Fréchet Differentiability of Parameter-Dependent Analytic Semigroups
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 tripl...
Saved in:
Published in | Journal of mathematical analysis and applications Vol. 232; no. 1; pp. 119 - 137 |
---|---|
Main Authors | , |
Format | Journal Article |
Language | English |
Published |
San Diego, CA
Elsevier Inc
01.04.1999
Elsevier |
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | 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. |
---|---|
ISSN: | 0022-247X 1096-0813 |
DOI: | 10.1006/jmaa.1998.6249 |