Timeline for null controllability of linear wave equation
Current License: CC BY-SA 3.0
15 events
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| May 5, 2013 at 21:36 | comment | added | reseacher | For the Deane's response, I can not see how to show this. I know from (Haraux 2004), that some "geometric conditions on the domain $\Omega$" are needed with additional assumption on the sign of $k(x)$. | |
| May 5, 2013 at 20:37 | comment | added | Igor Khavkine | @researcher: Deane's point stands independent of the sign of $k(x)$. So, unique initial data $(z,z_t)$ at $t=0$ is guaranteed for any $k(x)$ and $h(t)$. What is suspicious about your claim in reply to Deane's comment is the existence of such $h(t)$ for any given initial data set at $t=0$, provided that your notation means that $h(t)$ depends only on $t$. For example, what if $(z,z_t)$ have compact support at $t=0$? Then $h(t)$ will produce ripples arbitrarily far away that could not be canceled by the propagation of any part of the initial data. | |
| May 5, 2013 at 14:22 | comment | added | reseacher | This is showed by : A. Haraux, "An alternative functional approach to exact controllability of reversible systems", Portugallae Mathematca, Vol. 61 Fasc. 4, Nova Série, (2004). | |
| May 5, 2013 at 1:53 | comment | added | drbobmeister | @researcher: what leads you to suspect such control can be obtained if $k(x) \geq 0$? | |
| May 5, 2013 at 0:47 | comment | added | reseacher | I mean : For any initial state $(z(0),z_t(0))$, there exists a time function $h(t)$ such that $z(T)=z_t(T)=0$. I think that this is true if $k(x)\ge 0.$ But, I m looking for constraints on $h(t$ but without condition on the sign of $k(x)!$ | |
| May 4, 2013 at 23:54 | comment | added | Deane Yang | Another thought: Given $k$ and $h$, solving the PDE backwards in time, there is a unique solution for $t \in [0,T]$ such that $(z,z_t) = 0$ at $t = T$. Therefore, given $k$ and $h$, there exists unique initial data $(z,z_t)$ at $t = 0$ for which the solution satisfies $(z,z_t) = 0$ at $t = T$. | |
| May 4, 2013 at 23:48 | comment | added | Deane Yang | Do you want $(z, z_t)$ to vanish at $t=T$ for any solution to this equation or, say, for the unique solution that satisfies $(z,z_t) = 0$ at $t = 0$? | |
| May 4, 2013 at 20:55 | comment | added | reseacher | Concerning the question of Igor, we can suppose any constraint on the function "control" $h(t)$ | |
| May 4, 2013 at 20:35 | comment | added | drbobmeister | Looks like researcher edited his question. Cool! | |
| May 4, 2013 at 19:43 | comment | added | drbobmeister | I meant "progress", not "orogress" ! | |
| May 4, 2013 at 19:41 | comment | added | drbobmeister | This is an interesting question but the $z$-$y$ ambiguity needs resolution to allow orogress. Waiting for the OP to speak up . . . | |
| May 4, 2013 at 18:10 | comment | added | Igor Khavkine | Are you supposing by any chance that $h(t)$ vanishes in a neighborhood of $t=0$ or $T$? | |
| May 4, 2013 at 18:07 | history | edited | reseacher | CC BY-SA 3.0 |
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| May 4, 2013 at 17:43 | comment | added | András Bátkai | Is is correct that $z=y$? | |
| May 4, 2013 at 17:12 | history | asked | reseacher | CC BY-SA 3.0 |