International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 2, Pages 255-264
doi:10.1155/S0161171295000317
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 E. N. Chukwu and P. Smoczynski. 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.
Abstract
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 V′≤0 (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.