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A Hamiltonian Pertubation Approach to Construction of Geometric Integrators for Optimal Control Problems

  • M. D. S. AliyuEmail author
Short Communication
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Abstract

In this paper, we discuss computational methods for optimal control problems which preserve some geometric properties of the system. A Hamiltonian perturbation method is developed, which does not involve any discretization of either the cost function, or the dynamic equations. Instead, the approach relies mainly on an iterative successive approximation of the value-function, which converges uniformly to the solution. The approach is most effective when the Hamiltonian of the system is a polynomial function of the phase coordinates, and an example is presented to demonstrate this.

Keywords

Optimal control Hamiltonian system Symplectic manifold Pontryagin’s minimum principle Generating function Hamilton–Jacobi equation Geometric integrator 

Notes

References

  1. 1.
    Abraham, R., Marsden, J.: Foundations of Mechanics. Benjamin Cummings Publishing Co., San Francisco (1978)zbMATHGoogle Scholar
  2. 2.
    Ariadna, F.: High precision symplectic integrators for the solar system. Celest. Mech. Dyn. Astron. 116(2), 141–174 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1985)Google Scholar
  4. 4.
    Athans, M., Falb, P.L.: Optimal Control: An Introduction to the Theory and Its Applications. Dover Academic Publishers, New York (2007)Google Scholar
  5. 5.
    Barbero-Linan, M., De León, M., De Diego, D.M.: Lagrangian submanifolds and the Hamilton–Jacobi equation. Monatsh Math. 171, 269–290 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blanes, S., Casas, F., Ros, S.: Processing symplectic methods for near-integrable Hamiltonian systems. Celest. Mech. Dyn. Astron. 77, 17–35 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Blanes, S., Casas, F., Ros, J.: Symplectic integration with processing: a general study. SIAM J. Sci. Comput. 21, 711–727 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chin, S.A., Chen, C.R.: Forward symplectic integrators for solving gravitational few-body problems. Celest. Mech. Dyn. Astron. 91(3), 301–322 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chyba, M., Hairer, E., Vilmart, G.: The role of symplectic integrators in optimal control. Opt. Control Appl. Methods 30, 367–382 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    De León, M., De Diego, D.M., Merino, A.S.: Geometric numerical integration of nonholonomic systems and optimal control problems. In: Proceedings IFAC Conference on Lagrangian and Hamiltonian Methods for Nonlinear Control, Seville, Spain, vol. 36, no. 2, pp. 141–146 (2003)Google Scholar
  11. 11.
    De León, M., De Diego, D.M., Merino, A.S.: Geometric numerical integration of nonholonomic systems and optimal control problems. Eur. J. Control 10(5), 515–521 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    De León, M., De Diego, D.M., Merino, A.S.: Geometric integrators and nonholonomic mechanics. J. Math. Phys. 45(3), 1042–1064 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    De León, M., De Diego, D.M.: Variational integrators and time-dependent Lagrangian systems. Rep. Math. Phys. 49(2/3), 183–192 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    De León, M., De Diego, D.M., Merino, A.S.: Discrete variational integrators and optimal control theory. Adv. Comput. Maths 26(1–3), 251–268 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Evans, L.C.: Partial Differential Equations. Graduate Text in Maths. AMS, Providence (1998)Google Scholar
  16. 16.
    Ferraro, S., Jimenez, F., De Diego, D.M.: New developments on the geometric nonholonomic integrator. Nonlineraity 28(4), 871–900 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Forest, E., Ruth, R.D.: Fourth-order symplectic integration. Physica D 43, 105–117 (1990)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fritz, J.: Partial Differential Equations. Springer-Verlag Applied Mathematical Sciences Series, 3rd edn. Springer, New York (1978)zbMATHGoogle Scholar
  19. 19.
    Goldstein, H.: Classical Mechanics. Addison-Wesley, Boston (1950)zbMATHGoogle Scholar
  20. 20.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  21. 21.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numer. 12, 399–450 (2003)MathSciNetCrossRefGoogle Scholar
  22. 22.
    McLachlan, R.I., Atela, P.: The accuracy of symplectic integrators. Nonlinearity 5, 541–562 (1992)MathSciNetCrossRefGoogle Scholar
  23. 23.
    McLachlan, R.I.: On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16, 151–168 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Murua, A.: On order conditions for partitioned symplectic methods. SIAM J. Numer. Anal. 34(6), 2204–2211 (1997)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kirk, D.: Optimal Control Theory: An Introduction. Prentice Hall, Upper Saddle River (1972)Google Scholar
  26. 26.
    Sofroniou, M., Spaletta, G.: Symplectic Methods for Separable Hamiltonian Systems. In: Sloot, P.M.A. et al. (eds.)ICCS, LNCS 2331, pp. 506–515. Springer, Berlin (2002)zbMATHGoogle Scholar
  27. 27.
    Yoshida, H.: Symplectic integrators for Hamiltonian systems: basic theory. In: Sylvio F-.M. (ed.) Chaos, Resonance, and Collective Dynamical Phenomena in the Solar System: Proceedings of 152nd Symposium of the International Astronomical Union, Angra dos Reis, Brazil, 15–19 July, 1991. Kluwer Academic Publishers, Dordrecht, pp. 407–411 (1992)Google Scholar

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© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of Electrical EngineeringKing Faisal UniversityAl-AhsaSaudi Arabia

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