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

Let $ \left(H, \langle \cdot, \cdot \rangle \right) $ be a separable Hilbert space and $ A_{i}:D(A_i) \to H $ ($ i = 1, \cdots, 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] \to H $ and $ n $ time-dependent diffusion coefficients $ \alpha_{1}, \cdots, \alpha_{n}:[s, T] \to {\mathbb{R}}_+ $ that fulfill the initial-value problem

$ u'(t) + \alpha_{1}(t) A_{1}u(t) + \cdots + \alpha_{n}(t) A_{n}u(t) = 0, \quad s \leq t \leq T, \quad u(s) = x, $

and the additional conditions

$ \langle A_{1} u(t), u(t)\rangle = \varphi_{1}(t), \quad \cdots \quad, \langle A_{n} u(t), u(t)\rangle = \varphi_{n}(t), \quad s \leq t \leq T. $

Under suitable assumptions on the operators $ A_i $, $ i = 1, \cdots, n $, on the initial data $ x\in H $ and on the given functions $ \varphi_i $, $ i = 1, \cdots, 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 Lamé parameters in an elasticity problem on a plate.