A weak invariance principle and asymptotic stability for evolution equations with bounded generators (Q1805116)

Let \(B\) be a complex reflexive Banach space, and consider the linear evolution equation \(du/dt = Zu\), \(t > 0\), where \(Z : B \to B\) is a bounded linear operator. The authors of this interesting paper establish a weak asymptotic stability theorem for such evolution equations by using a Lyapunov function with a nonstrictly negative time derivative. To this end they assume, in addition, a certain observability hypothesis and use an extension of J. P. LaSalle's invariance principle which yields weak convergence. Then they apply their results to an integro-differential equation which arises in the theory of chemical processes. Finally, when \(B\) is a Hilbert space and \(Z\) is a normal operator the authors use spectral theory to prove a strong asymptotic stability theorem.