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## Archive of Issues

Russia Izhevsk
Year
2014
Issue
2
Pages
100-110
 Section Mathematics Title The characteristics of invariance of attainability set of control systems with random coefficients Author(-s) Hammady A.H.a Affiliations Udmurt State Universitya Abstract This article is continuation of works of L.I. Rodina and E.L. Tonkov in which expansion of the concept of invariance for sets concerning control systems and differential inclusions is entered. This expansion consists in research of the sets which are not invariant in “classical’’ sense, but possess the property of statistical invariance, and also in studying of statistical characteristics for attainability set of control system. We consider the characteristics connected with the invariance of the given set $\mathfrak M (\sigma)$ with respect to the control system which display the property of uniformity of stay for the attainability set of the system in $\mathfrak M (\sigma)$ on the finite time interval. We obtain estimates of these characteristics for systems with random coefficients in terms of Lyapunov functions, a derivative owing to differential inclusion and the dynamical system of shifts. In particular, we investigate the estimations with probability one for characteristics of control system which we will name a system with switchings. This system can be identified with a stationary random process whose set of states is finite; for this set there are given the initial probability distribution and the probabilities of finding in each state; the lengths of intervals between the moments of switching system from one state to another are random variables with a given distribution function. The example of estimation of the investigated characteristics for a linear control system with switchings is considered. Keywords control systems with stochastic coefficients, attainability set, dynamical system, differential inclusions UDC 517.935, 517.938 MSC 34A60, 37N35, 49J15 DOI 10.20537/vm140207 Received 29 March 2014 Language Russian Citation Hammady A.H. The characteristics of invariance of attainability set of control systems with random coefficients, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 2, pp. 100-110. References 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 (in Russian). Rodina L.I. The space ${\rm clcv}($ $\mathbb R$$n$ $)$ with the Hausdorff-Bebutov metric and statistically invariant sets of control systems, Proceedings of the Steklov Institute of Mathematics, 2012, vol. 278, issue 1, pp. 208-217. Rodina L.I. Estimation of statistical characteristics of attainability sets of controllable systems, Russian Mathematics, 2013, vol. 57, issue 11, pp. 17-27. Rodina L.I. Invariant and statistically weakly invariant sets of control systems, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2012, no. 2 (40), pp. 3-164 (in Russian). Kornfel'd I.P., Sinai Ya.G., Fomin S.V. Ergodicheskaya teoriya (The ergodic theory), Moscow: Nauka, 1980, 384 p. Panasenko E.A., Tonkov E.L. Invariant and stably invariant sets for differential inclusions, Proceedings of the Steklov Institute of Mathematics, 2008, vol. 262, issue 1, pp. 194-212. Clarke F. Optimization and nonsmooth analysis, Wiley, 1983. Translated under the title Optimizatsiya i negladkii analiz, Moscow: Nauka, 1988, 300 p. Rodina L.I. On some probability models of dynamics of population growth, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 4, pp. 109-124 (in Russian). Masterkov Yu. V., Rodina L. I. Sufficient conditions for the local controllability of systems with random parameters for an arbitrary number of system states, Russian Mathematics, 2008, vol. 52, issue 3, pp. 34-44. Shiryaev A.N. Veroyatnost' (Probability), Moscow: Nauka, 1989, 581 p. Korolyuk V.S., Portenko N.I., Skorokhod A.V., Turbin A.F. Spravochnik po teorii veroyatnostei i matematicheskoi statistike (Handbook of probability theory and mathematical statistics), Moscow: Nauka, 1985, 640 p. Full text