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## Archive of Issues

Russia Izhevsk; Yekaterinburg
Year
2012
Issue
4
Pages
68-79
 Section Mathematics Title About the attainability set of control system without assumption of compactness of geometrical restrictions on admissible controls Author(-s) Rodina L.I.a, Tonkov E.L.ab Affiliations Udmurt State Universitya, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesb Abstract We investigate the conditions under which the control system $\dot x=f(t,x,u),$ $u\in U (t,x)$ together with closure of set of shifts (concerning time $t$) of control system possesses property of uniform local or uniform global attainability on the given time interval. We do not suppose that function $(t,x)\to U(t,x),$ setting geometrical restrictions on admissible controls $u(t,x)\in U(t,x),$ has convex compact images and we do not suppose that differential inclusion corresponding to control system has convex images. Keywords statistically weakly invariant sets, controlled systems, attainability set, integral funnel, differential inclusion UDC 517.977.5 MSC 35F15, 37G10 DOI 10.20537/vm120406 Received 7 September 2012 Language Russian Citation Rodina L.I., Tonkov E.L. About the attainability set of control system without assumption of compactness of geometrical restrictions on admissible controls, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 4, pp. 68-79. References Anosov D.V. Lektsii po lineinoi algebre (Lectures on linear algebra), Moscow: Regular and Chaotic Dynamics, 1999, 105 p. Filippov A.F. Differential equations with discontinuous right-hand sides, Kluwer Academic Publishers, Dordrecht, 1988, 307 p. Zhukovskii E.S., Panasenko E.A. On one metric in the space of nonempty closed subsets of $\mathbb{R}$$n , Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 1, pp. 15–25. Panasenko E.A., Rodina L.I., Tonkov E.L. The space {\rm clcv}( \mathbb{R}$$n$ $)$ with the Hausdorff–Bebutov metric and differential inclusions, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 275, suppl. 1, pp. 121–136. Panasenko E.A. Dynamical system of translations in the space of multi-values functions with closed images, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 2, pp. 28–33. Bebutov M.V. Dynamical systems in the space of continuous function, Bull. Mat. Inst. Moscow State University, 1940, vol. 2, no. 5, pp. 1–52. Tonkov E.L. Globally controllable linear systems, Journal of Mathematical Sciences, 2006, vol. 139, issue 5, pp. 6976–6996. 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. Panasenko E.A., Rodina L.I., Tonkov E.L. Asymptotically stable statistically weakly invariant sets of control systems, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2010, vol. 16, no. 5, pp. 135–142. Rodina L.I. The statistically invariant sets of control systems with random parameters, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 2. pp. 68–87. Clarke F. Optimizatsiya i negladkii analiz (Optimization and the nonsmooth analysis), Moscow: Nauka, 1988, 300 p. Ioffe A.G., Tikhomirov V.M. Theory of extremal problems, North-Holland Publishing Company, Amsterdam–New York–Oxford, 1979, 460 p. Full text