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## Archive of Issues

Russia Izhevsk
Year
2012
Issue
2
Pages
34-43
 Section Mathematics Title Statistical characteristics of attainability set and periodic processes of control systems Author(-s) Rodina L.I.a Affiliations Udmurt State Universitya Abstract We investigate the statistical characteristics of attainability set $A(t,\sigma,X)$ of control system $$\dot x=f(h^t\sigma,x,u),\quad (t,\sigma,x,u)\in\mathbb R\times\Sigma\times\mathbb R^n\times\mathbb R^m, \qquad (1)$$ which is parametrized by means of topological dynamic system $(\Sigma,h^t).$ We obtained the lower estimations for such characteristics as the relative frequency of containing, the upper and lower relative frequencies of containing of attainability set of the system $(1)$ in the given set $M$ as well as new sufficient conditions of statistical invariance of the set $M$ with respect to control system. We received the conditions for system $(1)$ and set $X$ at which for given $\sigma\in\Sigma$ and $\varkappa_0\in (0,1]$ the relative frequency of containing of attainability set $A(t,\sigma,X)$ of systems $(1)$ in the set $M$ not less $\varkappa_0.$ Results of the work are illustrated by the example of control system which describes periodic processes in a chemical reactor. Keywords control systems, dynamical systems, differential inclusions, statistically invariant sets UDC 517.935, 517.938 MSC 34A60, 37N35, 49J15, 93B03 DOI 10.20537/vm120204 Received 30 March 2012 Language Russian Citation Rodina L.I. Statistical characteristics of attainability set and periodic processes of control systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 2, pp. 34-43. References Ushakov V.N., Malev A.G. On the question of the stability defect of sets in an approach game problem, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2010, vol. 16, no. 1, pp. 199-222. Ushakov V.N., Zimovets A.A. Invariance defect of sets with respect to differential inclusion, Vestn. Udmurt. Univ. Mat. Mekh. Komp. Nauki, 2011, no. 2, pp. 98-111. Rodina L.I., Tonkov E.L. Statistical characteristics of attainable set of controllable system, non-wandering, and minimal attraction center, Nelin. Dinam., 2009, vol. 5, no. 2, pp. 265-288. Rodina L.I., Tonkov E.L. The statistically weak invariant sets of control systems, Vestn. Udmurt. Univ. Mat. Mekh. Komp. Nauki, 2011, no. 1, pp. 67-86. Panasenko E.A., Rodina L.I., Tonkov E.L. Asymptotically stable statistically weakly invariant sets for controlled systems, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2010, vol. 16, no. pp. 135-142. Petrova V.V., Tonkov E.L. The admissibility of periodic processes and existence theorems for periodic solutions. I, Izv. Vyssh. Uchebn. Zaved. Mat., 1996, no. 11, pp. 65-72. Petrova V.V., Tonkov E.L. The admissibility of periodic processes and existence theorems for periodic solutions. II, Izv. Vyssh. Uchebn. Zaved. Mat., 1997, no. 6, pp. 17-24. Nemytskii V.V., Stepanov V.V. Kachestvennaya teoriya differentsial'nykh uravnenii (Qualitative theory of differential equations), Moscow: Gos. Izd. Tekh. Teor. Lit., 1949, 550 p. Rodina L.I. The statistically invariant sets of controllable systems with random parameters, Vestn. Udmurt. Univ. Mat. Mekh. Komp. Nauki, 2011, no. 2, pp. 68-87. Hartman Ph. Ordinary differential equations, New York-London-Sydney: John Wiley and Sons, 1964, 720 p. Translated under the title Obyknovennye differentsialnye uravneniya, Moscow: Mir, 1970, 720 p. Buntov S.D., Leonov N.I., Petrov N.N., Borisov A.V., Gryzlov A.A., Derr V.Ya., Karpov A.I., Kondratyev B.P., Popova S.N. Tonkov Evgeny Leonidovich. To seventieth anniversary, Vestn. Udmurt. Univ. Mat. Mekh. Komp. Nauki, 2010, no. 3, pp. 3-9. Full text