Frechet differentiability of parameter-dependent analytic semigroups

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:232

Article

Seubert, S ; Wade, JG

关键词: parameterized evolution equations;    Frechet differentiability;    analytic semigroups;    abstract elliptic operators;    Gelfand triples;    inverse Laplace transform;    resolvent perturbation;   

DOI  :  10.1006/jmaa.1998.6249

来源: Elsevier

【 摘 要 】

We study the dependence on a vector-valued parameter q of a collection of analytic semigroups {T(t;q), t greater than or equal to 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 map q H 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 Frechet derivative both as a contour integral and as the solution of a particular initial-value problem. (C) 1999 Academic Press.

【 授权许可】

Free