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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, Issue 11, Pages 54–61 (Mi vyuru46)  

This article is cited in 2 scientific papers (total in 2 papers)

Mathematical Modelling

On the Controllability of Linear Sobolev Type Equations with Relatively Sectorial Operator

O. A. Ruzakova, E. A. Oleynik

South Ural State University (Chelyabinsk, Russian Federation)

Abstract: $ \varepsilon $-controllability of linear first order differential equations not resolved with respect to the time derivative $L \dot{x} (t) = Mx (t) + Bu (t), \quad 0<t<T$ are studied. It is assumed that $\ker L \ne \{0 \}$ and the operator $M$ is strongly $(L, p)$-sectorial. These conditions guarantee the existence of an analytic semigroup in the sector of the resolution of the homogeneous equation $ L \dot{x} (t) = Mx (t) $. Using the theory of degenerate semigroups of operators with kernels the original equation is reduced to a system of two equations: regular, i.e. solved for the derivative (on the image of the semigroup of the homogeneous equation) and the singular (on the kernel of the semigroup) with a nilpotent operator at the derivative. Using the results of $\varepsilon$-controllability of the regular and singular equations, necessary and sufficient conditions of $\varepsilon $-controllability of the original equation of Sobolev type with respect to $p$-sectorial operator in terms of the operators are obtained. Abstract results are applied to the study of $\varepsilon$-controllability of a particular boundary-value problem, which is the linearization at zero phase–field equations describing the theory in the framework of mesoscopic phase transition.

Keywords: relatively $p$-sectorial operators, controllability.

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UDC: 517.9
Received: 15.11.2011

Citation: O. A. Ruzakova, E. A. Oleynik, “On the Controllability of Linear Sobolev Type Equations with Relatively Sectorial Operator”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 2012, no. 11, 54–61

Citation in format AMSBIB
\Bibitem{RuzOle12}
\by O.~A.~Ruzakova, E.~A.~Oleynik
\paper On the Controllability of Linear Sobolev Type Equations with Relatively Sectorial Operator
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2012
\issue 11
\pages 54--61
\mathnet{http://mi.mathnet.ru/vyuru46}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. O. A. Ruzakova, “Ob upravlyaemosti odnoi neklassicheskoi modeli matematicheskoi fiziki”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 6:4 (2014), 5–12  mathnet
    2. O. Tsyplenkova, “Optimal control of solutions to Cauchy problem for Sobolev type equation of higher order”, J. Comp. Eng. Math., 1:2 (2014), 62–67  mathnet  zmath  elib
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