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

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \left(H, \langle \cdot, \cdot \rangle \right) $\end{document}</tex-math></inline-formula> be a separable Hilbert space and <inline-formula><tex-math id="M2">\begin{document}$ A_{i}:D(A_i) \to H $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M3">\begin{document}$ i = 1, \cdots, n $\end{document}</tex-math></inline-formula>) be a family of nonnegative selfadjoint operators mutually commuting. We study the inverse problem consisting in the identification of the function <inline-formula><tex-math id="M4">\begin{document}$ u:[0, T] \to H $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> time-dependent <i>diffusion coefficients</i> <inline-formula><tex-math id="M6">\begin{document}$ \alpha_{1}, \cdots, \alpha_{n}:[s, T] \to {\mathbb{R}}_+ $\end{document}</tex-math></inline-formula> that fulfill the initial-value problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ 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, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the additional conditions</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \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. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Under suitable assumptions on the operators <inline-formula><tex-math id="M7">\begin{document}$ A_i $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ i = 1, \cdots, n $\end{document}</tex-math></inline-formula>, on the initial data <inline-formula><tex-math id="M9">\begin{document}$ x\in H $\end{document}</tex-math></inline-formula> and on the given functions <inline-formula><tex-math id="M10">\begin{document}$ \varphi_i $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ i = 1, \cdots, n $\end{document}</tex-math></inline-formula>, 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.</p>