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A weak invariance principle and asymptotic stability for evolution equations with bounded generators. (English) Zbl 0848.34040
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.
##### MSC:
 34G10 Linear differential equations in abstract spaces 34C11 Growth and boundedness of solutions to ordinary differential equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34D05 Asymptotic properties of solutions to ordinary differential equations 45J05 Integro-ordinary differential equations
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