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A Weak Invariance Principle and Asymptotic Stability for Evolution Equations with Bounded Generators

Resource type
Date created
1995
Authors/Contributors
Author: Chukwu, E.N.
Abstract
If V is a Lyapunov function of an equation du/dt u’ Zu in a Banach space thenasymptotic stability of an equilibrium point may be easily proved if it is known that sup(V’) < 0 onsufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability isproved for a bounded infinitesimal generator Z under a weaker assumption V’ < 0 (which aloneimplies ordinary stability only) if some observability condition, involving Z and the Frechet derivativeof V’, is satisfied. The proof is based on an extension of LaSalle’s invariance principle, which yieldsconvergence in a weak topology and uses a strongly continuous Lyapunov funcdon. The theory isillustrated with an example of an integro-differential equation of interest in the theory of chemicalprocesses. In this case strong asymptotic stability is deduced from the weak one and explicit sufficientconditions for stability are given. In the case of a normal infinitesimal generator Z in a Hilbertspace, strong asymptotic stability is proved under the following assumptions Z* + Z is weaklynegative definite and Ker Z 0 }. The proof is based on spectral theory.
Document
Published as
International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 2, Pages 255-264
http://dx.doi.org/10.1155/S0161171295000317
Publication title
International Journal of Mathematics and Mathematical Sciences
Document title
A Weak Invariance Principle and Asymptotic Stability for Evolution Equations with Bounded Generators
Date
1995
Volume
18
Issue
2
First page
255
Last page
264
Publisher DOI
10.1155/S0161171295000317
Copyright statement
Copyright is held by the author(s).
Permissions
You are free to copy, distribute and transmit this work under the following conditions: You must give attribution to the work (but not in any way that suggests that the author endorses you or your use of the work); You may not use this work for commercial purposes.
Scholarly level
Peer reviewed?
Yes
Language
English
Download file Size
689851.pdf 3.47 MB

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