Three-dimensional inverse acoustic scattering problem by the BC-method
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2024
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| _version_ | 1866909308414853120 |
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| author | Belishev, M. I. Vakulenko, A. F. |
| author_facet | Belishev, M. I. Vakulenko, A. F. |
| contents | Let $Σ:=[0,\infty)\times S^2$, $\mathscr F:=L_2(Σ)$. The {\it forward} acoustic scattering problem under consideration is to find $u=u^f(x,t)$ satisfying \begin{align} \label{Eq 01} &u_{tt}-Δu+qu=0, && (x,t) \in {\mathbb R}^3 \times (-\infty,\infty); \\ \label{Eq 02} &u \mid_{|x|<-t} =0 , && t<0;\\ \label{Eq 03} &\lim_{s \to -\infty} s\,u((-s+τ)\,ω,s)=f(τ,ω), && (τ,ω) \in Σ; \end{align} for a real valued compactly supported potential $q\in L_\infty(\Bbb R^3)$ and a control $f \in\mathscr F$. The response operator $R: \mathscr F\to\mathscr F$, \begin{align*} & (Rf)(τ,ω)\,:= \lim_{s \to +\infty} s\, u^f((s+τ)\,ω,s), \quad (τ,ω) \in Σ\end{align*} depends on $q$ {\it locally}: if $ξ>0$ and $f\in\mathscr F^ξ:=\{f\in\mathscr F\,|\,\,\,f\!\mid_{[0,ξ)}=0\}$ holds, then the values $(Rf)\!\mid_{τ\geqslantξ}$ are determined by $q\!\mid_{|x|\geqslantξ}$ (do not depend on $q\!\mid_{|x|<ξ}$).
The {\it inverse problem} is: for an arbitrarily fixed $ξ>0$, to determine $q\mid_{|x|\geqslantξ}$ from $X^ξR\upharpoonright\mathscr F^ξ$, where $X^ξ$ is the projection in $\mathscr F$ onto $\mathscr F^ξ$. It is solved by a relevant version of the boundary control method. The key point of the approach are recent results on the controllability of the system (\ref{Eq 01})--(\ref{Eq 03}). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_20191 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Three-dimensional inverse acoustic scattering problem by the BC-method Belishev, M. I. Vakulenko, A. F. Analysis of PDEs Mathematical Physics 35Q93, 35L05, 78A45 Let $Σ:=[0,\infty)\times S^2$, $\mathscr F:=L_2(Σ)$. The {\it forward} acoustic scattering problem under consideration is to find $u=u^f(x,t)$ satisfying \begin{align} \label{Eq 01} &u_{tt}-Δu+qu=0, && (x,t) \in {\mathbb R}^3 \times (-\infty,\infty); \\ \label{Eq 02} &u \mid_{|x|<-t} =0 , && t<0;\\ \label{Eq 03} &\lim_{s \to -\infty} s\,u((-s+τ)\,ω,s)=f(τ,ω), && (τ,ω) \in Σ; \end{align} for a real valued compactly supported potential $q\in L_\infty(\Bbb R^3)$ and a control $f \in\mathscr F$. The response operator $R: \mathscr F\to\mathscr F$, \begin{align*} & (Rf)(τ,ω)\,:= \lim_{s \to +\infty} s\, u^f((s+τ)\,ω,s), \quad (τ,ω) \in Σ\end{align*} depends on $q$ {\it locally}: if $ξ>0$ and $f\in\mathscr F^ξ:=\{f\in\mathscr F\,|\,\,\,f\!\mid_{[0,ξ)}=0\}$ holds, then the values $(Rf)\!\mid_{τ\geqslantξ}$ are determined by $q\!\mid_{|x|\geqslantξ}$ (do not depend on $q\!\mid_{|x|<ξ}$). The {\it inverse problem} is: for an arbitrarily fixed $ξ>0$, to determine $q\mid_{|x|\geqslantξ}$ from $X^ξR\upharpoonright\mathscr F^ξ$, where $X^ξ$ is the projection in $\mathscr F$ onto $\mathscr F^ξ$. It is solved by a relevant version of the boundary control method. The key point of the approach are recent results on the controllability of the system (\ref{Eq 01})--(\ref{Eq 03}). |
| title | Three-dimensional inverse acoustic scattering problem by the BC-method |
| topic | Analysis of PDEs Mathematical Physics 35Q93, 35L05, 78A45 |
| url | https://arxiv.org/abs/2407.20191 |