Transformation operators in controllability problems for the degenerate wave equation with variable coefficients
We study the control system $w_{tt} = \cfrac1{\rho } (kw_x) x + \gamma w,\; w(0, t) = u(t),\; x \in (0, l), t \in (0, T)$, in special modified spaces of the Sobolev type. Here, $\rho , k,$ and \gamma are given functions on $[0, l)$; $u \in L^{\infty} (0, T)$ is a control, and $T > 0$ is a...
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| Date: | 2018 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2018
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1623 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | We study the control system $w_{tt} = \cfrac1{\rho }
(kw_x) x + \gamma w,\; w(0, t) = u(t),\; x \in (0, l), t \in (0, T)$, in special modified spaces
of the Sobolev type. Here, $\rho , k,$ and \gamma are given functions on $[0, l)$; $u \in L^{\infty} (0, T)$ is a control, and $T > 0$ is a constant.
The functions $\rho$ and $k$ are positive on $[0, l)$ and may tend to zero or to infinity as $x \rightarrow l$. The growth of distributions from these spaces is determined by the growth of $\rho$ and $k$ as $x \rightarrow l$. Applying the method of transformation operators,
we establish necessary and sufficient conditions for the $L^{\infty}$ -controllability and approximate $L^{\infty}$ -controllability at a given
time and at a free time.
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