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This article is cited in 1 scientific paper (total in 1 paper)
Transformation operators and modified Sobolev spaces in controllability problems on a half-axis
L. V. Fardigola B. Verkin Institute for Low Temperature Physics and Engineering,
National Academy of Sciences of Ukraine, 47 Nauki Ave., Kharkiv, 61103, Ukraine
Abstract:
In the paper, the control system $ w_{tt}=\frac1\rho(k w_x){}_x+\gamma w$, $w_x(0,t)=u(t)$, $x>0$, $t\in(0,T)$, is considered in special modified spaces of Sobolev type. Here $\rho$, $k$, and $\gamma$ are given functions on $[0,+\infty)$; $u\in L^\infty(0,\infty)$ is a control; $T>0$ is a constant. The growth of distributions from these spaces depends on the growth of $\rho$ and $k$. With the aid of some transformation operators, it is proved that the control system replicates the controllability properties of the auxiliary system $ z_{tt}=z_{\xi\xi}-q^2z$, $z_\xi(0,t)=v(t)$, $\xi>0$, $t\in(0,T)$, and vise versa. Here $q\ge0$ is a constant and $v\in L^\infty(0,\infty)$ is a control. For the main system, necessary and sufficient conditions of the $L^\infty$-controllability and the approximate $L^\infty$-controllability are obtained from those known for the auxiliary system.
Key words and phrases:
wave equation, half-axis, controllability problem, transformation operator, modified space of Sobolev type.
Received: 20.09.2015
Citation:
L. V. Fardigola, “Transformation operators and modified Sobolev spaces in controllability problems on a half-axis”, Zh. Mat. Fiz. Anal. Geom., 12:1 (2016), 17–47
Linking options:
https://www.mathnet.ru/eng/jmag627 https://www.mathnet.ru/eng/jmag/v12/i1/p17
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