Nothing Special   »   [go: up one dir, main page]

Skip to main content
Log in

Multilevel Techniques for the Solution of HJB Minimum-Time Control Problems

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

The solution of minimum-time feedback optimal control problems is generally achieved using the dynamic programming approach, in which the value function must be computed on numerical grids with a very large number of points. Classical numerical strategies, such as value iteration (VI) or policy iteration (PI) methods, become very inefficient if the number of grid points is large. This is a strong limitation to their use in real-world applications. To address this problem, the authors present a novel multilevel framework, where classical VI and PI are embedded in a full-approximation storage (FAS) scheme. In fact, the authors will show that VI and PI have excellent smoothing properties, a fact that makes them very suitable for use in multilevel frameworks. Moreover, a new smoother is developed by accelerating VI using Anderson’s extrapolation technique. The effectiveness of our new scheme is demonstrated by several numerical experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Jurdjevic V, Geometric Control Theory, Cambridge University Press, Cambridge, 1996.

    Book  MATH  Google Scholar 

  2. Lee E B and Markus L, Foundations of Optimal Control Theory, Wiley, New York, 1967.

    MATH  Google Scholar 

  3. Sussmann H J and Willems J C, 300 years of optimal control: From the brachystochrone to the maximum principle, IEEE Control Systems, 1997, 17: 32–44.

    Article  Google Scholar 

  4. Bellman R, Dynamic Programming, Princeton University Press, Princeton, 1957.

    MATH  Google Scholar 

  5. Bardi M and Dolcetta I C, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Basel, 1997.

    Book  MATH  Google Scholar 

  6. Guo B and Wu T T, Numerical solution to optimal feedback control by dynamic programming approach: A local approximation algorithm, Journal of Systems Science and Complexity, 2017, 30(4): 782–802.

    Article  MathSciNet  MATH  Google Scholar 

  7. Chai L and Qian C, State feedback stabilization for a class of nonlinear time-delay systems via dynamic linear controllers, Journal of Systems Science and Complexity, 2014, 27(3): 453–462.

    Article  MathSciNet  MATH  Google Scholar 

  8. Falcone M and Ferretti R, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, Vol. 133, SIAM, 2014.

    MATH  Google Scholar 

  9. Howard R A, Dynamic programming, Manage. Sci., 1966, 12(5): 317–348.

    Google Scholar 

  10. Pollatschek M and Avi-Itzhak B, Algorithms for stochastic games with geometrical interpretation, Manage. Sci., 1969, 15(7): 399–415.

    Article  MathSciNet  MATH  Google Scholar 

  11. Puterman M L and Brumelle S L, On the convergence of policy iteration in stationary dynamic programming, Math. Oper. Res., 1979, 4(1): 60–69.

    Article  MathSciNet  MATH  Google Scholar 

  12. Alla A, Falcone M, and Kalise D, An efficient policy iteration algorithm for dynamic programming equations, SIAM J. Sci. Comput., 2015, 37(1): A181–A200.

    Article  MathSciNet  MATH  Google Scholar 

  13. Cacace S, Cristiani E, Falcone M, et al., A patchy dynamic programming scheme for a class of Hamilton-Jacobi-Bellman equations, SIAM J. Sci. Comput., 2012, 34(5): A2625–A2649.

    Article  MathSciNet  MATH  Google Scholar 

  14. Camilli F, Falcone M, Lanucara P, et al., A domain decomposition method for Bellman equations, Contemp. Math., 1994, 180: 477–477.

    Article  MathSciNet  MATH  Google Scholar 

  15. Festa A, Reconstruction of independent sub-domains for a class of Hamilton-Jacobi equations and application to parallel computing, ESAIM: Math. Model. Num., 2016, 50(4): 1223–1240.

    Article  MathSciNet  MATH  Google Scholar 

  16. Alla A, Haasdonk B, and Schmidt A, Feedback control of parametrized pdes via model order reduction and dynamic programming principle, Adv. Comput. Math., 2020, 46(1): 1–28.

    Article  MathSciNet  MATH  Google Scholar 

  17. Sun B, Tao Z Z, and Wang Y Y, Dynamic programming viscosity solution approach and its applications to optimal control problems, Mathematics Applied to Engineering, Modelling, and Social Issues, Springer, 2019, 363–420.

    Chapter  Google Scholar 

  18. Sun B and Guo B Z, Convergence of an upwind finite-difference scheme for Hamilton-Jacobi-Bellman equation in optimal control, IEEE Transactions on Automatic Control, 2015, 60(11): 3012–3017.

    Article  MathSciNet  MATH  Google Scholar 

  19. Alla A, Fabrini G, and Falcone M, Coupling MPC and DP methods for an efficient solution of optimal control problems, IFIP Conference on System Modeling and Optimization, Springer, 2015, 68–77.

