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On the Relation between Stability of Continuous- and Discrete-Time Evolution Equations via the Cayley Transform

Authors:	B Z Guo Hans Zwart

Institution:	(1) Academy of Mathematics and System Sciences, Academia Sinica, Institute of Systems Science, Beijing, 100080, China;(2) Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Abstract:	In this paper we investigate and compare the properties of the semigroup generated by A, and the sequence $A_{d} ^{n} ,\,n \in \mathbb{N},$ where A_d = (I + A) (I − A)⁻¹. We show that if A and A⁻¹ generate a uniformly bounded, strongly continuous semigroup on a Hilbert space, then A_d is power bounded. For analytic semigroups we can prove stronger results. If A is the infinitesimal generator of an analytic semigroup, then power boundedness of A_d is equivalent to the uniform boundedness of the semigroup generated by A.

Keywords:	34A30 34A45 34C10 39A11 47D60 93D05
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