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## Archive of Issues

Russia Izhevsk
Year
2013
Issue
1
Pages
35-48
 Section Mathematics Title The characteristics of attainability set connected with invariancy of control systems on the finite time interval Author(-s) Rodina L.I.a, Hammady A.H.a Affiliations Udmurt State Universitya Abstract We study the statistical characteristics of the attainability set $A(t,\sigma,X)$ of the control system which is parametrized by means of a topological dynamical system $(\Sigma,h^t).$ We obtain the lower estimates for characteristics connected with invariance of given set on a finite time interval. We also consider the following problem arising in many applications. Let numbers $\lambda_0\in (0,1]$ and $\vartheta>0$ are given. It is necessary to find the conditions which the control system and set $X$ should satisfy providing that for given $\sigma\in\Sigma$ relative frequency of containing of the attainability set $A(t,\sigma, X)$ in the given set $M$ on any interval of time length $\vartheta$ would be not less than $\lambda_0.$ Let's notice, that the characteristic $\vartheta$ is assumed given depending on an applying problems. In particular, if control process is periodic, then $\vartheta$ is the period of the process. Results are illustrated by examples of the control systems which describe different models of population growth. Keywords control systems, dynamical systems, differential inclusions, statistically invariant sets UDC 517.935, 517.938 MSC 34A60, 37N35, 49J15, 93B03 DOI 10.20537/vm130105 Received 12 November 2012 Language Russian Citation Rodina L.I., Hammady A.H. The characteristics of attainability set connected with invariancy of control systems on the finite time interval, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 1, pp. 35-48. References Rodina L.I., Tonkov E.L. Statistical characteristics of attainable set of control 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'yut. Nauki, 2011, no. 1, pp. 67–86. Rodina L.I. Statistical characteristics of attainable set and periodic processes of control systems, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 2, pp. 34–43. Rodina L.I. The space ${\rm clcv}($ $\mathbb{R}$$n ) with the Hausdorff–Bebutov metric and statistically invariant sets of control systems, Tr. Mat. Inst. Steklova, 2012, vol. 278, pp. 217–226. Rodina L.I. Invariant and statistically weakly invariant sets of controllable systems, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2012, no. 2 (40), pp. 3–164. Davydov A.A., Pastres R., Petrenko I.A. Optimization of the spatial distribution of pollution emission in 1D flow, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2010, vol. 16, no. 5, pp. 30–35. Dmitruk A.V. Principle of maximum for the general problem of optimum control with the phase and regular mixed restrictions, Optimality of operated dynamic systems: Transactions, Moscow: Institute of Systems Analysis, Russian Academy of Sciences, 1990, no. 14, pp. 26–42. Nemytskii V.V., Stepanov V.V. Kachestvennaya teoriya differentsial’nykh uravnenii (Qualitative theory of differential equations), Moscow: Gos. Izd. Tekh. Teor. Lit., 1949, 550 p. Panasenko E.A., Rodina L.I., Tonkov E.L. The space {\rm clcv}( \mathbb{R}$$n$ $)$ with the Hausdorff–Bebutov metric and differential inclusions, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 275, suppl. 1, pp. 121–136. Clarke F. Optimizatsiya i negladkii analiz (Optimization and the nonsmooth analysis),Moscow: Nauka, 1988, 300 p. Hartman Ph. Ordinary differential equations, New York–London–Sydney: John Wiley and Sons, 1964. Translated under the title Obyknovennye differentsialnye uravneniya, Moscow: Mir, 1970, 720 p. Kuzenkov O.A., Ryabova E.A. Matematicheskoe modelirovanie protsessov otbora (Mathematical modelling of processes of selection), Nizhni Novgorod: Nizhni Novgorod State University, 2007, 324 p. Modeli prirodnykh sistem (Models of natural systems), Ed.: Gurman V.I., Druzhinina I.P., Novosibirsk: Nauka, 1978, 224 p. Arnol’d V.I., Avets A. Ergodicheskie problemy klassicheskoi mekhaniki (Ergodic problems of classical mechanics), Izhevsk: Regular & Chaotic Dynamics, 1999, 284 p. Kornfel’d I.P., Sinai Ya.G., Fomin S.V. Ergodicheskaya teoria (The ergodic theory), Moscow: Nauka, 1980, 384 p. Katok A.B., Hasselblat B. Vvedenie v sovremennuyu teoriyu dinamicheskikh sistem (Introduction to the modern theory of dynamical systems), Moscow: Faktorial, 1999, 768 p. Perov A.I. Some remarks on differential inequalities, Izv. Vyssh. Uchebn. Zaved. Mat., 1965, no. 4, pp. 104–112. Full text