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Interpolation by vector-valued analytic functions, with applications to controllability. (English) Zbl 1137.46015

The authors study the following interpolation problem for vector-valued analytic functions from the Hardy space in the half-plane. Let \(H^2(\mathbb C_+,\mathcal H)\) denote the Hardy space of \(\mathcal H\)-valued functions in the half plane \(\mathbb C_+=\{z\in\mathbb C\,:\,\operatorname{Re} z>0\}\), where \(\mathcal H\) is an auxiliary Hilbert space. Given interpolation data \(z_k\in \mathbb C_+\), \(a_k\in \mathcal H\), and \(G_k\in L(\mathcal H)\), \(k=1,2,\dots \), one searches for a function \(f\in H^2(\mathbb C_+,\mathcal H)\) of minimal norm satisfying \(G_k f(z_k)=a_k\). Estimates for solutions are obtained in terms of Carleson constants for certain related measures. For example, Theorem 2.11 of the paper says that in the case where \(\dim \mathcal H <\infty\), the above interpolation problem is solvable for arbitrary \(a_k\in \text{Ran}(G_k)\) with \(\sum_k \| a_k\| ^2<\infty\) if and only if the measure
\[ \sum_k \frac{| 2\operatorname{Re} z_k| ^2}{| b_{\infty,k}| ^2}\| (G_k^{-1})^*\bar\Theta^I(z_k)\| ^2 \delta_{z_k} \]
satisfies the Carleson condition. Here,
\[ | b_{\infty,k}| =\prod_{j\neq k}\left| \frac{z_k-z_j}{z_k+\bar z_j}\right| \]
and \(\bar \Theta^I\) is the Blaschke-Potapov product corresponding to a certain shift-invariant subspace of \(H^2(\mathbb C_+,\mathcal H)\).
In the second part of the paper, the authors apply the obtained interpolation results to controllability properties of infinite-dimensional linear systems. They study systems of the form
\[ \dot x(t)=A x(t)+Bu(t), \;t\geq 0; \quad x(0)=x_0, \]
where \(A\) is the generator of an exponentially stable \(C_0\)-semigroup on a Hilbert space \(\mathcal H\) such that the eigenvectors of \(A\) form an orthonormal basis of \(\mathcal H\). Conditions for null-controllability and exact controllability are obtained and expressed in terms of Carleson conditions for certain measures.

MSC:

46E20 Hilbert spaces of continuous, differentiable or analytic functions
30E05 Moment problems and interpolation problems in the complex plane
47D06 One-parameter semigroups and linear evolution equations
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