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Nelin. Dinam., 2009, Volume 5, Number 2, Pages 265–288 (Mi nd93)  

This article is cited in 22 scientific papers (total in 23 papers)

Statistical characteristics of attainable set of controllable system, non-wandering, and minimal attraction center

L. I. Rodina, E. L. Tonkov

Udmurt State University

Abstract: In the terms of Lyapunov functions we obtain the conditions that allow to estimate the relative frequency of occurrence of the attainable set of a controllable system in a given set $\mathfrak M$. The set $\mathfrak M$ is called statistically invariant if the relative frequency of occurrence in $\mathfrak M$ is equal to one. We also derive the conditions of the statistically weak invariance of $\mathfrak M$ with respect to controllable system, that is, for every initial point from $\mathfrak M$, at least one solution of the controllable system is statistically invariant. We obtain the conditions for the attainable set to be non-wandering as well as the conditions of existence of the minimal attraction center.

Keywords: controllable systems, dynamical systems, differential inclusions, attainability, invariance, non-wandering, recurrence.

Full text: PDF file (411 kB)
UDC: 517.911/517.93
MSC: 37N30, 37N35, 49J15, 93B03
Received: 07.11.2008

Citation: L. I. Rodina, E. L. Tonkov, “Statistical characteristics of attainable set of controllable system, non-wandering, and minimal attraction center”, Nelin. Dinam., 5:2 (2009), 265–288

Citation in format AMSBIB
\Bibitem{RodTon09}
\by L.~I.~Rodina, E.~L.~Tonkov
\paper Statistical characteristics of attainable set of controllable system, non-wandering, and minimal attraction center
\jour Nelin. Dinam.
\yr 2009
\vol 5
\issue 2
\pages 265--288
\mathnet{http://mi.mathnet.ru/nd93}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Rodina L.I., Tonkov E.L., “Statisticheski invariantnye mnozhestva upravlyaemoi sistemy”, Vestn. Tambovskogo un-ta. Ser.: Estestvennye i tekhnicheskie nauki, 14:4 (2009), 788–790  elib
    2. E. A. Panasenko, L. I. Rodina, E. L. Tonkov, “Asimptoticheski ustoichivye statisticheski slabo invariantnye mnozhestva upravlyaemykh sistem”, Tr. IMM UrO RAN, 16, no. 5, 2010, 135–142  mathnet  elib
    3. S. D. Buntov, N. I. Leonov, N. N. Petrov, A. V. Borisov, A. A. Gryzlov, V. Ya. Derr, A. I. Karpov, B. P. Kondratev, S. N. Popova, “K semidesyatiletiyu Evgeniya Leonidovicha Tonkova”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 3, 3–9  mathnet  elib
    4. E. A. Panasenko, L. I. Rodina, E. L. Tonkov, “The space $\mathrm{clcv}(\mathbb R^n)$ with the Hausdorff–Bebutov metric and differential inclusions”, Proc. Steklov Inst. Math. (Suppl.), 275, suppl. 1 (2011), S121–S136  mathnet  crossref  isi  elib
    5. L. I. Rodina, E. L. Tonkov, “Statisticheski slabo invariantnye mnozhestva upravlyaemykh sistem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2011, no. 1, 67–86  mathnet
    6. L. I. Rodina, “Statisticheski invariantnye mnozhestva upravlyaemykh sistem so sluchainymi parametrami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2011, no. 2, 68–87  mathnet
    7. V. N. Ushakov, A. R. Matviichuk, A. V. Ushakov, “Approksimatsiya mnozhestv dostizhimosti i integralnykh voronok differentsialnykh vklyuchenii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2011, no. 4, 23–39  mathnet
    8. L. I. Rodina, “Statisticheskie kharakteristiki mnozhestva dostizhimosti i periodicheskie protsessy upravlyaemykh sistem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2012, no. 2, 34–43  mathnet
    9. L. I. Rodina, “Statisticheskie kharakteristiki mnozhestva dostizhimosti upravlyaemoi sistemy”, Izv. IMI UdGU, 2012, no. 1(39), 111–113  mathnet
    10. L. I. Rodina, “Invariantnye i statisticheski slabo invariantnye mnozhestva upravlyaemykh sistem”, Izv. IMI UdGU, 2012, no. 2(40), 3–164  mathnet
    11. L. I. Rodina, “The space $\mathrm{clcv}(\mathbb R^n)$ with the Hausdorff–Bebutov metric and statistically invariant sets of control systems”, Proc. Steklov Inst. Math., 278 (2012), 208–217  mathnet  crossref  mathscinet  isi  elib  elib
    12. L. I. Rodina, E. L. Tonkov, “O mnozhestve dostizhimosti upravlyaemoi sistemy bez predpolozheniya kompaktnosti geometricheskikh ogranichenii na dopustimye upravleniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2012, no. 4, 68–79  mathnet
    13. A. V. Ushakov, “Ob odnom variante priblizhennogo postroeniya razreshayuschikh upravlenii v zadache o sblizhenii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2012, no. 4, 94–107  mathnet
    14. L. I. Rodina, A. Kh. Khammadi, “Kharakteristiki mnozhestva dostizhimosti, svyazannye s invariantnostyu upravlyaemoi sistemy na konechnom promezhutke vremeni”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 1, 35–48  mathnet
    15. L. I. Rodina, “Estimation of statistical characteristics of attainability sets of controllable systems”, Russian Math. (Iz. VUZ), 57:11 (2013), 17–27  mathnet  crossref
    16. Ya. Yu. Larina, L. I. Rodina, “Statisticheskie kharakteristiki upravlyaemykh sistem, voznikayuschie v razlichnykh modelyakh estestvoznaniya”, Model. i analiz inform. sistem, 20:5 (2013), 62–77  mathnet
    17. L. I. Rodina, “O nekotorykh veroyatnostnykh modelyakh dinamiki rosta populyatsii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 4, 109–124  mathnet
    18. Ya. Yu. Larina, “Statisticheskie kharakteristiki sistem s pochti periodicheskimi koeffitsientami”, Aktualnye napravleniya nauchnykh issledovanii XXI veka: teoriya i praktika, 2:2 (9) (2014), 417–420  elib
    19. Ya. Yu. Larina, “Statisticheskie kharakteristiki upravlyaemykh sistem s impulsnym vozdeistviem”, Izv. IMI UdGU, 2015, no. 2(46), 99–105  mathnet  elib
    20. A. Kh. Khammadi, “O svoistvakh kharakteristik mnozhestva dostizhimosti upravlyaemoi sistemy”, Izv. IMI UdGU, 2015, no. 2(46), 216–227  mathnet  elib
    21. Ya. Yu. Larina, “O slaboi asimptoticheskoi ustoichivosti upravlyaemykh sistem s impulsnym vozdeistviem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 26:1 (2016), 68–78  mathnet  crossref  mathscinet  elib
    22. L. I. Rodina, A. Kh. Khammadi, “Kharakteristiki invariantnosti mnozhestva dostizhimosti upravlyaemoi sistemy”, Izv. IMI UdGU, 2016, no. 1(47), 44–53  mathnet  elib
    23. Ya. Yu. Larina, L. I. Rodina, “Statistical characteristics of continuous functions and statistically weakly invariant sets of controllable system”, Russian Math. (Iz. VUZ), 61:2 (2017), 28–35  mathnet  crossref  isi
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