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Brachistochrone problem and two-dimensional Goddard problem

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Abstract

The motion of a point mass in a vertical plane under the action of gravity, linear viscous friction, support reaction of the curve, and the thrust is considered. The angle of inclination and the thrust are considered as control variables. The amount of fuel is indicated. The goal of the control is to maximize the horizontal coordinate of a point in a given time. The novelty of the paper lies in the fact that the structure of the optimal thrust is determined and the optimal synthesis is constructed in the three-dimensional space "slope angle-velocity-mass". For the case of a motion without friction, it is shown that the optimal thrust control takes boundary values, and the trajectory consists of two arcs, at the beginning with maximum thrust, and ending with zero thrust. The optimal synthesis in the three-dimensional space “slope angle-velocity-mass” is constructed for a specific area of the variables. For the case of linear viscous friction, an arc with an intermediate thrust can be included in an extreme trajectory. Assuming that the intermediate (singular) thrust satisfies the constraints, it is shown that the optimal thrust program consists of two arcs, maximum thrust at the beginning and zero thrust at the end, or three arcs: maximum thrust at the beginning, then intermediate thrust and zero thrust at the end. The following combination of arcs is also possible: zero thrust at the beginning, then an intermediate thrust and again zero thrust at the end. The logic of thrust control is similar to the well-known solution of the Goddard problem. The results of numerical simulation illustrating the theoretical conclusions are presented. The results are also valid for the Brachistochrone problem, which is interrelated to the range maximization problem. The results of numerical simulation illustrating the theoretical conclusions are presented.

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Acknowledgements

The paper was recommended for publication in International Journal of Nonlinear Dynamics and Chaos in Engineering Systems by the organizing committee of the DSTA 2021. The paper develops the research stated in the brief communication of the DSTA 2021 conference. The work was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation within the framework of the Center "Supersonic" program (agreement 075-15-2022-331 of April 26, 2022).

Funding

The paper was prepared within of the Program of creation and development of the world-class research center Sverhzvuk in 2020–2025 under financial support of the Ministry of Science and Higher Education of the Russian Federation (Order of the Government of the Russian Federation dated 24 October 2020 N 2744-p). The work was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation within the framework of the Center "Supersonic" program (agreement 075–15-2022–331 of April 26, 2022).

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    O. Yu Cherkasov, E. V. Malykh & N. V. Smirnova

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  1. O. Yu Cherkasov
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  2. E. V. Malykh
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  3. N. V. Smirnova
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Contributions

OYC: General management of the work, formulation of the problem, preparation of the final version of the article, obtaining theoretical results, derivation of formulas. NVS: Numerical modeling, production of drawings, preparation of the draft version of the article, obtaining theoretical results, derivation of formulas. EVM: Numerical modeling, production of drawings, preparation of the draft version of the article, obtaining theoretical results, derivation of formulas.

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Correspondence to O. Yu Cherkasov.

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All Authors Cherkasov, Smirnova and Malykh, declare they have no financial interests.

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Cherkasov, O.Y., Malykh, E.V. & Smirnova, N.V. Brachistochrone problem and two-dimensional Goddard problem. Nonlinear Dyn 111, 243–254 (2023). https://doi.org/10.1007/s11071-022-07857-x

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