Recovering a large number of diffusion constants in a parabolic equation from energy measurements

Abstract

Let (H, •, •) be a separable Hilbert space and A i : D(A i) → H (i = 1, • • • , n) be a family of nonnegative and self-adjoint operators mutually commuting. We study the inverse problem consisting in the identification of a function u : [0, T ] → H and n constants α 1 , • • • , αn > 0 (diffusion coefficients) that fulfill the initial-value problem u (t) + α 1 A 1 u(t) + • • • + αnAnu(t) = 0, t ∈ (0, T), u(0) = x, and the additional conditions A 1 u(T), u(T) = ϕ 1 , • • • , Anu(T), u(T) = ϕn, where ϕ i are given positive constants. Under suitable assumptions on the operators A i and on the initial data x ∈ H, we shall prove that the solution of such a problem is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion constants in a heat equation and of the Lamé parameters in a elasticity problem on a plate.