Journal Article

Properties of a subalgebra of H<sup>∞</sup>(𝔻) and stabilization

Amol Sasane

inΒ IMA Journal of Mathematical Control and Information

Published on behalf of Institute of Mathematics and its Applications

Volume 25, issue 1, pages 1-21
Published in print March 2008 | ISSN: 0265-0754
Published online February 2007 | e-ISSN: 1471-6887 | DOI:Β http://dx.doi.org/10.1093/imamci/dnm010
Properties of a subalgebra of H∞(𝔻) and stabilization

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Let 𝔻 denote the open unit disc in β„‚. Let 𝕋 denote the unit circle and let S βŠ‚ T. We denote by AS(𝔻) the set of all functions f : 𝔻 βˆͺ S β†’ β„‚ that are holomorphic in 𝔻 and are bounded and continuous in 𝔻 βˆͺ S. Equipped with the supremum norm, AS(𝔻) is a Banach algebra, and it lies between the extreme cases of the disc algebra A(𝔻) and the Hardy space H∞(𝔻). We show that AS(𝔻) has the following properties:

P1. The corona theorem holds for AS(𝔻).

P2. The integral domain AS(𝔻) is not a BΓ©zout domain, but it is a Hermite ring.

P3. The stable rank of AS(𝔻) is 1.

P4. The Banach algebra AS(𝔻) has topological stable rank 2.

The classes AS(𝔻) 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

Journal Article.Β  0 words.Β 

Subjects: Mathematics

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