Locally smooth unitary groups and applications to boundary control of PDEs

Stephen W. Taylor

doi:10.3934/eect.2013.2.733

Article Contents

2013, Volume 2, Issue 4: 733-740. Doi: 10.3934/eect.2013.2.733

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Locally smooth unitary groups and applications to boundary control of PDEs

Stephen W. Taylor^1,

1.
Mathematics Department, The University of Auckland, Private Bag 92019, Auckland

Received: June 30, 2013

Revised: August 31, 2013

Published: November 2013

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Abstract

Let $\mathcal{P}$ be the projection operator for a closed subspace $\mathcal{S}$ of a Hilbert space $\mathcal{H}$ and let $U$ be a unitary operator on $\mathcal{H}$. We consider the questions
1. Under what conditions is $\mathcal{P}U\mathcal{P}$ a strict contraction?
2. If $g$, $h\in \mathcal{S}$, can we find $f\in \mathcal{H}$ such that $\mathcal{P}f=g$ and $\mathcal{P}Uf=h$?
The results are abstract versions and generalisations of results developed for boundary control of partial differential equations. We discuss how these results can be used as tools in the direct construction of boundary controls.

Keywords:

Mathematics Subject Classification: Primary: 35Q40, 35Q41, 35Q93; Secondary: 35Nxx, 35Lxx.

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