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Optimal control and implicit Hamiltonian systems

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Book cover Nonlinear control in the Year 2000

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 258))

Abstract

Optimal control problems naturally lead, via the Maximum Principle, to implicit Hamiltonian systems. It is shown that symmetries of an optimal control problem lead to symmetries of the corresponding implicit Hamiltonian system. Using the reduction theory described in [3,2] one can reduce the system to a lower dimensional implicit Hamiltonian system. It is shown that for symmetries coming from the optimal control problem, doing reduction and applying the Maximum Principle commutes.

Furthermore, it is stressed that implicit Hamiltonian systems give rise to more general symmetries than only those coming from the corresponding optimal control problem. It is shown that corresponding to these symmetries, there exist conserved quantities, i.e. functions of the phase variables (that is, q and p) which are constant along solutions of the optimal control problem. See also [19,17].

Finally, the results are extended to the class of constrained optimal control problems, which are shown to also give rise to implicit Hamiltonian systems.

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References

  1. R. Abraham and J.E. Marsden. Foundations of Mechanics. Benjamin / Cummings Publishing Company, second edition, 1978.

    Google Scholar 

  2. G. Blankenstein and A.J. van der Schaft. Reduction of implicit Hamiltonian systems with symmetry. In Proceedings of the 5th European Control Conference, ECC'99, Karlsruhe, 1999.

    Google Scholar 

  3. G. Blankenstein and A.J. van der Schaft. Symmetry and reduction in implicit generalized Hamiltonian systems. Memorandum 1489, University of Twente, Faculty of Mathematical Sciences, June 1999. Accepted for publication in Reports on Mathematical Physics.

    Google Scholar 

  4. A.E. Bryson and Y.-C. Ho. Applied Optimal Control: Optimization, Estimation and Control. John Wiley, second edition, 1975.

    Google Scholar 

  5. T. Courant. Dirac manifolds. Trans. American Math. Soc., 319:631–661, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Dalsmo and A.J. van der Schaft. On representations and integrability of mathematical structures in energy-conserving physical systems. SIAM J. Cont. Opt., 37(1):54–91, 1999.

    Article  MATH  Google Scholar 

  7. I. Dorfman. Dirac Structures and Integrability of Nonlinear Evolution Equations. Chichester: John Wiley, 1993.

    Google Scholar 

  8. W.H. Fleming and R.W. Rishel. Deterministic and Stochastic Optimal Control. Springer-Verlag, 1975.

    Google Scholar 

  9. J.W. Grizzle. The Structure and Optimization of Nonlinear Control Systems possessing Symmetries. PhD thesis, University of Texas, 1983.

    Google Scholar 

  10. J.W. Grizzle and S.I. Marcus. Optimal Control of Systems Processing Symmetries. IEEE Trans. Automatic Control, 29(11):1037–1040, 1984.

    Article  MathSciNet  Google Scholar 

  11. M. Kummer. On the Constuction of the Reduced Phase Space of a Hamiltonian System with Symmetry. Indiana Univ. Math. J., 30(2):281–291, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  12. P. Libermann and C.-M. Marle. Symplectic Geometry and Analytical Mechanics. Reidel, Dordrecht, 1987.

    MATH  Google Scholar 

  13. J.E. Marsden. Lectures on Mechanics. Cambridge University Press, 1992.

    Google Scholar 

  14. J.E. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry. Rep. Math. Phys., 5:121–130, 1974.

    Article  MATH  MathSciNet  Google Scholar 

  15. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mischenko. The Mathematical Theory of Optimal Processes. Pergamon Press, 1964.

    Google Scholar 

  16. W.J. Satzer. Canonical Reduction of Mechanical Systems Invariant under Abelian Group Actions with an Application to Celestial Mechanics. Indiana Univ. Math. J., 26(5):951–976, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  17. H.J. Sussmann. Symmetries and Integrals of Motion in Optimal Control. In A. Fryszkowski, B. Jacubczyk, W. Respondek, and T. Rzezuchowski, editors, Geometry in Nonlinear Control and Differential Inclusions, pages 379–393. Banach Center Publications, Math. Inst. Polish Academy of Sciences, Warsaw, 1995.

    Google Scholar 

  18. H.J. Sussmann. An introduction to the coordinate-free Mamimum Principle. In B. Jacubczyk and W. Respondek, editors, Geometry of Feedback and Optimal Control, pages 463–557. Marcel Dekker, New York, 1997.

    Google Scholar 

  19. A.J. van der Schaft. Symmetries in optimal control. SIAM J. Control and Optimization, 25(2):245–259, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  20. A.J. van der Schaft. Implicit Hamiltonian Systems with Symmetry. Rep. Math. Phys., 41:203–221, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  21. A.J. van der Schaft and B.M Maschke. Interconnected mechanical systems, part I: geometry of interconnection and implicit Hamiltonian systems. In A. Astolfi, D.J.N Limebeer, C. Melchiorri, A. Tornambè, and R.B. Vinter, editors, Modelling and Control of Mechanical Systems, pages 1–15. Imperial College Press, 1997.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematical Sciences, Department of Systems, Signals and Control, University of Twente, P.O.Box 217, 7500 AE, Enschede, The Netherlands

    Guido Blankenstein & Arjan van der Schaft

Authors
  1. Guido Blankenstein
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  2. Arjan van der Schaft
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Editor information

Alberto Isidori (Professor)Françoise Lamnabhi-Lagarrigue (Docteur D’état)Witold Respondek (Professor)

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© 2001 Springer-Verlag London Limited

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Blankenstein, G., van der Schaft, A. (2001). Optimal control and implicit Hamiltonian systems. In: Isidori, A., Lamnabhi-Lagarrigue, F., Respondek, W. (eds) Nonlinear control in the Year 2000. Lecture Notes in Control and Information Sciences, vol 258. Springer, London. https://doi.org/10.1007/BFb0110216

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  • DOI: https://doi.org/10.1007/BFb0110216

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  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-363-8

  • Online ISBN: 978-1-84628-568-4

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