Computational Mathematics and Modeling

, Volume 26, Issue 3, pp 299–335 | Cite as

Generalized Chaplygin Problem: Theoretical Analysis and Numerical Experiments

Mathematical Modeling
First Online:
  • 64 Downloads

We investigate a modified Chaplygin problem with flight path enclosing the maximum area. A twodimensional controlled plant with simple motion and control region in the form of a smooth planar convex compactum with interior point O should describe, in a given time, a closed curve enclosing a plane region of maxi-mum area. The initial and the final points of the trajectory coincide. The direction of the velocity vector at the initial time is given. The problem is solved by the Pontryagin maximum principle. A procedure to construct a programmed optimal control is described. An optimal control law is described in synthesis form (feedback form). The theoretical analysis relies on the apparatus of support and distance functions. The optimal trajectory can be derived from the polar of the control region by simple linear transformations: multiplication by a positive number, rotation through a right angle, and parallel translation. Numerical experiments have been carried out. The discussion is illustrated with graphs.

Keywords

optimal control Pontryagin maximum principle maximum-principle boundary-value problem programmed control feedback support and distance functions numerical experiments 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Fizmatgiz, Moscow (1961).Google Scholar
  2. 2.
    Yu. N. Kiselev, S. N. Avvakumov, and M. V. Orlov, Optimal Control. Linear Theory and Applications [in Russian], MAKS Press, Moscow (2007).Google Scholar
  3. 3.
    L. S. Pontryagin, Ordinary Differential Equations [in Russian], Nauka, Moscow (1961).Google Scholar
  4. 4.
    Yu. N. Kiselev, “Generalized Chaplygin’s problem,” in: Proc. Tenth Crimean Autumn Math. School, Vol. 10 (2000), pp. 160–163.Google Scholar
  5. 5.
    M. A. Lavrent’ev and L. A. Lyusternik, A Course in Variational Calculus [in Russian], GONTI-NKTI, Moscow-Leningrad (1938).Google Scholar
  6. 6.
    E. V. Mukhammetzhanov and I. A. Mochalov, The Fuzzy Chaplygin Problem, Nauka i Obrazovanie, e-journal (05 May 2012).Google Scholar
  7. 7.
    Yu. N. Kiselev, “Controlled systems with an integral invariant,” Diff. Uravn., 32, No. 4, 44–51 (1996).Google Scholar
  8. 8.
    Yu. N. Kiselev, “Construction of exact solutions for nonlinear time-optimal problems of special form,” Fund. Prikla. Mat., 3, No. 3, 847–868 (1997).MATHGoogle Scholar
  9. 9.
    Yu. N. Kiselev, “Solution methods for a linear time-optimal problem,” Trudy Mat. Inst. V. A. Steklova RAN, 185, 106–115 (1988).MATHGoogle Scholar
  10. 10.
    S. N. Avvakumov, Yu. N. Kiselev, and M. V. Orlov, “Methods for optimal control problems using the Pontryagin maximum principle,” Trudy Mat. Inst. V. A. Steklova RAN, 211, 3–31 (1995).MathSciNetGoogle Scholar
  11. 11.
    L. Cesari, Optimization – Theory and Applications, Springer, New York (1983).CrossRefMATHGoogle Scholar
  12. 12.
    Yu. N. Kiselev and S. N. Avvakumov, “Construction of an analytical solution of the Cauchy problem for a two-dimensional Hamiltonian system,” Trudy IMM UrO RAN, 19, No. 4, 131–141 (2013).MathSciNetGoogle Scholar
  13. 13.
    Yu. N. Kiselev, Dynamic Programming Method in Continuous Controlled Systems, Krymskaya Elektronnaya Biblioteka [http://libkruz.com/books/3062.html]
  14. 14.
    S. N. Avvakumov and Yu. N. Kiselev, “Some optimal control algorithms,” Trudy Inst. Matem. Mekhan., 12, No. 2, 3–17 (2006).MathSciNetGoogle Scholar
  15. 15.
    S. N. Avvakumov and Yu. N. Kiselev, “Support functions of some special sets, constructive smoothing procedures, and geometrical difference,” Probl. Dynam. Upravl., MAKS Press, Moscow, 1, 24–110 (2005).Google Scholar
  16. 16.
    Yu. N. Kiselev, Linear Theory of Time-Optimal Control with Perturbations [in Russian], MGU, Moscow (1986).Google Scholar
  17. 17.
    Yu. N. Kiselev, “Fast-converging algorithms for the linear time-optimal problem,” Kibernetika, No. 6, 47–57 (1990).Google Scholar
  18. 18.
    Yu. N. Kiselev and M. V. Orlov, “Numerical algorithms for linear time-optimal control,” Zh. Vychisl. Matem. i Mat. Fiz., 31, No. 12, 1763–1771 (1991).MATHMathSciNetGoogle Scholar
  19. 19.
    Yu. N. Kiselev and M. V. Orlov, “Potential method in linear time-optimal problem,” Diff. Uravn., 32, No. 1, 44–51 (1996).MathSciNetGoogle Scholar
  20. 20.
    Yu. N. Kiselev and M. V. Orlov, “Methods for the maximum-principle nonlinear boundary-value problem in the time-optimal problem,” Diff. Uravn., 31, No. 11, 1843–1850 (1995).MathSciNetGoogle Scholar
  21. 21.
    S. N. Avvakumov, “Smooth approximation of convex compacta,” Trudy Inst. Matem. Mekhan., 4, 184–200 (1996).MATHMathSciNetGoogle Scholar
  22. 22.
    Yu. N. Kiselev, “Construction of optimal feedback in smooth linear time-optimal problem,” Diff. Uravn., 26, No. 2, 232–237 (1990).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Faculty of Computational Mathematics and CyberneticsMoscow State UniversityMoscowRussia

Personalised recommendations