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• W3095354691 abstract "Abstract The inverse coefficient problem of recovering the potential q ( x ) in the damped wave equation <?CDATA $mleft(xright){u}_{tt}+mu left(xright){u}_{t}={left(rleft(xright){u}_{x}right)}_{x}+qleft(xright)u$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mfenced close=) open=(> <mml:mrow> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> </mml:math> , ( x , t ) ∈ Ω T ≔ (0, ℓ ) × (0, T ) subject to the boundary conditions r (0) u x (0, t ) = f ( t ), u ( ℓ , t ) = 0, from the Dirichlet boundary measured output ν ( t ) ≔ u (0, t ), t ∈ (0, T ] is studied. A detailed microlocal analysis of regularity of the direct problem solution in the subdomains defined by the characteristics as well as along these characteristics is provided. Based on this analysis, necessary regularity results and energy estimates are derived. It is proved that the Dirichlet boundary measured output uniquely determines the potential q ( x ) in the interval [0, h ( T /2)] and this solution belongs to C (0, h ( T /2)) with T < T *, where h ( z ) is the root of the equation <?CDATA $z={int }_{0}^{hleft(zright)}sqrt{mleft(xright)/rleft(xright)}enspace mathrm{d}x$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mo>∫</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>h</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mrow> </mml:msubsup> <mml:msqrt> <mml:mrow> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mrow> </mml:msqrt> <mml:mspace width=0.17em /> <mml:mi mathvariant=normal>d</mml:mi> <mml:mi>x</mml:mi> </mml:math> , <?CDATA ${T}^{{ast}}=2{int }_{0}^{ell }sqrt{mleft(xright)/rleft(xright)}enspace mathrm{d}x$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>*</mml:mo> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:msubsup> <mml:mrow> <mml:mo>∫</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:msubsup> <mml:msqrt> <mml:mrow> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mrow> </mml:msqrt> <mml:mspace width=0.17em /> <mml:mi mathvariant=normal>d</mml:mi> <mml:mi>x</mml:mi> </mml:math> . Moreover, the global uniqueness theorem is proved. Compactness, invertibility and Lipschitz continuity of the Neumann-to-Dirichlet operator <?CDATA ${{Phi}}^{f}left[cdot right]:mathcal{Q}subset Cleft(0,ell right){mapsto}{L}^{2}left(0,Tright)$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:msup> <mml:mrow> <mml:mi mathvariant=normal>Φ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mrow> <mml:mo>⋅</mml:mo> </mml:mrow> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mi mathvariant=script>Q</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>↦</mml:mo> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:math> , Φ f [ q ]( t ) ≔ u (0, t ; q ) is proved. This allows us to prove an existence of a quasi-solution of the inverse problem defined as a minimum of the Tikhonov functional <?CDATA $Jleft(qright){:=}left(1/2right)enspace {Vert}{{Phi}}^{f}left[cdot right]-nu {{Vert}}_{{L}^{2}left(0,Tright)}^{2}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mi>J</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>≔</mml:mo> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mspace width=0.17em /> <mml:mo stretchy=false>‖</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=normal>Φ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mrow> <mml:mo>⋅</mml:mo> </mml:mrow> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mi>ν</mml:mi> <mml:msubsup> <mml:mrow> <mml:mo stretchy=false>‖</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> as well as its Fréchet differentiability. An explicit formula for the Fréchet gradient is derived by making use of the unique solution to corresponding adjoint problem. The proposed approach is leads to very effective gradient based computational identification algorithm." @default.
• W3095354691 created "2020-11-09" @default.
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• W3095354691 date "2020-10-23" @default.
• W3095354691 modified "2023-09-24" @default.
• W3095354691 title "Recovering a potential in damped wave equation from Neumann-to-Dirichlet operator" @default.
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