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Quadratic convergence algorithms for the problem of capture of a point by a family of convex bodies

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Abstract

We consider the nonlinear optimal control problem with an integral functional in which the integrand function is the characteristic function of a given closed set in the phase space. The approximation method is applied to prove the necessary conditions of optimality in the form of the Pontryagin maximum principle without any prior assumptions on the behavior of the optimal trajectory. Similarly to the case of phase-constrained problems, we derive conditions of nondegeneracy and pointwise nontriviality of the maximum principle.

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References

  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. Yu. N. Kiselev, Linear Theory of Time-Optimal Control with Perturbations [in Russian], Izd. MGU, Moscow (1986).

    Google Scholar 

  3. Yu. N. Kiselev, Optimal Control [in Russian], Izd. MGU, Moscow (1988).

    Google Scholar 

  4. L. W. Neustadt, “Synthesizing time-optimal control systems,” J. Math. Anal. Appl., 1, No. 3, 4, 484–493 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  5. J. O. Eaton, “An iterative solution to time optimal control,” J. Math. Anal. Appl., 5, No. 2, 329–344 (1962).

    Article  MATH  MathSciNet  Google Scholar 

  6. Yu. N. Kiselev, “Fast converging algorithms for the linear time-optimal control problem,” Kibernetika, No. 6, 47–57 (1990).

  7. S. N. Avakumov, Yu. N. Kiselev, and M. V. Orlov, “Methods of solving optimal control problems by the Pontryagin maximum principle,” Trudy MIAN im. V. A. Steklova, 211, 3–31 (1995).

    Google Scholar 

  8. Yu. N. Kiselev and M. V. Orlov, “Numerical algorithms for linear time-optimal control,” Zh. Vychisl. Matem. i Mat. Fiziki, No. 12, 1763–1771 (1991).

  9. B. N. Pshenichnyi, “Numerical method for computing time-optimal control for linear systems,” Zh. Vychisl. Matem. i Mat. Fiziki, No. 1, 52–60 (1964).

  10. T. Fujisawa and Y. Yasuda, “An iterative procedure for solving the time-optimal regulator problem,” J. SIAM Control, 5, No. 4, 501–512 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  11. V. I. Blagodatskikh, An Introduction to Optimal Control [in Russian], Vysshaya Shkola, Moscow (2001).

    Google Scholar 

  12. R. P. Fedorenko, Approximate Methods for Optimal Control Problems [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  13. N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods [in Russian], Nauka, Moscow (1987).

    MATH  Google Scholar 

  14. N. N. Kalitkin, Numerical Methods [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  15. A. V. Pogorelov, Differential Geometry [in Russian], 7th ed., Nauka, Moscow (1979).

    Google Scholar 

  16. Yu. N. Kiselev, “Approximation of a smooth convex compactum by osculating ellipsoids,” in: Proc. 9th Int. Conf. on Automatic Control “Avtomatika-2001” [in Russian], Vol. 1, Izd. Odessa Nat. Tech. Univ., Odessa (2001), pp. 31–32.

    Google Scholar 

  17. Yu. N. Kiselev, “Projecting a point on a convex compactum,” in: Proc. 9th Int. Conf. on Automatic Control “Avtomatika-2001” [in Russian], Vol. 1, Izd. Odessa Nat. Tech. Univ., Odessa (2001), pp. 30–31.

    Google Scholar 

  18. N. V. Khabarov, “Quadratic-convergence algorithms to project a point on a smooth convex compactum,” in: Abstracts of papers VZMSh (2001), pp. 269–270.

  19. N. V. Khabarov, “Quadratic-convergence algorithms for time-optimal control problem,” in: Abstracts of papers VVMSh (2001), pp. 164–165.

  20. N. V. Khabarov, “Algorithms to solve the time-optimal control problem by projecting the terminal state onto the reachability set,” in: Proc. XXIVth Conf. of Young Scientists of the Faculty of Mechanics and Mathematics of the Lomonosov University [in Russian], Izd. MGU, Moscow (2002), pp. 182–184.

    Google Scholar 

  21. N. V. Khabarov, “Algorithm to solve the time-optimal control problem using osculating ellipsoids,” in: Proc. Voronezh Spring Mathematical School “Pontryagin Readings-XIII” [in Russian], Izd. Voronezh Univ., Voronezh (2002), pp. 156–157.

    Google Scholar 

  22. Yu. N. Kiselev and N. V. Khabarov, “Quadratic convergence of the projection scheme in the problem of capture of a point by a family of complex bodies,” Issledovano v Rossii: Electronic Journal, 170, 2068–2077 (2003) [http://zhurnal.ape.relarn.ru/articles/2003/170.pdf].

    Google Scholar 

  23. Yu. N. Kiselev and N. V. Khabarov, “Osculating ellipsoids in algorithms to solve the linear time-optimal control problem,” in: Proc. 10th Int. Conf. on Automatic Control “Avtomatika-2003” [in Russian], Vol. 1, Izd. Sevastopol’ Nat. Tech. Univ., Sevastopol’ (2003), pp. 49–50.

    Google Scholar 

  24. N. V. Khabarov, A Method to Solve the Linear Optimal Control Problem with a “Difference Norm” Terminal Functional Using Osculating Ellipsoids [in Russian], Moscow (2003). Manuscript deposited in VINITI 20.10.2003, No. 1832-V2003.

  25. N. V. Khabarov, A Method to Solve the Linear Time-Optimal Control Problem Using Osculating Ellipsoids [in Russian], Moscow (2003). Manuscript deposited in VINITI 20.10.2003, No. 1834-V2003.

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Authors
  1. Yu. N. Kiselev
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  2. N. V. Khabarov
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Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 241–256, 2004.

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Kiselev, Y.N., Khabarov, N.V. Quadratic convergence algorithms for the problem of capture of a point by a family of convex bodies. Comput Math Model 19, 57–72 (2008). https://doi.org/10.1007/s10598-008-0006-7

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