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Asymptotics of the optimal time in a time-optimal problem with two small parameters

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Abstract

A time-optimal problem of control of a small-mass point by a force of bounded magnitude in an unresisting medium is considered. An asymptotic expansion of the optimal time and optimal control is constructed with respect to two independent small parameters: the mass of the point and the perturbation of the initial conditions. It is shown that the asymptotics of the optimal time in this problem is complicated even for cases of general position.

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References

  1. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Wiley, New York, 1962).

    Google Scholar 

  2. N. N. Krasovskii, Theory of Motion Control (Nauka, Moscow, 1968) [in Russian].

    Google Scholar 

  3. E. B. Lee and L. Markus, Foundations of Optimal Control Theory (Wiley, New York, 1967; Nauka, Moscow, 1972).

    MATH  Google Scholar 

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

    Google Scholar 

  5. P. V. Kokotović and A. H. Haddad, “Controllability and time-optimal control of systems with slow and fast models,” IEEE Trans. Automat. Control 20(1), 111–113 (1975).

    Article  MATH  Google Scholar 

  6. A. B. Vasil’eva and M. G. Dmitriev, “Singular perturbations in optimal control problems,” J. Sov. Math. 34(3), 1579–1629 (1986).

    Article  MATH  Google Scholar 

  7. A. L. Dontchev, Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems (Springer, Berlin, 1983; Mir, Moscow, 1987).

    Book  MATH  Google Scholar 

  8. T. R. Gichev and A. L. Donchev, “Convergence of the solution of the linear singularly perturbed problem of time-optimal response,” J. Appl. Math. Mech. 43(3), 502–511 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  9. A. I. Kalinin and K. V. Semenov, “Asymptotic optimization method for linear singularly perturbed systems with multidimensional control,” Comp. Math. Math. Phys. 44(3), 407–417 (2004)

    MathSciNet  Google Scholar 

  10. A. R. Danilin and A. M. Il’in, “The asymptotics of the solution of a time-optimal problem with perturbed initial conditions,” J. Comput. Systems Sci. Internat. 33(6), 67–74 (1995).

    MathSciNet  Google Scholar 

  11. A. R. Danilin and A. M. Il’in, “On the structure of the solution of a perturbed time-optimality problem,” Fundam. Prikl. Mat. 4(3), 905–926 (1998).

    MATH  MathSciNet  Google Scholar 

  12. A. R. Danilin and O. O. Kovrizhnykh, “Time-optimal control of a small mass point without environmental resistance,” Dokl. Math. 88(1), 465–467 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  13. A. R. Danilin and O. O. Kovrizhnykh, “On the dependence of the time-optimal problem for a linear system on two parameters,” Vestn. Chelyab. Gos. Univ. 242, No. 27, 46–60 (2011).

    MathSciNet  Google Scholar 

  14. V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1974; Springer, New York, 1978).

    Google Scholar 

  15. A. R. Danilin and O. O. Kovrizhnykh, “Asymptotic representation of a solution to a singular perturbation linear time-optimal problem,” Proc. Steklov Inst. Math. 281(Suppl. 1), S22–S35 (2013).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    A. R. Danilin & O. O. Kovrizhnykh

  2. Institute of Mathematics and Computer Science, Ural Federal University, pr. Lenina 51, Yekaterinburg, 620000, Russia

    A. R. Danilin & O. O. Kovrizhnykh

Authors
  1. A. R. Danilin
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  2. O. O. Kovrizhnykh
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Correspondence to A. R. Danilin.

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Original Russian Text © A.R. Danilin, O.O. Kovrizhnykh, 2014, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2014, Vol. 20, No. 1.

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Danilin, A.R., Kovrizhnykh, O.O. Asymptotics of the optimal time in a time-optimal problem with two small parameters. Proc. Steklov Inst. Math. 288 (Suppl 1), 46–53 (2015). https://doi.org/10.1134/S0081543815020066

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  • DOI: https://doi.org/10.1134/S0081543815020066

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