Reconstruction of two constant coefficients in linear anisotropic diffusion model. (7th October 2016) - Mola, Gianluca; Okazawa, Noboru; Yokota, Tomomi
Abstract: Let ( H, ⟨ ·, · ⟩ ) be a complex Hilbert space and A : D ( A ) → H and B : D ( B ) → H be nonnegative and selfadjoint operators. We study the inverse problem consisting in the identification of the function u : [ 0, T ] → H and two constants α, β ∊ R + = ( 0, ∞ ) ( diffusion coefficients ) that fulfill the initial-value problem u ′ ( t ) + α Au ( t ) + β Bu ( t ) = 0, t ∊ ( 0, T ), u ( 0 ) = x, and the additional conditions ⟨ Au ( T ), u ( T ) ⟩ = φ and ⟨ Bu ( T ), u ( T ) ⟩ = ψ . Under suitable assumptions on the operators A and B, and on the data x ∊ H and φ, ψ > 0, we shall construct a solution and prove its uniqueness and continuous dependence on the data. Applications are considered.