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International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 2, Pages 255-264

A weak invariance principle and asymptotic stability for evolution equations with bounded generators

1Department of Mathematics, Box 8205, North Carolina State University, Raleigh 27695-8205, N. C., USA
2Department of Mathematics and Statistics, Simon Fraser University, B.C., Burnaby V5A 1S6, Canada

Received 7 October 1991; Revised 6 April 1993

Copyright © 1995 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


If V is a Lyapunov function of an equation du/dt=u=Zu in a Banach space then asymptotic stability of an equilibrium point may be easily proved if it is known that sup(V)<0 on sufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability is proved for a bounded infinitesimal generator Z under a weaker assumption V0 (which alone implies ordinary stability only) if some observability condition, involving Z and the Frechet derivative of V, is satisfied. The proof is based on an extension of LaSalle's invariance principle, which yields convergence in a weak topology and uses a strongly continuous Lyapunov function. The theory is illustrated with an example of an integro-differential equation of interest in the theory of chemical processes. In this case strong asymptotic stability is deduced from the weak one and explicit sufficient conditions for stability are given. In the case of a normal infinitesimal generator Z in a Hilbert space, strong asymptotic stability is proved under the following assumptions Z*+Z is weakly negative definite and Ker Z={0}. The proof is based on spectral theory.