IMA Journal of Mathematical Control and Information Advance Access published online on February 27, 2007
IMA Journal of Mathematical Control and Information, doi:10.1093/imamci/dnm010
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Properties of a subalgebra of H
(
) and stabilization
Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK
Email: a.j.sasane{at}lse.ac.uk
Received on 10 May 2006; Accepted on 14 January 2007
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Abstract |
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Let 
denote the open unit disc in 
. Let 
denote the unit circle and let S 
T. We denote by AS() the set of all functions f : 
S 
that are holomorphic in and are bounded and continuous in S. Equipped with the supremum norm, AS() is a Banach algebra, and it lies between the extreme cases of the disc algebra A() and the Hardy space H
(). We show that AS() has the following properties:
- P1. The corona theorem holds for AS().
- P2. The integral domain AS(
) is not a Bézout domain, but it is a Hermite ring.
- P3. The stable rank of AS(
) is 1.
- P4. The Banach algebra AS(
) has topological stable rank 2.
- P2. The integral domain AS(
) serve as appropriate transfer function classes for infinite-dimensional systems that are not exponentially stable, but stable only in some weaker sense. Consequences of the above properties to stabilizing controller synthesis using a coprime factorization approach are discussed.
Keywords: function algebras; coprime factorization; stabilization; infinite-dimensional systems.