Recovering a potential in damped wave equation from Dirichlet-to-Neumann operator Full article
| Journal |
Inverse Problems
ISSN: 0266-5611 |
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| Output data | Year: 2021, Volume: 37, Number: 3, Pages: 035005 Pages count : 33 DOI: 10.1088/1361-6420/abdb41 | ||||||
| Tags | recovering a potential, damped wave equation, regularity of the solution, uniqueness of the inverse problem solution, Dirichlet-to-Neumann operator, Fr´echet gradient | ||||||
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Abstract:
The inverse problem of recovering the potential q(x) in the damped wave
equation m(x)utt + μ(x)ut = (r(x)ux)x + q(x)u, (x, t) ∈ ΩT :=(0, ) × (0, T)
subject to the boundary conditions u(0, t) = ν(t), u(, t) = 0, from the Neumann
boundary measured output f(t) :=r(0)ux(0, t), t ∈ (0, T] is studied. The
approach proposed in this paper allows us to derive behavior of the direct
problem solution in the subdomains defined by characteristics of the wave
equation and along the characteristic lines, as well. Based on these results, a
local existence theorem and the stability estimate are proved. The compactness
and Lipschitz continuity of the Dirichlet-to-Neumann operator are derived.
Fr´echet differentiability of the Tikhonov functional is proved and an explicit
gradient formula is derived by means of an appropriate adjoint problem. It is
proved that this gradient is Lipschitz continuous.
Cite:
Romanov V.
, Hasanov A.
Recovering a potential in damped wave equation from Dirichlet-to-Neumann operator
Inverse Problems. 2021. V.37. N3. P.035005. DOI: 10.1088/1361-6420/abdb41 WOS Scopus OpenAlex
Recovering a potential in damped wave equation from Dirichlet-to-Neumann operator
Inverse Problems. 2021. V.37. N3. P.035005. DOI: 10.1088/1361-6420/abdb41 WOS Scopus OpenAlex
Dates:
| Submitted: | Oct 30, 2020 |
| Accepted: | Jan 13, 2021 |
Identifiers:
| Web of science: | WOS:000624472400001 |
| Scopus: | 2-s2.0-85102971334 |
| OpenAlex: | W3120967468 |