Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis
In the paper, the control system wₜₜ =1/ρ(kwₓ)ₓ + γw, wₓ(0, t) = u(t), x > 0, t belongs (0, T), is considered in special modified spaces of Sobolev type Here ρ, k, and γ are given functions on [0, +∞); u belongs L∞(0, ∞) is a control; T > 0 is a constant. The growth of distributions from thes...
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| Datum: | 2016 |
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| Format: | Artikel |
| Sprache: | English |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2016
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| Schriftenreihe: | Журнал математической физики, анализа, геометрии |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/140546 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis / L.V. Fardigola // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 1. — С. 17-47. — Бібліогр.: 40 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | In the paper, the control system wₜₜ =1/ρ(kwₓ)ₓ + γw, wₓ(0, t) = u(t), x > 0, t belongs (0, T), is considered in special modified spaces of Sobolev type Here ρ, k, and γ are given functions on [0, +∞); u belongs L∞(0, ∞) is a control; T > 0 is a constant. The growth of distributions from these spaces depends on the growth of ρ and k. With the aid of some transformation operators, it is proved that the control system replicates the controllability properties of the auxiliary system zₜₜ = zξξ − q²z, zξ(0, t) = v(t), ξ > 0, t belongs (0, T), and vise versa. Here q ≥ 0 is a constant and v belongs L∞(0, ∞) is a control. For the main system, necessary and sufficient conditions of the L∞-controllability and the approximate L∞-controllability are obtained from those known for the auxiliary system. |
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