    Google Scholar 

  20. Fabrini G, Falcone M, and Volkwein S, Coupling MPC and HJB for the computation of POD-based feedback laws, European Conference on Numerical Mathematics and Advanced Applications, Springer, 2017, 941–949.

    MATH  Google Scholar 

  21. Annunziato M, Borzi A, Nobile F, et al., On the connection between the Hamilton-Jacobi-Bellman and the Fokker-Planck control frameworks, J. Appl. Math, 2014, 5(16): 2476–2484.

    Article  Google Scholar 

  22. Anderson D, Iterative procedures for nonlinear integral equations, J. Assoc. Comput. Mach., 1965, 12: 547–560.

    Article  MathSciNet  MATH  Google Scholar 

  23. Walker H F and Ni P, Anderson acceleration for fixed-point iterations, SIAM J. Numer. Anal., 2011, 49(4): 1715–1735.

    Article  MathSciNet  MATH  Google Scholar 

  24. Walker H F, Anderson acceleration: Algorithms and implementations, WPI Math. Sciences Dept. Report MS-6-15-50, 2011.

    Google Scholar 

  25. Saad Y, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester, UK, 1992.

    MATH  Google Scholar 

  26. Toth A and Kelley C, Convergence analysis for Anderson acceleration, SIAM J. Numer. Anal., 2015, 53(2): 805–819.

    Article  MathSciNet  MATH  Google Scholar 

  27. Hoppe R H, Multi-grid methods for Hamilton-Jacobi-Bellman equations, Numer. Math., 1986, 49(2–3): 239–254.

    Article  MathSciNet  MATH  Google Scholar 

  28. Han D and Wan J W, Multigrid methods for second order Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Sci. Comput., 2013, 35(5): S323–S344.

    Article  MathSciNet  MATH  Google Scholar 

  29. Akian M and Detournay S, Multigrid methods for two-player zero-sum stochastic games, Numer. Linear Algebra Appl., 2012, 19(2): 313–342.

    Article  MathSciNet  MATH  Google Scholar 

  30. Hackbusch W, Multi-Grid Methods and Applications, Springer, New York, 2003.

    MATH  Google Scholar 

  31. Borzì A, Introduction to Multigrid Methods, Lecture Notes, 2011.

    Google Scholar 

  32. Fleming W and Rishel R, Deterministic and Sthochastic Optimal Control, Springer Verlag, New York, 1975.

    Book  MATH  Google Scholar 

  33. Bardi M and Falcone M, An approximation scheme for the minimum time function, SIAM J. Control Optim., 1990, 28(4): 950–965.

    Article  MathSciNet  MATH  Google Scholar 

  34. Falcone M, Numerical solution of dynamic programming equations, Optimal Control and Viscosity Solutions of HJB Equations, Birkhäuser, Berlin, 1997.

    Google Scholar 

  35. Santos M S and Rust J, Convergence properties of policy iteration, SIAM J. Control Optim., 2004, 42(6): 2094–2115.

    Article  MathSciNet  MATH  Google Scholar 

  36. Gander W, Gander M J, and Kwok F, Scientific Computing — An Introduction Using Maple and Matlab, Vol. 11, Springer Science & Business, 2014.

    Book  MATH  Google Scholar 

  37. Ulbrich M, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces, SIAM, Philadelphia, PA, 2011.

    Book  MATH  Google Scholar 

  38. Henson V E, Multigrid methods for nonlinear problems: An overview, Tech. rep., Lawrence Livermore National Lab., CA (US), 2002.

    Google Scholar 

  39. Borzì A, Ciaramella G, and Sprengel M, Formulation and Numerical Solution of Quantum Control Problems, SIAM, Philadelphia, PA, 2017.

    Book  MATH  Google Scholar 

  40. Chen Y, Hou T, and Yi N, Variational discretization for optimal control problems governed by parabolic equations, Journal of Systems Science and Complexity, 2013, 26(6): 902–924.

    Article  MathSciNet  MATH  Google Scholar 

  41. Tang Y and Chen Y, Variational discretization for parabolic optimal control problems with control constraints, Journal of Systems Science and Complexity, 2012, 25(5): 880–895.

    Article  MathSciNet  MATH  Google Scholar 

  42. Gubisch M and Volkwein S, Proper orthogonal decomposition for linear-quadratic optimal control, Model Reduction and Approximation: Theory and Algorithms, SIAM, Philadelphia, PA, 2017.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Gabriele Ciaramella or Giulia Fabrini.

Additional information

This paper was recommended for publication by Editor LIU Kang-Zhi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ciaramella, G., Fabrini, G. Multilevel Techniques for the Solution of HJB Minimum-Time Control Problems. J Syst Sci Complex 34, 2069–2091 (2021). https://doi.org/10.1007/s11424-021-0253-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-021-0253-7

Keywords