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Generic well-posedness in a multidimensional hyperbolic inverse problem. (English) Zbl 0867.35115

Summary: Let $y\left(f\right)$ and $u\left(p,a,h\right)$ be the solutions, respectively, to the following initial/boundary value problems in a bounded domain ${\Omega }\subset {ℝ}^{n}$ $\left(n\ge 1\right)$ with a smooth boundary $\partial {\Omega }$:

$\frac{{\partial }^{2}y}{\partial {t}^{2}}\left(x,t\right)={\Delta }y\left(x,t\right)-p\left(x\right)y\left(x,t\right)-f\left(x\right)\lambda \left(x,t\right),\phantom{\rule{1.em}{0ex}}x\in {\Omega },\phantom{\rule{1.em}{0ex}}t>0\phantom{\rule{2.em}{0ex}}\left(1\right)$
$y\left(x,0\right)=\frac{\partial y}{\partial t}\left(x,0\right)=0,\phantom{\rule{1.em}{0ex}}x\in {\Omega },\phantom{\rule{1.em}{0ex}}y\left(x,t\right)=0,\phantom{\rule{1.em}{0ex}}x\in \partial {\Omega },\phantom{\rule{1.em}{0ex}}t>0$
$\frac{{\partial }^{2}u}{\partial {t}^{2}}\left(x,t\right)={\Delta }u\left(x,t\right)-p\left(x\right)u\left(x,t\right),\phantom{\rule{1.em}{0ex}}x\in {\Omega },\phantom{\rule{1.em}{0ex}}t>0\phantom{\rule{2.em}{0ex}}\left(2\right)$
$u\left(x,0\right)=a\left(x\right),\phantom{\rule{1.em}{0ex}}\frac{\partial u}{\partial t}\left(x,0\right)=0,\phantom{\rule{1.em}{0ex}}x\in {\Omega },\phantom{\rule{1.em}{0ex}}u\left(x,t\right)=h\left(x,t\right),\phantom{\rule{1.em}{0ex}}\in \partial {\Omega },\phantom{\rule{1.em}{0ex}}t>0·$

For a given ${\Gamma }\subset \partial {\Omega }$ and a sufficiently large $T<\infty$, by the exact controllability method, we get stability estimates for two inverse problems:

(I) Determine $f\left(x\right)$ $\left(x\in {\Omega }\right)$ from $\left(\partial y\left(f\right)/\partial n\right)\left(x,t\right)$ ($x\in {\Gamma }$, $0) provided that $p\left(x\right)$ and $\lambda \left(x,t\right)$ are given functions.

(II) Determine $p\left(x\right)$ $\left(x\in {\Omega }\right)$ from $\left(\partial u\left(p,a,h\right)/\partial n\right)\left(x,t\right)$ $\left(x\in {\Gamma }$, $0) provided that $a\left(x\right)$ and $h\left(x,t\right)$ are given functions.

##### MSC:
 35R30 Inverse problems for PDE 35L15 Second order hyperbolic equations, initial value problems
##### Keywords:
exact controllability method; stability estimates