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Stochastic Reachability Concepts

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Stochastic Reachability Analysis of Hybrid Systems

Part of the book series: Communications and Control Engineering ((CCE))

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Abstract

This chapter provides characterisations of stochastic reachability thought of as a stochastic hybrid control problem. First, we define formally the stochastic reachability problem as a measure of those system trajectories that visit in finite or infinite time horizon a given target set. We prove that this concept is well defined. Then, we link stochastic reachability with other mathematical concepts, which come with richer theoretical and numerical characterisations. Such concepts are: hitting distributions, exit distributions, occupation measures, occupation time distribution, Choquet capacity and reduced function and balayage. These concepts belong to the Markov process equipment and each one is interesting by itself and has its own applications. These connections make stochastic reachability an extremely powerful and flexible tool that could have applications not only in verification problems, but also in problems like reliability, resilience, fault-tolerance, performability or other optimisation problems.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4471-2795-6_12

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Notes

  1. 1.

    In fact, N is the set of irregular points of A, which is a semipolar set, see [108].

References

  1. Bect, J.: A unifying formulation of the Fokker–Planck–Kolmogorov equation for general stochastic hybrid systems. Nonlinear Anal. Hybrid Syst. 4(2), 357–370 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bect, J., Baili, H., Fleury, G.: Generalized Fokker–Planck equation for piecewise-diffusion processes with boundary hitting resets. In: Proceedings of the 17th International Symposium on the Mathematical Theory of Networks and Systems (2006)

    Google Scholar 

  3. Bemporad, A., Di Cairano, S.: Modelling and optimal control of hybrid systems with event uncertainty. In: Morari, M. Thiele, L. (eds.) Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 3414, pp. 151–167. Springer, Berlin (2005)

    Chapter  Google Scholar 

  4. Bujorianu, M.L.: Capacities and Markov processes. Libertas Math. 24, 201–210 (2004)

    MATH  MathSciNet  Google Scholar 

  5. Bujorianu, M.L., Lygeros, J.: Reachability questions in piecewise deterministic Markov processes. In: Maler, O., Pnueli, A. (eds.) Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 2623, pp. 126–140. Springer, Berlin (2003)

    Chapter  Google Scholar 

  6. Bujorianu, M.L.: Extended stochastic hybrid systems and their reachability problem. In: Alur, R., Pappas, G. (eds.) Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 2993, pp. 234–249. Springer, Berlin (2004)

    Chapter  Google Scholar 

  7. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–291 (1953)

    Article  MathSciNet  Google Scholar 

  8. Chutinan, A., Krogh, B.H.: Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 1569, pp. 76–90. Springer, Berlin (1999)

    Chapter  Google Scholar 

  9. Dempster, D.: Upper and lower probabilities induced by a multi-valued mapping. Ann. Math. Stat. 38, 325–339 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  10. Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time I. Commun. Pure Appl. Math. 28, 1–47 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  11. Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time II. Commun. Pure Appl. Math. 28, 279–301 (1975)

    Article  MATH  Google Scholar 

  12. Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time III. Commun. Pure Appl. Math. 29, 389–461 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  13. Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time IV. Commun. Pure Appl. Math. 36, 183–212 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  14. Durbin, J., Williams, D.: The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Probab. 29, 291–304 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. El Karoui, N., Lepeltier, J.-P., Millet, A.: A probabilistic approach to the reduite in optimal stopping. Probab. Math. Stat. 13(1), 97–121 (1992)

    MATH  Google Scholar 

  16. Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)

    Book  MATH  Google Scholar 

  17. Fine, T.L.: Lower probability models for uncertainty and non-deterministic processes. J. Stat. Plan. Inference 20, 389–411 (1988)

    Article  MATH  Google Scholar 

  18. Fitzsimmons, P.J.: Markov processes with equal capacities. J. Theor. Probab. 12, 271–292 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Grecea, V.: On some results concerning the reduite and balayage. Math. Nachr. 223, 66–75 (2001)

    Article  MathSciNet  Google Scholar 

  20. Hoekstra, J.M., Ruigrok, R.C.J., van Gent, R.N.H.W.: Free flight in a crowded airspace? In: 3rd USA/Europe Air Traffic Management R&D Seminar Napoli (2000)

