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Zbl 0867.35115
Puel, J.-P.; Yamamoto, M.
Generic well-posedness in a multidimensional hyperbolic inverse problem. (English)
[J] J. Inverse Ill-Posed Probl. 5, No.1, 55-83 (1997). ISSN 0928-0219; ISSN 1569-3945/e

Summary: Let $y(f)$ and $u(p,a,h)$ be the solutions, respectively, to the following initial/boundary value problems in a bounded domain $\Omega\subset\bbfR^n$ $(n\ge 1)$ with a smooth boundary $\partial\Omega$: $${\partial^2y\over\partial t^2} (x,t)= \Delta y(x,t)-p(x)y(x,t)-f(x)\lambda(x,t),\quad x\in\Omega,\quad t>0\tag1$$ $$y(x,0)={\partial y\over\partial t} (x,0)=0,\quad x\in\Omega,\quad y(x,t)=0,\quad x\in\partial\Omega,\quad t>0$$ $${\partial^2u\over\partial t^2} (x,t)=\Delta u(x,t)-p(x)u(x,t),\quad x\in\Omega,\quad t>0\tag2$$ $$u(x,0)=a(x),\quad{\partial u\over\partial t} (x,0)=0,\quad x\in\Omega,\quad u(x,t)=h(x,t),\quad \in\partial\Omega,\quad t>0.$$ 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:\par (I) Determine $f(x)$ $(x\in\Omega)$ from $(\partial y(f)/\partial n)(x,t)$ ($x\in\Gamma$, $0<t<T$) provided that $p(x)$ and $\lambda(x,t)$ are given functions.\par (II) Determine $p(x)$ $(x\in\Omega)$ from $(\partial u(p,a,h)/\partial n)(x,t)$\ $(x\in\Gamma$, $0<t<T$) provided that $a(x)$ and $h(x,t)$ are given functions.

MSC 2000:: *35R30 Inverse problems for PDE
35L15 Second order hyperbolic equations, initial value problems

Keywords: exact controllability method; stability estimates

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Zentralblatt MATH Berlin [Germany]