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Extended Stochastic Hybrid Systems and Their Reachability Problem

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Hybrid Systems: Computation and Control (HSCC 2004)

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Abstract

In this paper we generalize a model for stochastic hybrid systems. First, we prove that this model is a right Markov process and it satisfies some mathematical properties. Second, we propose a method based on the theory of Dirichlet forms to study the reachability problem associated with these systems.

Work Supported by European Comission under COLUMBUSIST-2001-38314.

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References

  1. Albeverio, S., Ma, M.: A General Correspondence Between Dirichlet Forms and Right Processes. Bull. Amer. Math. Soc. 26(2), 245–252 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley - Interscience, New York (1974)

    MATH  Google Scholar 

  3. Bass, R.F.: Adding and Substracting Jumps from Markov Processes. Trans. of Amer. Math. Soc. 255, 363–376 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bensoussan, A., Menaldi, J.L.: Stochastic Hybrid Control. Journal of Mathematical Analysis and Applications 249, 261–288 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Benveniste, A., Jacod, J.: Systèmes de Lévy de Processus de Markov. Invent. Math. 21, 183–198 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bertsekas, D.P., Shreve, S.E.: Stochastic Optimal Control: The Discrete-Time Case. Athena Scientific, Belmont (1996)

    Google Scholar 

  7. Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Academic Press, New York (1968)

    MATH  Google Scholar 

  8. Bouleau, N., Hirsch, F.: Propriétés d’Absolue Continuité dans les Espaces de Dirichlet et Applications aux Equations Différentielles Stochastiques. In: Sém. de Probabilités, LNM, vol. 20, Springer, Heidelberg (1986)

    Google Scholar 

  9. Bujorianu, M.L., Lygeros, J.: Reachability Questions in Piecewise Deterministic Markov Processes. In: [18], pp. 126–140

    Google Scholar 

  10. Chaffin, M.S., Berry, J.D.: Navier-Stokes and Potential Theory Solutions for a Helicopter Fuselage. NASA TM-4566 ATCOM-TR-94-A-013 (1994)

    Google Scholar 

  11. Chen, Z.Q., Ma, Z.-M., Rockner, M.: Quasi-Homeomorphisms of Dirichlet Forms. Nagoya Math. J. 136, 1–15 (1994)

    MATH  MathSciNet  Google Scholar 

  12. Davis, M.H.A.: Markov Models and Optimization. Chapman & Hall, London (1993)

    MATH  Google Scholar 

  13. Dutertre, B.: Elements of Mathematical Analysis in PVS. TPHOLs 1996, Turku, Finland (1996)

    Google Scholar 

  14. Fukushima, M.: Dirichlet Forms and Markov Processes. N. Holland, Amsterdam (1980)

    MATH  Google Scholar 

  15. Hu, J., Lygeros, J., Sastry, S.: Towards a Theory of Stochastic Hybrid Systems. In: Lynch, N., Krogh, B. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 160–173. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  16. Iscoe, I., McDonald, D.: Induced Dirichlet Forms and Capacitary Inequalities. Ann. Prob. 18(3), 1195–1221 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  18. Maler, O., Pnueli, A. (eds.): Proceedings Hybrid Systems: Computation and Control, 6th International Workshop, HSCC 2003. LNCS, vol. 2623 (2003)

    Google Scholar 

  19. Meyer, P.A.: Probability and Potentials. Blaisdell, Waltham Mass (1966)

    MATH  Google Scholar 

  20. Meyer, P.A.: Renaissance, Recollectments, Mélanges, Ralentissement de Processus de Markov. Ann. Inst. Fourier 25, 465–497 (1975)

    MATH  Google Scholar 

  21. Pola, G., Bujorianu, M.L., Lygeros, J., Di Benedetto, M.D.: Stochastic Hybrid Models: An Overview with applications to Air Traffic Management. ADHS, Analysis and Design of Hybrid System, Saint-Malo (2003)

    Google Scholar 

  22. Rogers, L.C.G., Pitman, J.W.: Markov Functions. Ann. Prob. 9(4), 573–582 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  23. Soner, H.M., Touzi, N.: A stochastic Representation for the Level Set Equations. Comm. Partial Diff. Eq. 27(9-10), 2031–2053 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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Bujorianu, M.L. (2004). Extended Stochastic Hybrid Systems and Their Reachability Problem. In: Alur, R., Pappas, G.J. (eds) Hybrid Systems: Computation and Control. HSCC 2004. Lecture Notes in Computer Science, vol 2993. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24743-2_16

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  • DOI: https://doi.org/10.1007/978-3-540-24743-2_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-21259-1

  • Online ISBN: 978-3-540-24743-2

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