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On a Problem of Calculating the Solvability Set for a Linear System with Uncertainty

  • CONTROL THEORY
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Abstract

We consider a linear-convex control system defined by a set of differential equations with continuous matrix coefficients. The system may have control parameters, as well as uncertainties (interference) the possible values of which are subject to strict pointwise constraints. For this system, over a finite period of time, taking into account the constraints, we study the problem of guaranteed hitting the target set from a given initial position despite the effect of uncertainty. The main stage of solving the problem is the construction of an alternating integral and a solvability set. To construct the latter, the greatest computational complexity is the calculation of the geometric difference between the target set and the set determined by the uncertainty. A two-dimensional example of this problem is considered for which a method is proposed for finding the solvability set without the need to calculate the convex hull of the difference of the support functions of the sets.

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REFERENCES

  1. Pontryagin, L.S., On linear differential gamees. II, Dokl. Akad. Nauk SSSR, 1967, vol. 175, no. 4, pp. 910–912.

    Google Scholar 

  2. Pontryagin, L.S., Linear differential games of pursuit, Math. USSR-Sb., 1981, vol. 40, no. 3, pp. 285–303.

    Article  Google Scholar 

  3. Ukhobotov, V.I., On one control problem in the presence of uncertainty and possible breakdown, Tr. Inst. Mat. Mekh. UrO RAN, 2019, vol. 25, no. 3, pp. 265–278.

    Google Scholar 

  4. Kaplunova, E.P. and Tochilin, P.A., Target control problem for a quadrocopter while moving in a horizontal plane avoiding obstacles, Vestn. Mosk. Univ. Ser. 15. Vychisl. Mat. Kibern., 2021, no. 4, pp. 21–36.

  5. Krasovskii, N.N. and Subbotin, A.I., Pozitsionnye differentsial’nye igry (Positional Differential Games), Moscow: Nauka, 1974.

    Google Scholar 

  6. Kurzhanskii, A.B., Alternating Pontryagin integral in the theory of control synthesis, Proc. Steklov Inst. Math., 1999, vol. 224, pp. 212–225.

    Google Scholar 

  7. Fleming, W.H. and Soner, H.M., Controlled Markov Processes and Viscosity Solutions, New York: Springer, 1993.

    Google Scholar 

  8. Kurzhanski, A.B. and Váliy, I., Ellipsoidal Calculus for Estimation and Control, Boston: Birkhäuser, 1997.

  9. Polovinkin, E.C. and Balashov, M.V., Elementy vypuklogo i sil’no vypuklogo analiza (Elements of Convex and Strongly Convex Analysis), Moscow: Nauka, 2007.

    Google Scholar 

  10. Polovinkin, E.C., Mnogoznachnyi analiz i differentsial’nye vklyucheniya (Set-Valued Analysis and Differential Inclusions), Moscow: Fizmatlit, 2014.

    Google Scholar 

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Funding

This work was financially supported by the Russian Science Foundation, project 22-11-00042, https://rscf.ru/en/project/22-11-00042/.

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Authors and Affiliations

  1. Yandex.Technologies, Moscow, 119021, Russia

    A. A. Melnikova

  2. Lomonosov Moscow State University, Moscow, 119991, Russia

    P. A. Tochilin

  3. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 117997, Russia

    P. A. Tochilin

  4. Shenzhen MSU-BIT University, Shenzhen, Guangdong, 518172, China

    P. A. Tochilin

Authors
  1. A. A. Melnikova
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  2. P. A. Tochilin
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Corresponding authors

Correspondence to A. A. Melnikova or P. A. Tochilin.

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Translated by V. Potapchouck

CONFLICT OF INTEREST. The authors of this work declare that they have no conflicts of interest.

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Melnikova, A.A., Tochilin, P.A. On a Problem of Calculating the Solvability Set for a Linear System with Uncertainty. Diff Equat 59, 1538–1546 (2023). https://doi.org/10.1134/S00122661230110083

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  • DOI: https://doi.org/10.1134/S00122661230110083

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