Fréchet Differentiability of Parameter-Dependent Analytic Semigroups

Abstract

We study the dependence on a vector-valued parameterqof a collection of analytic semigroups {T(t;q),t≥0} arising, for example, from a collection of diffusion-convection equations whose infinitesimal generators are abstract elliptic operators defined in terms of sesquilinear forms in a “Gelfand triple” or “pivot space” framework. Within a mathematical framework slightly more general than the one set forth below, Banks and Ito [Banks, H. T. and Ito, K., “A unified framework for approximation in inverse problems for distributed parameter systems,” Control—Theory and Advanced Technology,4(1988), pp. 73–90] have shown, as an application of the Trotter-Kato Theorem, that the mapq↦T(t;q) is continuous in the strong operator topology. In this paper, we establish the analyticity of this map in the uniform operator topology, and exhibit its Fréchet derivative both as a contour integral and as the solution of a particular initial-value problem.