Towards a Theory of Stochastic Hybrid Systems

  • Jianghai Hu
  • John Lygeros
  • Shankar Sastry
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1790)


In this paper, we present a scheme of stochastic hybrid system which introduces randomness to the deterministic framework of the traditional hybrid systems by allowing the flow inside each invariant set of the discrete state variables to be governed by stochastic differential equation (SDE) rather than the deterministic ones. The notion of embedded Markov chains is proposed for such systems and some illustrative example from high way model is presented. As an important application, these ideas are then applied to the state space discretization of one dimensional SDE to obtain the natural discretized stochastic hybrid system together with its embedded MC. The invariant distribution and exit probability from interval of the MC are studied and it is shown that they converge to their counterparts for the solution process of the original SDE as the discretization step goes to zero. As a result, the discretized stochastic hybrid system provides a useful tool for studying various sample path properties of the SDE.


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  1. 1.
    E. Altman and V. Gaitsgory. Asymptotic optimization of a nonlinear hybrid system governed by a Markov decision process. SIAM Journal of Control and Optimization, 35(6):2070–2085, 1997.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Gopal K. Basak, Arnab Bisi, and Mrinal K. Ghosh. Ergodic control of random singular diffusions. In IEEE Conference on Decision and Control, pages 2545–2550, Kobe, Japan, 1996.Google Scholar
  3. 3.
    Richard Durrett. Probability: theory and examples, 2nd edition. Duxbury Press, 1996.Google Scholar
  4. 4.
    Richard Durrett. Stochastic calculus: A practical introduction. CRC Press, 1996.Google Scholar
  5. 5.
    J.A. Filar and V. Gaitsgory. Control of singularly perturbed hybrid stochastic systems. In IEEE Conference on Decision and Control, pages 511–516, Kobe, Japan, 1996.Google Scholar
  6. 6.
    John Lygeros. Hybrid systems: modeling, analysis and control. preprint, 1999.Google Scholar
  7. 7.
    Bernt Oksendal. Stochastic Differential Equations, an introduction with application. Fifth edition. Springer-Verlag, 1998.Google Scholar
  8. 8.
    L. Perko. Differential equation and dynamical systems, 2nd edition. Springer-Verlag, 1996.Google Scholar
  9. 9.
    E. Skafidas, R.J. Evans, and I.M. Mareels. Optimal controller switching for stochastic systems. In IEEE Conference on Decision and Control, pages 3950–3955, San Diego, CA, 1997.Google Scholar
  10. 10.
    Ching-Chih Tsai. Composite stabilization of singularly perturbed stochastic hybrid systems. International Journal of Control, 71(6):1005–1020, 1998.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ching-Chih Tsai and Abraham H. Haddad. Averaging, aggregation and optimal control of singulayly perturbed stochastic hybrid systems. International Journal of Control, 68(1):31–50, 1997.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Jianghai Hu
    • 1
  • John Lygeros
    • 1
  • Shankar Sastry
    • 1
  1. 1.Electrical Engineering and Computer SciencesUniversity of California, BerkeleyBerkeley

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