    Google Scholar 

  21. Huber, P.J., Strassen, V.: Minimax tests and the Neyman–Pearson lemma for capacities. Ann. Stat. 1, 251–263 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hunt, G.A.: Markov processes and potentials I. Ill. J. Math. 1(1), 44–93 (1957)

    MATH  Google Scholar 

  23. Hunt, G.A.: Markov processes and potentials II. Ill. J. Math. 1(3), 316–369 (1957)

    Google Scholar 

  24. Hunt, G.A.: Markov processes and potentials III. Ill. J. Math. 2(2), 151–213 (1958)

    Google Scholar 

  25. Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  26. Kruzynski, M., Chelminiak, P.: Mean first-passage time in the stochastic theory of biochemical processes. J. Stat. Phys. 110, 137–181 (2003)

    Article  Google Scholar 

  27. Krystul, J., Blom, H.A.P.: Sequential Monte Carlo simulation of rare event probability in stochastic hybrid systems. Preprint, 16th IFAC World Congress (2005)

    Google Scholar 

  28. Krystul, J., Bagchi, A.: Approximation of first passage times of switching diffusion. In: Proc. 16th International Symposium on Mathematical Theory of Networks and Systems (2004)

    Google Scholar 

  29. Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability analysis. In: Lynch, N., Krogh, B. (eds.) Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 1790, pp. 202–214. Springer, Berlin (2000)

    Chapter  Google Scholar 

  30. Lygeros, J., Tomlin, C., Sastry, S.: Controllers for reachability specifications of hybrid systems. Automatica 35, 349–370 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  31. Ma, M., Rockner, M.: The Theory of (Non-Symmetric) Dirichlet Forms and Markov Processes. Springer, Berlin (1990)

    Google Scholar 

  32. Meyer, P.A.: Processus de Markov. Lecture Notes in Mathematics, vol. 26. Springer, Berlin (1967)

    Book  MATH  Google Scholar 

  33. Nardo, E.D., Nobile, A.G., Pirozzi, E., Ricciardi, L.M.: A computational approach to first passage time problems for Gauss–Markov processes. Adv. Appl. Probab. 33, 453–482 (2001)

    Article  MATH  Google Scholar 

  34. Nobile, A.G., Ricciardi, L.M., Sacerdote, L.: Exponential trends of Ornstein–Uhlenbeck first-passage-time densities. J. Appl. Probab. 22, 360–369 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  35. Perron, O.: Eine neue Behandlung der ersten Randwerteaufgabe für Δu=0. Math. Z. 18, 42–54 (1923)

    Article  MATH  MathSciNet  Google Scholar 

  36. Pola, G., Bujorianu, M.L., Lygeros, J., Di Benedetto, M.D.: Stochastic hybrid models: an overview with applications to air traffic management. In: Proc. of the Analysis and Design of Hybrid Systems, pp. 45–50. IFAC Elsevier, Amsterdam (2003)

    Google Scholar 

  37. Puri, A., Varaiya, P.: Verification of hybrid systems using abstractions. In: Antsaklis, P.J. (ed.) Hybrid Systems II. Lecture Notes in Computer Science, vol. 999, pp. 359–369. Springer, Berlin (1995)

    Chapter  Google Scholar 

  38. Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  39. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)

    MATH  Google Scholar 

  40. Varaiya, P.: Reach set computation using optimal control problems in multidimensional systems. In: Proc. of KIT Workshop on Verification of Hybrid Systems (1998)

    Google Scholar 

  41. Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman & Hall, London (1991)

    Book  MATH  Google Scholar 

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  1. School of Mathematics, University of Manchester, Manchester, UK

    Luminita Manuela Bujorianu

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  1. Luminita Manuela Bujorianu
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Bujorianu, L.M. (2012). Stochastic Reachability Concepts. In: Stochastic Reachability Analysis of Hybrid Systems. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-2795-6_5

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  • DOI: https://doi.org/10.1007/978-1-4471-2795-6_5

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