNCS, pp. 286–300. Springer (2011)Google Scholar
  47. 47.
    Möhlmann, E., Hagemann, W., Theel, O.E.: Hybrid tools for hybrid systems—proving stability and safety at once. In: Sankaranarayanan, S., Vicario, E. (eds.) Formal Modeling and Analysis of Timed Systems—13th International Conference, FORMATS 2015, Madrid, Spain, September 2–4, 2015, Proceedings, Volume 9268 of LNCS, pp. 222–239. Springer (2015)Google Scholar
  48. 48.
    Möhlmann, E., Theel, O.E.: Stabhyli: a tool for automatic stability verification of non-linear hybrid systems. In: Belta, C., Ivančić, F. (eds.) Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control, HSCC 2013, April 8–11, 2013, Philadelphia, PA, USA, pp. 107–112. ACM (2013)Google Scholar
  49. 49.
    Navarro-López, E.M., Carter, R.: Hybrid automata: an insight into the discrete abstraction of discontinuous systems. Int. J. Syst. Sci. 42(11), 1883–1898 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Navarro-López, E.M., Carter, R.: Deadness and how to disprove liveness in hybrid dynamical systems. Theor. Comput. Sci. 642(C), 1–23 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Navarro-López, E.M., Suárez, R.: Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings, Vol. 2, pp. 1454–1460. IEEE (2004)Google Scholar
  52. 52.
    Nedialkov, N.S.: Interval tools for ODEs and DAEs. In: 12th GAMM—IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006), Duisburg (2006)Google Scholar
  53. 53.
    Neher, M., Jackson, K.R., Nedialkov, N.S.: On Taylor model based integration of ODEs. SIAM J. Numer. Anal. 45(1), 236–262 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Nishida, T., Mizutani, K., Kubota, A., Doshita, S.: Automated phase portrait analysis by integrating qualitative and quantitative analysis. In: Dean, T.L., McKeown, K.R. (eds.) Proceedings of the 9th National Conference on Artificial Intelligence, Anaheim, CA, USA, July 14–19, 1991, Vol. 2, pp. 811–816. AAAI Press/The MIT Press (1991)Google Scholar
  55. 55.
    Paulson, L.C.: MetiTarski: Past and future. In: Beringer, L., Felty, A.P. (eds.) Interactive Theorem Proving—Third International Conference, ITP 2012, Princeton, NJ, USA, August 13–15, 2012. Proceedings, Volume 7406 of LNCS, pp. 1–10. Springer (2012)Google Scholar
  56. 56.
    Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reason. 41(2), 143–189 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Platzer, A., Clarke, E.M.: Computing differential invariants of hybrid systems as fixedpoints. In: Gupta and Malik [28], pp. 176–189Google Scholar
  58. 58.
    Platzer, A., Quesel, J.: KeYmaera: a hybrid theorem prover for hybrid systems (system description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) Automated Reasoning, 4th International Joint Conference, IJCAR 2008, Sydney, Australia, August 12–15, 2008, Proceedings, Volume 5195 of LNCS, pp. 171–178. Springer (2008)Google Scholar
  59. 59.
    Podelski, A., Wagner, S.: Model checking of hybrid systems: from reachability towards stability. In: Hespanha, J.P., Tiwari, A. (eds.) Hybrid Systems: Computation and Control, 9th International Workshop, HSCC 2006, Santa Barbara, CA, USA, March 29–31, 2006, Proceedings, Volume 3927 of LNCS, pp. 507–521. Springer (2006)CrossRefzbMATHGoogle Scholar
  60. 60.
    Podelski, A., Wagner, S.: Region stability proofs for hybrid systems. In: Raskin, J., Thiagarajan, P.S. (eds.) Formal Modeling and Analysis of Timed Systems, 5th International Conference, FORMATS 2007, Salzburg, Austria, October 3–5, 2007, Proceedings, Volume 4763 of LNCS, pp. 320–335. Springer (2007)Google Scholar
  61. 61.
    Podelski, A., Wagner, S.: A sound and complete proof rule for region stability of hybrid systems. In: Bemporad et al. [4], pp. 750–753Google Scholar
  62. 62.
    Prabhakar, P., Soto, M.G.: Abstraction based model-checking of stability of hybrid systems. In: Sharygina and Veith [69], pp. 280–295CrossRefGoogle Scholar
  63. 63.
    Prajna, S., Jadbabaie, A.: Safety verification of hybrid systems using barrier certificates. In: Alur, R., Pappas, G.J. (eds.) Hybrid Systems: Computation and Control, 7th International Workshop, HSCC 2004, Philadelphia, PA, USA, March 25–27, 2004, Proceedings, Volume 2993 of LNCS, pp. 477–492. Springer (2004)CrossRefzbMATHGoogle Scholar
  64. 64.
