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

If V is a Lyapunov function of an equation d u / d t = u ′ = Z u 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.