Semigroup-theoretic approach to identificationof linear diffusion coefficients
Gianluca Mola, Noboru Okazawa, Jan W. Prüss, Tomomi Yokota · Discrete and Continuous Dynamical Systems - S · 2016
Let $X$ be a complex Banach space and$A:\,D(A) \to X$ a quasi-$m$-sectorial operatorin $X$. This paper is concerned with theidentification of diffusion coefficients$ u > 0$ in the initial-value problem:\[ (d/dt)u(t) + { u}Au(t) = 0,\quad t \in (0,T), \quad u(0) = x \in X,\]with additional condition $\|u(T)\| = \rho$,where $\rho >0$ is known. Except forthe additional condition, the solution to theinitial-value problem is given by$u(t) := e^{-t\,{ u}A} x\in C([0,T];X) \cap C^{1}((0,T];X)$.Therefore, the identification of $ u$ is reducedto solving the equation$\|e^{-{ u}TA}x\| = \rho$.It will be shown that the unique root$ u = u(x,\rho)$depends on $(x,\rho)$ locally Lipschitzcontinuously if the datum $(x,\rho)$ fulfillsthe restriction $\|x\|> \rho$. This extendsthose results inMola [6](2011).