Belishev, M. I. Wave bases in multidimensional inverse problems. (English. Russian original) Zbl 0698.35164 Math. USSR, Sb. 67, No. 1, 23-42 (1990); translation from Mat. Sb. 180, No. 5, 584-602 (1989). Let \(\Omega \subset {\mathbb{R}}^ n\) (n\(\geq 1)\) be a bounded domain with boundary \(\partial \Omega =\Gamma \in C^ 2\). Consider the problem \[ \rho (x)u_{tt}-\Delta u=0,\quad (x,t)\in \Omega \times [0,T];\quad u|_{t<0}=0,\quad \partial_{\nu}u|_{\Gamma \times [0,T]}=f(\gamma,t). \] Here \(\rho \in C^ 2(\Omega)\), \(0<\rho_ 1\leq \rho (x)\leq \rho_ 2\), \(\nu =\nu (\gamma)\) (\(\gamma\in \Gamma)\) is the outer normal to \(\Gamma\). Consider the operator \(R: f(\gamma,t)\to u^ f(\gamma,t),\) where \(u^ f\) is the solution of the above problem, \(\gamma\in \Gamma\). The inverse problem considered in the paper is to recover \(\rho\) (x) given the operator R. The author proposes a procedure for recovering \(\rho\) (x) for x near the boundary and a suitable modification of that procedure when x is arbitrary. Some ideas and results from boundary value control theory are used. The main tools in the recovering procedure are the bases of waves coming from sources located on the boundary. Reviewer: P.Stefanov Cited in 14 Documents MSC: 35R30 Inverse problems for PDEs 35L05 Wave equation 93C20 Control/observation systems governed by partial differential equations 93B05 Controllability Keywords:unknown coefficient; control theory; bases of waves; sources PDFBibTeX XMLCite \textit{M. I. Belishev}, Math. USSR, Sb. 67, No. 1, 23--42 (1990; Zbl 0698.35164); translation from Mat. Sb. 180, No. 5, 584--602 (1989) Full Text: DOI