| Abstract: |
| Let $\left(H, \langle \cdot, \cdot \rangle \right)$ be a Hilbert space and $A_{i}:D(A_i) \to H$ ($i = 1,\cdots,n$) be a family of nonnegative and self-adjoint operators mutually commuting. We study the inverse problem consisting in the identification of the function $u:[0,\infty) \to H$ and $n$ constants $\lambda_{1},\cdots,\lambda_{n} > 0$
that fulfill the initial-value problem
$$
\partial_{t}^{\alpha}u(t) + \lambda_{1} A_{1}u(t) + \cdots + \lambda_{n} A_{n}u(t) = 0, \quad t > 0, \quad u(0) = x,
$$
and the additional energy measurements at $t=T$
$$
\left\langle A_{1} u(T),u(T)\right\rangle = \varphi_{1},
\quad \cdots \quad,
\left\langle A_{n} u(T),u(T)\right\rangle = \varphi_{n}.
$$
Here, $\partial_{t}^{\alpha}$ denotes the \emph{Caputo fractional derivative} of order $0 < \alpha < 1$, which accounts for models of subdiffusion phenomena.
Under suitable assumptions on the initial datum $x \in H$, we shall provide a uniqueness result. Existence is also achieved, assuming that the overdeterminating conditions $\varphi_{1},\cdots,\varphi_{n} > 0$ belong to a proper set, which is described in concrete cases by means of numerical simulations for different values of $\alpha$.
Finally, we prove the convergence of the solution $(u,\lambda_{1},\cdots,\lambda_{n})$ as $\alpha \to 1^{-}$, thus linking the fractional derivative model with the classic derivative one.
Applications to heat conduction and plane elasticity are considered. |
|