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

Let $\left(H, \left\langle { \cdot , \cdot } \right\rangle \right)$ be a separable Hilbert space and $A_{i}:D(A_i) \to H$ ( $i = 1,···,n$ ) be a family of nonnegative and self-adjoint operators mutually commuting. We study the inverse problem consisting in the identification of a function $u:[0,T] \to H$ and $n$ constants $α_{1},···,α_{n} > 0$ (diffusion coefficients) that fulfill the initial-value problem

$ u'(t) + α_{1} A_{1}u(t) + ··· + α_{n} A_{n}u(t) = 0, ~~~t ∈ (0,T), ~~~u(0) = x,$

and the additional conditions

$\left\langle A_{1} u(T),u(T)\right\rangle = \varphi_{1}, ~~~··· ~~~,\left\langle A_{n} u(T),u(T)\right\rangle = \varphi_{n},$

where $\varphi_{i}$ 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.