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Solving the Soft Convergence Problem for Controlled Oscillatory Systems Based on the Time Dilation Principle

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Cybernetics and Systems Analysis Aims and scope

Abstract

The authors consider a game problem of soft meeting of controlled oscillating systems, i.e., their simultaneous convergence in geometric coordinates and velocities. Applying Pontryagin’s first direct method [1] to solve this problem is noted to be impossible since the condition underlying this method is not satisfied. This condition is an instantaneous advantage of the pursuer (the one who strives to achieve this meeting) over the evader (the one who tries to avoid it). In the method, we apply the principle of time dilation, which weakens this condition and makes it possible to terminate the game in a finite time. The paper outlines the problem solution method that employs a certain time dilation function. Also, an algorithm, versions of constructing pursuer’s control, and an example of computer implementation of the process of convergence on the plane are provided.

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References

  1. R. F. Isaacs, Differential Games, Wiley Interscience, New York–London–Sydney (1965).

  2. G. Siouris, Missile Guidance and Control Systems, Springer, New York (2004). https://doi.org/https://doi.org/10.1007/b97614.

    Article  Google Scholar 

  3. A. Friedman, Differential Games, Courier Corporation (2013).

  4. Y. Yavin and M. Pachter, Pursuit-Evasion Differential Games, Elsevier (2014).

  5. L. S. Pontryagin, Selected Scientific Works, Vol 2, Differential Equations. Theory of Operators. Optimal Control. Differential Games [in Russian], Nauka, Moscow (1988).

  6. A. A. Chikrii, Conflict-Controlled Processes, Springer Sci. and Busines Media, Dordrecht–Boston–London (2013).

  7. D. Sonnevend, “On one type of player superiority,” Doklady AN SSSR, Vol. 208, No. 3, 520–523 (1973).

    MATH  Google Scholar 

  8. M. S. Nikol’skii, “Application of the first direct method in linear differential games,” Izv. AN SSSR, Ser. Tekhn. Kibern., No. 10, 51–56 (1972).

  9. G. Ts. Chikrii, “Using the effect of information delay in differential pursuit games,” Cybern. Syst. Analysis, Vol. 43, No. 2, 233–245 (2007). https://doi.org/https://doi.org/10.1007/s10559-007-0042-x.

    Article  MATH  Google Scholar 

  10. G. Ts. Chikrii, “Principle of time stretching in evolutionary games of approach,” J. Autom. Inform. Sci., Vol. 48, Iss. 5, 12–26 (2016). https://doi.org/10.1615/JAutomatInfScien.v48.i5.20.

  11. G. Ts. Chikrii, “Principle of time stretching for motion control in condition of conflict,” in: Advanced Control Systems: Theory and Applications, Ch. 3, River Publ. (2021), pp. 52–82.

  12. G. Ts. Chikrii, “Principle of time dilation in game problems of dynamics,” in: Recent Developments in Automatic Control Systems, Ch. 5, River Publ. (2023), pp. 113–129.

  13. G. Ts. Chikrii and A. O. Chikrii, “Time dilation principle in dynamic game problems,” Cybern. Syst. Analysis, Vol. 58, No. 1, 36–44 (2022). https://doi.org/10.1007/s10559-022-00433-6.

  14. G. Ts. Chikrii and V. M. Kuzmenko, “Implementation of convergence of oscillating systems based on the principle of time dilation,” Probl. Keruv. ta Inform., No. 1, 25–36 (2022). http://doi.org/https://doi.org/10.34229/1028-0979-2022-1-3.

  15. A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Publishers, Dordrecht–Boston (1988). https://doi.org/https://doi.org/10.1007/978-94-015-7793-9.

    Article  Google Scholar 

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Authors and Affiliations

  1. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    G. Ts. Chikrii & V. M. Kuzmenko

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  1. G. Ts. Chikrii
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  2. V. M. Kuzmenko
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Correspondence to G. Ts. Chikrii.

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Translated from Kibernetyka ta Systemnyi Analiz, No. 3, May–June, 2023, pp. 83–94

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Chikrii, G.T., Kuzmenko, V.M. Solving the Soft Convergence Problem for Controlled Oscillatory Systems Based on the Time Dilation Principle. Cybern Syst Anal 59, 428–438 (2023). https://doi.org/10.1007/s10559-023-00577-z

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