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## Archive of Issues

Russia Izhevsk; Yekaterinburg
Year
2013
Issue
4
Pages
132-145
 Section Mathematics Title Turnpike processes of control systems on smooth manifolds Author(-s) Tonkov E.L.ab Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Udmurt State Universityb Abstract We consider the so-called standard control systems. These are systems of differential equations defined on smooth manifolds of finite dimension that are uniformly continuous and time-bound on the real axis and locally Lipschitz in the phase variables. In addition, we assume that the compact set is given, which defines geometric constraints on the admissible controls and moreover, the non-degeneracy condition holds. This condition means that for each point of the phase manifold and for all times there exists a control such that the value of vector field is contained in the Euclidean space that is tangent to the phase manifold at a given point. Using the modified method of the Lyapunov function and constructing omega-limit set of the corresponding dynamical system of shifts, we give propositions about the existence of admissible control processes that are bounded on the positive semiaxis, and the assertion of uniform local controllability of the corresponding turnpike process. Keywords turnpike processes, manifolds of finite dimension, uniform local controllability, omega-limit sets, Lyapunov functions UDC 515.163.1, 517.977.1 MSC 34A26, 34H05, 34A60 DOI 10.20537/vm130413 Received 30 November 2013 Language Russian Citation Tonkov E.L. Turnpike processes of control systems on smooth manifolds, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 4, pp. 132-145. References Panasyuk A.I., Panasyuk V.I. Asimptoticheskaya magistral'naya optimizatsiya upravlyaemykh system (Asymptotic turnpike optimization of control systems), Minsk: Nauka i Tekhnika, 1986, 296 p. Anosov D.V. Lektsii po lineinoi algebre (Lectures on linear algebra), Moscow: Regular and Chaotic Dynamics, 1999, 105 p. Engelking R. General topology, Warsaw: PWN, 1977. Translated under the title Obshchaya topologiya, Moscow: Mir, 1986, 751 p. Dubrovin B.A., Novikov S.P., Fomenko A.T. Sovremennaya geometriya (Modern geometry - methods and applications), Moscow: URSS, 1986, 759 p. Agrachev A.A., Sachkov Yu.L. Control theory from the geometric viewpoint, Berlin: Springer-Verlag, 2004. Arnold V.I. Ordinary differential equations, Berlin-Heidelberg-New York: Springer-Verlag, 1992. 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. 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. Skvortsov V.A. Borel set, Mathematical encyclopedia, Moscow: Soviet Encyclopedia, 1977, vol. 1, p. 535. Filippov A.F. Differential equations with discontinuous righthand sides, Dordrecht: Kluwer Academic Publishers, 1988. Full text