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Spiral-Like Extremals near a Singular Surface in a Rocket Control Problem

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Abstract

In this paper, we consider the minimum time problem for a space rocket whose dynamics is given by a control-affine system with drift. The admissible control set is a disc. We study extremals in the neighbourhood of singular points of the second order. Our approach is based on applying the method of a descending system of Poisson brackets and the Zelikin – Borisov method for resolution of singularities to the Hamiltonian system of Pontryagin’s maximum principle. We show that in the neighbourhood of any singular point there is a family of spiral-like solutions of the Hamiltonian system that enter the singular point in a finite time, while the control performs an infinite number of rotations around the circle.

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Fig. 1
Fig. 2

Notes

  1. The parameter \(c\) can be chosen arbitrarily close to \(1\).

References

  1. Chukanov, S. V. and Milyutin A. A., Qualitative Study of Singularities for Extremals of Quadratic Optimal Control Problem, Russian J. Math. Phys., 1994, vol. 2, no. 1, pp. 31–48.

    MathSciNet  MATH  Google Scholar 

  2. Farkas, M., Periodic Motions, Appl. Math. Sci., vol. 104, New York: Springer, 1994.

    MATH  Google Scholar 

  3. Goh, B. S., Optimal Singular Rocket and Aircraft Trajectories, in 2008 Chinese Control and Decision Conference (Yantai, China, Jul 2008), pp. 1531–1536.

  4. Hartman, Ph., Ordinary Differential Equations, New York: Wiley, 1964.

    MATH  Google Scholar 

  5. Kelley, H. J., Singular Extremals in Lawden’s Problem of Optimal Rocket Flight, AIAA J., 1963, vol. 1, no. 7, pp. 1578–1580.

    Article  Google Scholar 

  6. Kupka, I. A. K., The Ubiquity of Fuller’s Phenomenon, in Nonlinear Controllability and Optimal Control, Monogr. Textbooks Pure Appl. Math., vol. 133, New York: Dekker, 1990, pp. 313–350.

    Google Scholar 

  7. Lawden, D. F., Optimal Trajectories for Space Navigation, London: Butterworths, 1963.

    MATH  Google Scholar 

  8. Manita, L. A. and Ronzhina, M. I., Optimal Spiral-Like Solutions near a Singular Extremal in a Two-Input Control Problem, Discrete Contin. Dyn. Syst. Ser. B, 2022, vol. 27, no. 6, pp. 3325–3343.

    Article  MathSciNet  MATH  Google Scholar 

  9. Park, C., Necessary Conditions for the Optimality of Singular Arcs of Spacecraft Trajectories Subject to Multiple Gravitational Bodies, Adv. Space Res., 2013, vol. 51, no. 11, pp. 2125–2135.

    Article  Google Scholar 

  10. Robbins, H. M., Optimality of Intermediate-Thrust Arcs of Rocket Trajectories, AIAA J., 1965, vol. 3, no. 6, pp. 1094–1098.

    Article  MathSciNet  MATH  Google Scholar 

  11. Ronzhina, M. I., Manita, L. A., and Lokutsievskiy, L. V., Solutions of a Hamiltonian System with Two-Dimensional Control in a Neighbourhood of a Singular Second-Order Extremal, Russian Math. Surveys, 2021, vol. 76, no. 5, pp. 936–938; see also: Uspekhi Mat. Nauk, 2021, vol. 76, no. 5(461), pp. 201-202.

    Article  MathSciNet  MATH  Google Scholar 

  12. Ronzhina, M. I., Manita, L. A., and Lokutsievskiy, L. V., Neighborhood of the Second-Order Singular Regime in Problems with Control in a Disk, Proc. Steklov Inst. Math., 2021, vol. 315, pp. 209–222; see also: Tr. Mat. Inst. Steklova, 2021, vol. 315, pp. 222-236.

    Article  MathSciNet  MATH  Google Scholar 

  13. Seywald, H. and Kumar, R. R., Singular Control in Minimum Time Spacecraft Reorientation, J. Guid. Control Dynam., 1993, vol. 16, no. 4, pp. 686–694.

    Article  Google Scholar 

  14. Shen, H. and Tsiotras, P., Time-Optimal Control of Axi-Symmetric Rigid Spacecraft Using Two Controls, J. Guid. Control Dynam., 1999, vol. 22, no. 5, pp. 682–694.

    Article  Google Scholar 

  15. Witt, E., Die Unterringe der freien Lieschen Ringe, Math. Z., 1956, vol. 64, pp. 195–216.

    Article  MathSciNet  MATH  Google Scholar 

  16. Zelikin, M. I. and Borisov, V. F., Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering, Boston: Birkhäuser, 1994.

    MATH  Google Scholar 

  17. Zelikin, M. I., Lokutsievskii, L. V., and Hildebrand, R., Typicality of Chaotic Fractal Behavior of Integral Vortices in Hamiltonian Systems with Discontinuous Right Hand Side, J. Math. Sci., 2017, vol. 221, no. 1, pp. 1–136.

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhu, J., Trélat, E., and Cerf, M., Planar Tilting Maneuver of a Spacecraft: Singular Arcs in the Minimum Time Problem and Chattering, Discrete Contin. Dyn. Syst. Ser. B, 2016, vol. 21, no. 4, pp. 1347–1388.

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhu, J., Trélat, E., and Cerf, M., Minimum Time Control of the Rocket Attitude Reorientation Associated with Orbit Dynamics, SIAM J. Control Optim., 2016, vol. 54, no. 1, pp. 391–422.

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhu, J., Trélat, E., and Cerf, M., Geometric Optimal Control and Applications to Aerospace, Pac. J. Math. Ind., 2017, vol. 9, Art. 8, 41 pp.

    Article  MathSciNet  MATH  Google Scholar 

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ACKNOWLEDGMENTS

The authors thank Prof. M. I. Zelikin and Prof. L. V. Lokutsievskiy for useful discussions. We are also grateful to the anonymous reviewers for useful comments that helped to improve the text.

Funding

This work is supported by the Russian Science Foundation (grant No. 20-11-20169).

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, 119991, Moscow, Russia

    Mariya I. Ronzhina & Larisa A. Manita

Authors
  1. Mariya I. Ronzhina
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  2. Larisa A. Manita
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Corresponding authors

Correspondence to Mariya I. Ronzhina or Larisa A. Manita.

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The authors declare that they have no conflicts of interest.

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MSC2010

49J15, 49N60, 34H05

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Ronzhina, M.I., Manita, L.A. Spiral-Like Extremals near a Singular Surface in a Rocket Control Problem. Regul. Chaot. Dyn. 28, 148–161 (2023). https://doi.org/10.1134/S1560354723020028

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