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Locally smooth unitary groups and applications to boundary control of PDEs

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


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