Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T01:21:55.952Z Has data issue: false hasContentIssue false
  • English
  • Français

The uniform exponential stability of a class of linear differential–difference equations in a Hilbert space

Published online by Cambridge University Press:  14 November 2011

Richard Datko
Richard Datko
Affiliation:
Department of Mathematics, Georgetown University, Washington, D.C., U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

A necessary and sufficient condition is given for the uniform exponential stability of certain autonomous differential–difference equations whose phase space is a Hilbert space. It is shown that this property is preserved when the delays depend homogeneously on a nonnegative parameter.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 89 , Issue 3-4 , 1981 , pp. 201 - 215
Copyright
Copyright © Royal Society of Edinburgh 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Henry, D.. Linear autonomous neutral functional differential equations. J. Differential Equations 15 (1974), 106–128.CrossRefGoogle Scholar
2Datko, R.. A procedure for determination of the exponential stability of certain differentialdifference equations. Quart. Appl. Math. 36 (1978), 279–292.CrossRefGoogle Scholar
3Datko, R.. Linear autonomous neutral differential equations in a Banach space. J. Differential Equations 25 (1977), 258–274.CrossRefGoogle Scholar
4Hille, E. and Phillips, R. S.. Functional Analysis and Semi-groups. A.M.S. Colloquium Publications, Vol. 31 (Providence, R.I: A.M.S., 1957).Google Scholar
5Cruz, M. A. and Hale, J. K.. Stability of functional differential equations of neutral type. J. Differential Equations 7 (1970), 334–355.CrossRefGoogle Scholar
6Datko, R.. Representation of solutions and stability of linear differential-differenceequations in a Banach space. J. Differential Equations 29 (1978), 105–166.CrossRefGoogle Scholar
7Datko, R.. The Perron condition for linear neutral differential-difference equations in a Hilbert space. Submitted for publication.Google Scholar
8Dunford, N. and Schwartz, J. T.. Linear Operators, 1 (New York: Wiley, 1958).Google Scholar