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Construction of Reachable Sets of Two-Dimensional Nonlinear Controlled System By The Pixel Method

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We describe the pixel method for the construction of reachable sets in various optimal control problems using parallel computation technology. A program has been developed for the construction of reachable sets in nonlinear controlled systems by the pixel method. Sample cases have been examined with known analytical representations for the reachable set boundary. The sample library has been expanded. For problems with gravitational-type nonlinearity we analyze the comparison of the constructed approximation with the corresponding analytical representation. The approximation accuracy is shown to depend on the selection of input parameters.

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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], Nauka, Moscow (1983).

    Google Scholar 

  2. Yu. N. Kiselev, S. N. Avvakumov, and M. V. Orlov, Optimal Control. Linear Theory and Applications: A Textbook [in Russian], MAKS Press, Moscow (2007).

  3. A.V. Boreskov and A. A. Kharlamov, Application Principles for the CUDA Technology [in Russian], DMK-Press, Moscow (2010).

    Google Scholar 

  4. Kh. G. Guseinov, A. N. Moiseev, and V. N. Ushakov, “Approximation of reachable regions of controlled systems,” Prikl. Mat. Mekhan., 62, No. 2, 179–187 (1998).

    MathSciNet  Google Scholar 

  5. R. P. Fedorenko, Approximate Solution of Optimal Control Problems [in Russian], Nauka, Moscow (1978).

    MATH  Google Scholar 

  6. E. V. Vinnikov, “Numerical construction of reachable sets of nonlinear controlled systems in the Matlab environment,” in: Modern Methods in the Theory of Boundary-Value Problems: Proceeding of the Spring Mathematical School “Pontryagin Readings – XX” (2009), pp. 35–36.

  7. A. O. Novikova, “Construction of reachable sets by the pixel method for some controlled systems,” Sbornik Statei Molodykh Uchenykh Fakul’teta VMK MGU, No. 9, 136–153 (2012).

    Google Scholar 

  8. A. O. Novikova, “Computation and visualization of control systems reachable sets with parallel processing on graphics processing units,” in: 13 th Viennese Workshop on Optimal Control and Dynamics Games: Operations Research and Control Systems, TU Wien (2015), p. 57.

  9. E. B. Lee and L. Markus, Foundations of Optimal Control Theory [Russian translation], Nauka, Moscow (1972).

  10. Yu. N. Kiselev, “Construction of exact solutions for nonlinear time-optimal problems of special type,” Fundamental’naya i Prikladnaya Matematika, 3, No. 3, 847–868 (1997).

    MATH  Google Scholar 

  11. E. K. Kostousova and V. I. Pochinskii, “On problems of putting a carrier rocket in given elliptical orbits,” Trudy Inst. Mat. Mekhan. Uro RAN, No. 3, 201–216 (2011).

    Google Scholar 

  12. E. K. Kostousova and V. I. Pochinskii, “On problems of putting a payload at a given point of an orbit,” Avtomat. Telemekh., No. 4, 18–31 (2012).

    MathSciNet  Google Scholar 

  13. S. A. Ashmanov, Mathematical Models and Methods in Economics [in Russian], MGU, Moscow (1980).

    Google Scholar 

  14. S. A. Ashmanov, Introduction to Mathematical Economics [in Russian], Nauka, Moscow (1984).

    MATH  Google Scholar 

  15. F. L. Chernous’ko, “Ellipsoidal approximation of reachable sets of a linear system with an uncertain matrix,” Prikl. Matem. Mekhan., 60, No. 6, 940–950 (1996).

    MathSciNet  Google Scholar 

  16. Yu. N. Kiselev and M. V. Orlov, “Optimal resource allocation problem in a two-sector economic model with the Cobb–Douglas production function,” in: Modern Methods in the Theory of Boundary-Value Problems: Proceeding of the Spring Mathematical School “Pontryagin Readings – XX” (2009), pp. 85–86.

  17. R. Baier and E. Farkhi, “Optimization-based subdivision algorithm for reachable sets,” in: 13 th Viennese Workshop on Optimal Control and Dynamics Games: Operations Research and Control Systems, TU Wien (2015), p. 57.

  18. V. A. Skvortsov, Examples of Metric Spaces [in Russian], MTsN-MO, Moscow (2002).

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia

    A. O. Novikova

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  1. A. O. Novikova
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Correspondence to A. O. Novikova.

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Translated from Prikladnaya Matematika i Informatika, No. 50, 2015, pp. 62–82.

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Novikova, A.O. Construction of Reachable Sets of Two-Dimensional Nonlinear Controlled System By The Pixel Method. Comput Math Model 27, 444–459 (2016). https://doi.org/10.1007/s10598-016-9335-0

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  • DOI: https://doi.org/10.1007/s10598-016-9335-0

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