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Russia Izhevsk; Yekaterinburg
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 Sciences^{a}, Udmurt State University^{b} 
Abstract  We consider the socalled standard control systems. These are systems of differential equations defined on smooth manifolds of finite dimension that are uniformly continuous and timebound 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 nondegeneracy 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 omegalimit 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, omegalimit 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. 132145. 
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