The controllability in a~filled domain for a~multidimensional wave equation with a~singular boundary control

S. A. Avdonin ; M. I. Belishev ; S. A. Ivanov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 23, Tome 210 (1994), pp. 7-21

Résumé

A local approach to Inverse Problems (so-called Bl-method) induces the corresponding Boundary Control Problem to describe a reachable set of “waves” $u^f(\cdot,T)$, где $u^f(x,t)$ being a solution of the problem: $u_{tt}-\Delta u=0$ in $\Omega\times(0,T)$, $u|_{t0}=0$, $u|_{\partial\Omega\times(0,T)}=f$ with singular controls $f$. The following result is established. Let $\Omega^T=\{x\in\Omega\colon\operatorname{dist}(x,\partial\Omega)$ be a subdomain of $\Omega\subset\mathbb R^n$ ($\operatorname{diam}\Omega\infty$) filled by waves to the final moment $t=T$; $T_*=\inf\{T\colon\Omega^T=\Omega\}$ be time of filling of the whole $\Omega$. Denote by $D_m=\operatorname{Dom}((-\Delta)^{m/2})$, where $-\Delta$ is Laplace operator defined on $\operatorname{Dom}(-\Delta)=H^2(\Omega)\cap H^1_0(\Omega)$; $D_{-m}=D'_m$; $D_{-m}(\Omega^T)=\{y\in D_{-m}\colon\operatorname{supp}y\subset\Omega^T\}$. The authors prove that if $T$ then the reachable set $R^T_m=\{u^f(\cdot,T)\colon f\in L_2((0,T);H^{-m}(\partial\Omega))\}$ is dense in $D_{-m}(\Omega^T)$ ($\forall m>0$), but it does not content the class $C^\infty_0(\Omega^T)$. The examples of $a\in C^\infty_0(\Omega^T)$, $a\not\in R^T_m$ are demonstrated. Bibliography: 19 titles.