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.
