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Constructing Reachable Sets for Control Systems of Second Order of Accuracy with Respect to Time Step

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ABSTRACT

The paper investigates the pixel method for constructing reachable sets for a dynamic control system. Sufficient conditions for a control system have been obtained under which the explicit second order Runge–Kutta method (the modified Euler method) provides the second order of accuracy with respect to the time step when constructing reachable sets even if the class of admissible controls includes discontinuous functions.

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REFERENCES

  1. Sethi, S.P. and Thompson, G.L., Optimal Control Theory: Applications to Management Science and Economics, 2nd ed., Springer, 2005.

  2. Filkenstein, E.A. and Gornov, A.Yu., Algorithm for Quasi-Uniform Filling of Reachable Set of Nonlinear Control System, Izv. Irkutsk Gos. Univ. Ser. Mat., 2017, vol. 19, pp. 217–223.

  3. Gornov, A.Yu. and Filkenstein, E.A., Algorithm for Piecewise-Linear Approximation of the Reachable Set Boundary, Automat. Remote Control, 2015, vol. 76, no. 3, pp. 385–393.

  4. Ferretti, R., High-Order Approximations of Linear Control Systems via Runge–Kutta Schemes, Computing, 1997, vol. 58, no. 4, pp. 351–364.

  5. Veliov, V.M., Second Order Discrete Approximation to Linear Differential Inclusions, SIAM J. Num. An., 1992, vol. 29, no. 2, pp. 439–451.

  6. Guang, D.H. and Mingzhu, L., Input-to-State Stability of Runge–Kutta Methods for Nonlinear Control Systems, J. Comput. Appl. Math., 2007, vol. 205, no. 1, pp. 633–639.

  7. Baier, R., Selection Strategies for Set-Valued Runge–Kutta Methods, Proc. of the Numerical Analysis and Its Applications (NAA 2004), Lect. Notes. Comp. Sci., 2005, vol. 3401, pp. 149–157.

  8. Novikova, A.O., Construction of Reachable Sets for Two-Dimensional Nonlinear Control Systems by Pixel Method, Prikl. Mat. Informat., 2015, no. 50, pp. 62–82.

  9. Novikova, A.O., Computation and Visualization of Control Systems Reachable Sets with Parallel Processing on Graphics Processing Units, 13th Viennese Workshop on Optimal Control and Dynamic Games: Operations Research and Control Systems, TU Wien, 2015, pp. 57/58.

  10. Zorich, V.A., Matematicheskii analiz. Chast’ I (Mathematical Analysis. Part I), 6th ed., Moscow: MTsNMO, 2012.

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ACKNOWLEDGMENTS

The author thanks V.N. Ushakov for setting the problem and V.G. Pimenov for helpful discussions.

Funding

This work was supported by RFBR (project no. 18-01-00221) and by RF Ministry of Education and Science (Government Decree no. 211, contract no. 02.A03.21.0006).

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Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, 620990, Yekaterinburg, Russia

    A. A. Ershov

  2. Eltsin Ural Federal University, 620002, Yekaterinburg, Russia

    A. A. Ershov

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  1. A. A. Ershov
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Correspondence to A. A. Ershov.

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Ershov, A.A. Constructing Reachable Sets for Control Systems of Second Order of Accuracy with Respect to Time Step. Numer. Analys. Appl. 13, 306–320 (2020). https://doi.org/10.1134/S1995423920040023

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  • DOI: https://doi.org/10.1134/S1995423920040023

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