A Toolbox of Hamilton-Jacobi Solvers for Analysis of Nondeterministic Continuous and Hybrid Systems

  • Ian M. Mitchell
  • Jeremy A. Templeton
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3414)

Abstract

Hamilton-Jacobi partial differential equations have many applications in the analysis of nondeterministic continuous and hybrid systems. Unfortunately, analytic solutions are seldom available and numerical approximation requires a great deal of programming infrastructure. In this paper we describe the first publicly available toolbox for approximating the solution of such equations, and discuss three examples of how these equations can be used in system analysis: cost to go, stochastic differential games, and stochastic hybrid systems. For each example we briefly summarize the relevant theory, describe the toolbox implementation, and provide results.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Isaacs, R.: Differential Games. John Wiley, Chichester (1967)MATHGoogle Scholar
  2. 2.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997)Google Scholar
  3. 3.
    Souganidis, P.E.: Two-player, zero-sum differential games and viscosity solutions. In: Bardi, M., Raghavan, T.E.S., Parthasarathy, T. (eds.) Stochastic and Differential Games: Theory and Numerical Methods. Annals of International Society of Dynamic Games, vol. 4, pp. 69–104. Birkhäuser (1999)Google Scholar
  4. 4.
  5. 5.
    Tomlin, C., Mitchell, I., Bayen, A., Oishi, M.: Computational techniques for the verification of hybrid systems. Proceedings of the IEEE 91, 986–1001 (2003)CrossRefGoogle Scholar
  6. 6.
    Mitchell, I., Bayen, A., Tomlin, C.J.: A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. Submitted to IEEE Transactions on Automatic Control (2004)Google Scholar
  7. 7.
    Mitchell, I.M.: A toolbox of level set methods. Technical Report TR-2004-09, Department of Computer Science, University of British Columbia, Vancouver, BC, Canada (2004)Google Scholar
  8. 8.
    Hespanha, J.P.: Stochastic hybrid systems: Application to communication networks. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 397–401. Springer, Heidelberg (2004)Google Scholar
  9. 9.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Heidelberg (2002)Google Scholar
  10. 10.
    Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society 27, 1–67 (1992)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Crandall, M.G., Evans, L.C., Lions, P.L.: Some properties of viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society 282, 487–502 (1984)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Øksendal, B.: Stochastic Differential Equations: an Introduction with Applications, 6th edn. Springer, Heidelberg (2003)Google Scholar
  13. 13.
    Mangel, M.: Decision and Control in Uncertain Resource Systems. Academic Press, Orlando (1985)MATHGoogle Scholar
  14. 14.
    Evans, L.C., Souganidis, P.E.: Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana University Mathematics Journal 33, 773–797 (1984)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Osher, S.: A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations. SIAM Journal of Mathematical Analysis 24, 1145–1152 (1993)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Athans, M., Falb, P.L.: Optimal Control. McGraw-Hill, New York (1966)MATHGoogle Scholar
  17. 17.
    Broucke, M., Benedetto, M.D.D., Gennaro, S.D., Sangiovanni-Vincentelli, A.: Optimal control using bisimulations: Implementation. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 175–188. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Tsai, Y.H.R., Cheng, L.T., Osher, S., Zhao, H.K.: Fast sweeping methods for a class of Hamilton-Jacobi equations. SIAM Journal on Numerical Analysis 41, 673–694 (2003)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sethian, J.A., Vladimirsky, A.: Ordered upwind methods for static Hamilton-Jacobi equations: Theory and algorithms. SIAM Journal on Numerical Analysis 41, 325–363 (2003)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Cardaliaguet, P., Quincampoix, M., Saint-Pierre, P.: Optimal times for constrained nonlinear control problems without local controllability. Applied Mathematics and Optimization 35, 1–22 (1997)MathSciNetGoogle Scholar
  21. 21.
    Falcone, M.: Numerical solution of dynamic programming equations. In: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser (1997) Appendix A of [2]Google Scholar
  22. 22.
    Ishii, H.: On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Communications on Pure and Applied Mathematics 42, 15–45 (1989)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Fleming, W.H., Souganidis, P.E.: On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana University Mathematics Journal 38, 293–313 (1989)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hespanha, J.P.: A model for stochastic hybrid systems with application to communication networks. Submitted to the International Journal of Hybrid Systems (2004)Google Scholar
  25. 25.
    Ghosh, M.K., Arapostathis, A., Marcus, S.I.: Ergodic control of switching diffusions. SIAM Journal of Control and Optimization 35, 1952–1988 (1997)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Filar, J.A., Gaitsgory, V., Haurie, A.B.: Control of singularly perturbed hybrid stochastic systems. IEEE Transactions on Automatic Control 46, 179–190 (2001)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Kushner, H.J., Dupuis, P.: Numerical Methods for Stochastic Control Problems in Continuous Time. Springer, Berlin (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ian M. Mitchell
    • 1
  • Jeremy A. Templeton
    • 2
  1. 1.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Mechanical EngineeringStanford UniversityStanford

Personalised recommendations