Inverse Problem Related to Boundary Shape Identification for a Hyperbolic Differential Equation

<jats:p>In this paper, we are interested in the inverse problem of the determination of the unknown part <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M1"> <mo>∂</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <msub> <mi mathvariant="normal">Γ</mi> <mrow> <mn>0</mn> </mrow> </msub> </math> </jats:inline-formula> of the boundary of a uniformly Lipschitzian domain <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M2"> <mi mathvariant="normal">Ω</mi> </math> </jats:inline-formula> included in <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M3"> <msup> <mrow> <mi>ℝ</mi> </mrow> <mrow> <mi>N</mi> </mrow> </msup> </math> </jats:inline-formula> from the measurement of the normal derivative <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M4"> <msub> <mrow> <mo>∂</mo> </mrow> <mrow> <mi>n</mi> </mrow> </msub> <mi>v</mi> </math> </jats:inline-formula> on suitable part <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M5"> <msub> <mi mathvariant="normal">Γ</mi> <mrow> <mn>0</mn> </mrow> </msub> </math> </jats:inline-formula> of its boundary, where <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M6"> <mi>v</mi> </math> </jats:inline-formula> is the solution of the wave equation <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M7"> <msub> <mrow> <mo>∂</mo> </mrow> <mrow> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mi>v</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mrow> </mfenced> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mrow> </mfenced> <mo>+</mo> <mi>p</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> <mi>v</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> <mo>=</mo> <mn>0</mn> </math> </jats:inline-formula> in <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M8"> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mfenced open="(" close=")" separators="|"> <mrow> <mn>0</mn> <mo>,</mo> <mi>T</mi> </mrow> </mfenced> </math> </jats:inline-formula> and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M9"> <mi mathvariant="normal">Γ</mi> </math> </jats:inline-formula> of <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M10"> <mo>∂</mo> <mi mathvariant="normal">Ω</mi> </math> </jats:inline-formula>. From necessary conditions, we estimate a Lagrange multiplier <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M11"> <mi>k</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="normal">Ω</mi> </mrow> </mfenced> </math> </jats:inline-formula> which appears by derivation with respect to the domain. By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. The Lipschitz stability is established by increasing of the energy of the system. Some numerical simulations are made to illustrate the optimal shape. </jats:p>