Fréchet Differentiability of Parameter-Dependent Analytic Semigroups

• Published in: Journal of mathematical analysis and applications Vol. 232; no. 1; pp. 119 - 137
• Main Authors: Seubert, S, Wade, J.G
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.04.1999; Elsevier
• Subjects: abstract elliptic operators; analytic semigroups; Calculus of variations and optimal control; Exact sciences and technology; Fréchet differentiability; Gelfand triples; Group theory; Group theory and generalizations; inverse Laplace transform; Mathematical analysis; Mathematics; parameterized evolution equations; resolvent perturbation; Sciences and techniques of general use; parameterized evolution equations; resolvent perturbation; Fréchet differentiability; abstract elliptic operators; Gelfand triples; analytic semigroups; inverse Laplace transform; Differential equation; Initial value problem; Uniform topology; Evolution equation; Convection diffusion equation; Semigroup; Banach space; Partial differential equation; Inverse problem; Perturbation method; Boundary value problem; Fokker Planck equation; Gelfand triple; Population dynamics; Hilbert space; Elliptic operator; Laplace transformation; Regularity; Neumann series; Gateaux derivative
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1006/jmaa.1998.6249

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.