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 Zh. Vychisl. Mat. Mat. Fiz., 2014, Volume 54, Number 6, Pages 928–941 (Mi zvmmf10046)

Constructive observability inequalities for weak generalized solutions of the wave equation with elastic restraint

A. A. Dryazhenkov, M. M. Potapov

Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119992, Russia

Abstract: Problems with one-sided boundary controls and a homogeneous Robin boundary condition set on the uncontrolled end are considered in the class of strong generalized solutions of the variable coefficient wave equation. In the adjoint class of weak generalized solutions of the dual problems with one-sided observations, new constructive observability inequalities are obtained that differ from previously known ones by an optimal threshold time. It is shown that, in the considered functional classes, the estimated constants degenerate as the time interval length approaches the threshold. Numerical illustrations are given showing that the stability of approximate solutions to control problems can be substantially enhanced by taking into account a priori information contained in the resulting observability inequalities.

Key words: wave equation, control problems, observation problems, threshold time, observability inequality, approximate solutions.

DOI: https://doi.org/10.7868/S0044466914060064

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English version:
Computational Mathematics and Mathematical Physics, 2014, 54:6, 939–952

Bibliographic databases:

UDC: 519.626
MSC: 93C20 (65M06 93B05 93B07 93B40)

Citation: A. A. Dryazhenkov, M. M. Potapov, “Constructive observability inequalities for weak generalized solutions of the wave equation with elastic restraint”, Zh. Vychisl. Mat. Mat. Fiz., 54:6 (2014), 928–941; Comput. Math. Math. Phys., 54:6 (2014), 939–952

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. A. Dryazhenkov, M. M. Potapov, “Numerical solution of the positional boundary control problem for the wave equation with unknown initial data”, Proc. Steklov Inst. Math. (Suppl.), 299, suppl. 1 (2017), 22–30
2. D. A. Ivanov, M. M. Potapov, “Approximations to time-optimal boundary controls for weak generalized solutions of the wave equation”, Comput. Math. Math. Phys., 57:4 (2017), 607–625
3. Andrey A. Dryazhenkov, Mikhail M. Potapov, “A stable method for linear equation in Banach spaces with smooth norms”, Ural Math. J., 4:2 (2018), 56–68
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