Reconstruction of boundary controls in parabolic systems / Korotkii A. I.,Mikhailova D. O. // PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS. - 2013. - V. 280, l. 1. - P. 98-118.

ISSN/EISSN:

0081-5438 / 1531-8605

Type:

Article

Abstract:

In the paper, an inverse dynamic problem is considered. It consists in reconstructing a priori unknown boundary controls in dynamical systems described by boundary value problems for partial differential equations of parabolic type. The source information for solving the inverse problem is the results of approximate measurements of the states of the observed system's motion. The problem is solved in the static case; i.e., to solve it, we use all the measurement data accumulated during some specified observation interval. The problem under consideration is ill-posed. To solve it, we propose the Tikhonov method with a stabilizer containing the sum of the mean-square norm and total time variation of the control. The use of such nondifferentiable stabilizer allows us to obtain more precise results than the approximation of the desired control in the Lebesgue spaces. In particular, this method provides the pointwise and piecewise uniform convergences of regularized approximations and makes possible the numerical reconstruction of the subtle structure of the desired control. In the paper, the subgradient projection method for obtaining a minimizing sequence for the Tikhonov functional is described and substantiated. Also, we demonstrate the two-stage finitedimensional approximation of the problem and present the results of numerical simulation.

Author keywords:

dynamical system; control; reconstruction; observation; measurement; inverse problem; regularization; the Tikhonov method; variation; piecewise uniform convergence ILL-POSED PROBLEMS; NONSMOOTH SOLUTIONS; APPROXIMATION; REGULARIZATION

DOI:

10.1134/S0081543813020090

Web of Science ID:

ISI:000317236500009

Соавторы в МНС:

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Другие поля

Поле	Значение
Month	APR
Publisher	MAIK NAUKA/INTERPERIODICA/SPRINGER
Address	233 SPRING ST, NEW YORK, NY 10013-1578 USA
Language	English
EISSN	1531-8605
Keywords-Plus	ILL-POSED PROBLEMS; NONSMOOTH SOLUTIONS; APPROXIMATION; REGULARIZATION
Research-Areas	Mathematics
Web-of-Science-Categories	Mathematics, Applied; Mathematics
Author-Email	korotkii@imm.uran.ru darso@rambler.ru
Funding-Acknowledgement	Ural Branch of the Russian Academy of Sciences {[}09-P-1-1009]; Russian Foundation for Basic Research {[}11-01-00073]
Funding-Text	This work was supported by the Ural Branch of the Russian Academy of Sciences (project no. 09-P-1-1009) within the Program for Fundamental Research of the Presidium of the Russian Academy of Sciences ``Fundamental Problems of Nonlinear Dynamics{''} and by the Russian Foundation for Basic Research (project no. 11-01-00073).
Number-of-Cited-References	44
Usage-Count-Last-180-days	1
Usage-Count-Since-2013	15
Journal-ISO	Proc. Steklov Inst. Math.
Doc-Delivery-Number	121IL