    Ratschan, S., She, Z.: Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions. SIAM J. Control Optim. 48(7), 4377–4394 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Rebiha, R., Moura, A.V., Matringe, N.: Generating invariants for non-linear hybrid systems. Theor. Comput. Sci. 594, 180–200 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Richardson, D.: Some undecidable problems involving elementary functions of a real variable. J. Symb. Logic 33(4), 514–520, 12 (1968)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Sankaranarayanan, S.: Automatic invariant generation for hybrid systems using ideal fixed points. In: Johansson, K.H., Yi, W. (eds.) Proceedings of the 13th ACM International Conference on Hybrid Systems: Computation and Control, HSCC 2010, Stockholm, Sweden, April 12–15, 2010, pp. 221–230. ACM (2010)Google Scholar
  68. 68.
    Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Constructing invariants for hybrid systems. Form. Methods Syst. Des. 32(1), 25–55 (2008)CrossRefzbMATHGoogle Scholar
  69. 69.
    Sharygina, N., Veith, H. (eds.): Computer Aided Verification—25th International Conference, CAV 2013, Saint Petersburg, Russia, July 13–19, 2013. Proceedings, Volume 8044 of LNCS. Springer (2013)Google Scholar
  70. 70.
    Sogokon, A., Ghorbal, K., Jackson, P.B., Platzer, A.: A method for invariant generation for polynomial continuous systems. In: Jobstmann, B., Leino, K.R.M. (eds.) Verification, Model Checking, and Abstract Interpretation—17th International Conference, VMCAI 2016, St. Petersburg, FL, USA, January 17–19, 2016. Proceedings, Volume 9583 of LNCS, pp. 268–288. Springer (2016)Google Scholar
  71. 71.
    Sogokon, A., Ghorbal, K., Johnson, T.T.: Operational models for piecewise-smooth systems. ACM Trans. Embed. Comput. Syst. 16(5), 185:1–185:19 (2017)Google Scholar
  72. 72.
    Sogokon, A., Jackson, P.B.: Direct formal verification of liveness properties in continuous and hybrid dynamical systems. In: Bjørner, N., de Boer, F.S. (eds.) FM 2015: Formal Methods—20th International Symposium, Oslo, Norway, June 24–26, 2015, Proceedings, Volume 9109 of LNCS, pp. 514–531. Springer (2015)Google Scholar
  73. 73.
    Strzeboński, A.W.: Cylindrical decomposition for systems transcendental in the first variable. J. Symb. Comput. 46(11), 1284–1290 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Taly, A., Tiwari, A.: Deductive verification of continuous dynamical systems. In: Kannan, R., Kumar, K.N. (eds.) IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2009, December 15–17, 2009, IIT Kanpur, India, Volume 4 of LIPIcs, pp. 383–394. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2009)Google Scholar
  75. 75.
    Tiwari, A.: Generating box invariants. In: Egerstedt, M., Mishra, B. (eds.) Hybrid Systems: Computation and Control, 11th International Workshop, HSCC 2008, St. Louis, MO, USA, April 22–24, 2008. Proceedings, Volume 4981 of LNCS, pp. 658–661. Springer (2008)Google Scholar
  76. 76.
    Vannelli, A., Vidyasagar, M.: Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems. Automatica 21(1), 69–80 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Wang, S., Zhan, N., Zou, L.: An improved HHL prover: an interactive theorem prover for hybrid systems. In: Butler, M.J., Conchon, S., Zaïdi, F. (eds.) Formal Methods and Software Engineering—17th International Conference on Formal Engineering Methods, ICFEM 2015, Paris, France, November 3–5, 2015, Proceedings, Volume 9407 of LNCS, pp. 382–399. Springer (2015)Google Scholar
  78. 78.
    Xue, B., Easwaran, A., Cho, N., Fränzle, M.: Reach-avoid verification for nonlinear systems based on boundary analysis. IEEE Trans. Autom. Control 62(7), 3518–3523 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Zhao, H., Yang, M., Zhan, N., Gu, B., Zou, L., Chen, Y.: Formal verification of a descent guidance control program of a lunar lander. In: Jones, C.B., Pihlajasaari, P., Sun, J. (eds.) FM 2014: Formal Methods—19th International Symposium, Singapore, May 12–16, 2014. Proceedings, Volume 8442 of LNCS, pp. 733–748. Springer (2014)Google Scholar
  80. 80.
    Zhao, H., Zhan, N., Kapur, D.: Synthesizing switching controllers for hybrid systems by generating invariants. In: Theories of Programming and Formal Methods—Essays Dedicated to Jifeng He on the Occasion of His 70th Birthday, pp. 354–373 (2013)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.University of EdinburghEdinburghUK
  3. 3.Vanderbilt UniversityNashvilleUSA

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