Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements

Let (H, <.,.>) be a separable Hilbert spaces and A_i : D(A_i) --->H (i=1,...n) be a family of nonnegative selfadjoint operators mutually commuting. We study the inverse problem consisting in the identification of the function u : [0,T] ---> H and n time-dependent diffusion coefficients \alpha_1, ..., \alpha_n : [s,T] --->R_+ that fulffill the initial-value problem u'(t) + \alpha_1(t) A_1 u (t) + ... + \alpha_n(t) A_n u (t) = 0, s\le t \le T, u(s)=x, and the additional conditions =\phi_1 (t), .... , = \phi_n(t), s\le t\le T. Under suitable assumptions on the operators A_i, i=1, ... , n, on the initial data x\in H and on the given functions \phi_i, i=1, ... , n, we shall prove that the solution of such a problem exists, is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion coefficients in a heat equation and of the Lame' parameters in an elasticity problem on a plate.