Boundary controllability problems for the equation of oscillation of an inhomogeneous string on a semiaxis
We consider a wave equation on a semiaxis, namely, $w_{tt}(x,t) = w_{xx}(x,t) — q(x)w(x,t), x > 0$. The equation is controlled by one of the following two boundary conditions: $w(0,t) = u_0(t)$ and $w_x(0,t) = u_1(t), t \in (0,T)$, where $u_0, u_1$ are controls. In both cases, the potentia...
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| Datum: | 2012 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2012
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2593 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | We consider a wave equation on a semiaxis, namely, $w_{tt}(x,t) = w_{xx}(x,t) — q(x)w(x,t), x > 0$.
The equation is controlled by one of the following two boundary conditions: $w(0,t) = u_0(t)$ and $w_x(0,t) = u_1(t), t \in (0,T)$, where $u_0, u_1$ are controls.
In both cases, the potential q satisfies the condition $q \in C[0, \infty)$, the controls belong to the class $L^{\infty}$ and the time $T >$ 0 is fixed. These control systems are considered in Sobolev spaces.
Using the operators adjoint to the transformation operators for the Sturm - Liouville problem,
we obtain necessary and sufficient conditions for the null-controllability and approximate null-controllability of these systems. The controls that solve these problems are found in explicit form. |
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