Ivan A. Finogenko     (Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of Russian Academy of Sciences, 134 Lermontova st., Irkutsk, 664033, Russian Federation)
Alexander N. Sesekin     (Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russia; Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)


Nonlinear control systems presented as differential inclusions with positional impulse controls are investigated. By such a control we mean some abstract operator with the Dirac function concentrated at each time. Such a control ("running impulse"), as a generalized function, has no meaning and is formalized as a sequence of correcting impulse actions on the system corresponding to a directed set of partitions of the control interval. The system responds to such control by discontinuous trajectories, which form a network of so-called "Euler's broken lines." If, as a result of each such correction, the phase point of the object under study is on some given manifold (hypersurface), then a slip-type effect is introduced into the motion of the system, and then the network of "Euler's broken lines" is called an impulse-sliding mode. The paper deals with the problem of approximating impulse-sliding modes using sequences of continuous delta-like functions. The research is based on Yosida's approximation of set-valued mappings and some well-known facts for ordinary differential equations with impulses.


Positional impulse control, Differential inclusion, Impulse-sliding mode.

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  1. Aubin J.-P., Ekeland I. Applied Nonlinear Analysis. NY: Willey & Sons Inc., 1984. 532 p.
  2. Barbashin E.A. Funkcii Liapunova [Lyapunov functions]. Moscow: Nauka, 1970. 240 p. (in Russian)
  3. Borisovich Yu.G., Gel’man B.D., Myshkis A.D., Obukhovskii V.V. Vvedenie v teoriyu mnogoznachnykh otobrazhenii i differencial’nykh vklyuchenii [Introduction to the Theory of Set-Valued Mappings and Differential Inclusions]. M.: URSS, 2005. 216 p. (in Russian)
  4. Dykhta V.A., Samsonyuk O.N. Optimal'noe impul’snoe upravlenie s prilozheniyami [Optimal impulse control with applications]. M.: FIZMATLIT, 2003. 256 p. (in Russian)
  5. Filippov A.F. Differential Equations with Discontinuous Righthand Sides. Math. Appl. Ser., vol. 18. Netherlands: Springer Science+Business Media, 1988. 304 p. DOI: 10.1007/978-94-015-7793-9
  6. Filippov A.F. On approximate computation of solutions of ordinary differential equations with discontinuous right hand sides. Vestnik Moskov. Univ. Ser. 15. Vychisl. Mat. Kibernet., 2001. No. 2. P. 18–20.
  7. Finogenko I.A. On the right Lipschitz condition for differential equations with piecewise continuous right-hand sides. Differ. Equ., 2003. Vol. 39, No. 8. P. 1124–1131. DOI: 10.1023/B:DIEQ.0000011286.59758.0d
  8. Finogenko I.A. On continuous approximations and right-sided solutions of differential equations with piecewise continuous right-hand sides. Differ. Equ., 2005. Vol. 41, No. 5. P. 677–686. DOI: 10.1007/s10625-005-0202-6
  9. Finogenko I.A., Ponomarev D.V. On differential inclusions with positional discontinuous and pulse controls. Trudy Inst. Mat. i Mekh. UrO RAN, 2013. Vol. 19, no. 1. P. 284–299. (in Russian)
  10. Finogenko I.A., Sesekin A.N. Impulse position control for differential inclusions. AIP Conf. Proc., 2018. Vol. 2048, No. 1. Art. no. 020008. DOI: 10.1063/1.5082026
  11. Finogenko I.A., Sesekin A.N. Positional impulse and discontinuous controls for differential inclusions. Ural Math. J., 2020. Vol. 6, No. 2. P. 68–75. DOI: 10.15826/umj.2020.2.007
  12. Kurzweil J. Generalized ordinary differential equations. Czechoslovak Math. J., 1958. Vol. 8, No. 3. P. 360–388.
  13. Miller B.M., Rubinovich E.Ya. Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations. Autom. Remote Control, 2013. Vol. 74. P. 1969–2006. DOI: 10.1134/S0005117913120047
  14. Zavalishchin S.T., Sesekin A.N. Dynamic Impulse Systems: Theory and Applications. Dordrecht: Kluwer Academic Publishers,  1997. 256 p. DOI: 10.1007/978-94-015-8893-5


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