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 Trudy Inst. Mat. i Mekh. UrO RAN, 2012, Volume 18, Number 2, Pages 154–169 (Mi timm816)

Reconstruction of boundary controls in hyperbolic systems

A. I. Korotkiiab, E. I. Gribanovab

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
b Ural Federal University

Abstract: The inverse problem of dynamics consisting in reconstructing of a priori unknown controls generating the observed motion of dynamic system is considered. The dynamic system is defined by a boundary value problem for the equation with private derivatives of the hyperbolic type, controls are on a boundary of the object. The source information for solving the inverse problem is results of approximate measurements of the current phase positions of the observed systems motion. The problem is solved in static case, when to solve the problem, all the cumulative during the definite observation period totality of results of measurements is used. This problem is ill-posed and to solve this problem, Tikhonovs method with a stabilizer containing the sum of root mean square norm and total variation of control in time is used. This way provides not only the convergence of regularized approximations in Lebesgue spaces but piecewise uniform convergence. This permits numerical reconstruction of desired controls subtle structure. In this paper subgradient projection method of the obtaining minimizing sequence for the Tikhonovs functional is justified, two-stage finite-dimensional approximation of the problem is described. Results of numerical experiments are presented.

Keywords: dynamical system, boundary control, reconstruction, observation, measurement, inverse problem, regularization, Tikhonov's method, variation, piecewise uniform convergence.

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Document Type: Article
UDC: 517.9

Citation: A. I. Korotkii, E. I. Gribanova, “Reconstruction of boundary controls in hyperbolic systems”, Trudy Inst. Mat. i Mekh. UrO RAN, 18, no. 2, 2012, 154–169

Citation in format AMSBIB
\Bibitem{KorGri12} \by A.~I.~Korotkii, E.~I.~Gribanova \paper Reconstruction of boundary controls in hyperbolic systems \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2012 \vol 18 \issue 2 \pages 154--169 \mathnet{http://mi.mathnet.ru/timm816} \elib{http://elibrary.ru/item.asp?id=17736194} 

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This publication is cited in the following articles:
1. A. I. Korotkii, N. A. Artemova, N. A. Vaganova, O. O. Kovrizhnykh, L. I. Rubina, O. N. Ulyanov, O. V. Ushakova, M. Yu. Filimonov, I. A. Tsepelev, “O razrabotkakh analiticheskikh i chislennykh metodov resheniya zadach mekhaniki sploshnoi sredy”, Tr. IMM UrO RAN, 19, no. 2, 2013, 203–215
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