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The Nevanlinna-Pick problem for matrix-valued functions. (English) Zbl 0608.47020

The Nevanlinna-Pick interpolation problem is to find an analytic function in a disc or half-plane which minimises the supremum norm subject to a set of interpolation conditions. A version of this problem for analytic matrix-valued functions has become topical in the study of robust control [e.g. B. Francis, A course in \(H^{\infty}\) control theory, Lect. Notes Control Inf. Sci. 88 (1987)]: given matrix polynomials A, B, C find an analytic matrix function Q in the open unit disc such that the supremum norm of \(A+BQC\) is minimised. In the scalar case the solution is unique, but for matrix-valued A, B, C there are typically infinitely many Q’s attaining the minimum norm. Here it is shown that there is a unique Q achieving the following stronger minimisation condition.
Let G be an \(m\times n\)-matrix-valued \(L^{\infty}\) function on the unit circle. Let \[ s_ j^{\infty}(G)= \sup_{| z| =1}s_ j(G(z)), \] where \(s_ j(\cdot)\) denotes jth singular value of a matrix, and let \[ s^{\infty}(G)=(s_ 0^{\infty}(G),s_ 1^{\infty}(G),s_ 2^{\infty}(G),...). \] The problem is to minimise \(s^{\infty}(A+BQC)\), over bounded analytic matrix-valued Q, with respect to the lexicographic ordering.
The unique solution Q has the property that all the singular values of \((A+BQC)(z)\) are essentially constant on the unit circle. The proof exhibits a constructive high level algorithm (in operator theoretic terms) for finding Q.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
41A05 Interpolation in approximation theory
49J27 Existence theories for problems in abstract spaces
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