Modified Sobolev Spaces in Controllability Problems for the Wave Equation on a Half-Plane
The 2-d wave equation wtt = Δw, t belongs (0, T), on the half-plane x1 > 0 controlled by the Neumann boundary condition wx1(0, x2, t) = δ(x2)u(t) is considered in Sobolev spaces, where T > 0 is a constant and u L∞(0, T) is a control. This control system is transformed into a control system f...
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| Datum: | 2015 |
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| Format: | Artikel |
| Sprache: | English |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2015
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| Schriftenreihe: | Журнал математической физики, анализа, геометрии |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/117982 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Modified Sobolev Spaces in Controllability Problems for the Wave Equation on a Half-Plane / L.V. Fardigola // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 1. — С. 18-44— Бібліогр.: 19 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | The 2-d wave equation wtt = Δw, t belongs (0, T), on the half-plane x1 > 0 controlled by the Neumann boundary condition wx1(0, x2, t) = δ(x2)u(t) is considered in Sobolev spaces, where T > 0 is a constant and u L∞(0, T) is a control. This control system is transformed into a control system for the 1-d wave equation in modified Sobolev spaces introduced and studied in the paper, and they play the main role in the study. The necessary and sufficient conditions of (approximate) L∞-controllability are obtained for the 1-d control problem. It is also proved that the 2-d control system replicates the controllability properties of the 1-d control system and vise versa. Finally, the necessary and suffcient conditions of (approximate) L∞- controllability are obtained for the 2-d control problem. |
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