Spectral Nevanlinna-Pick and Caratheodory-Fejer problems for $n\geq3$

Nikolov, Nikolai; Pflug, Peter; Thomas, Pascal J.

doi:10.1512/iumj.2011.60.4310

Cited by 11 publications

(13 citation statements)

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“…, W M ∈ Ω n lie off an explicitly defined set S n Ω n , which is of zero Lebesgue measure, the problem ( * ) is equivalent to an associated Nevanlinna-Pick problem for G n . (Also see [12] for an improvement of (a) when n = 2, 3. ) For most of the systems alluded to above, the associated E comprises matrices whose diagonal blocks are either scalar matrices or rank-one matrices.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

A Family of Domains Associated with $${\mu}$$ μ -Synthesis

Bharali¹

2014

Integr. Equ. Oper. Theory

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Abstract. We introduce a family of domains -which we call the µ1, n-quotients -associated with an aspect of µ-synthesis. We show that the natural association that the symmetrized polydisc has with the corresponding spectral unit ball is also exhibited by the µ1, n-quotient and its associated unit "µE-ball". Here, µE is the structured singular value for the case E = {[w] ⊕ (zIn−1) ∈ C n×n : z, w ∈ C}, n = 2, 3, 4, . . . Specifically: we show that, for such an E, the Nevanlinna-Pick interpolation problem with matricial data in a unit "µE -ball", and in general position in a precise sense, is equivalent to a Nevanlinna-Pick interpolation problem for the associated µ1, n-quotient. Along the way, we present some characterizations for the µ1, n-quotients.

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Section: Introduction and Main Resultsmentioning

confidence: 98%

A Family of Domains Associated with $${\mu}$$ μ -Synthesis

Bharali¹

2014

Integr. Equ. Oper. Theory

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“…If any of the W j are scalar matrices then the corresponding interpolation conditions can be removed by the standard process of Schur reduction, and so we may suppose that all the W j are nonscalar. Alternatively, if some W j are scalar, one may still reduce to an interpolation problem for Hol(D, Γ), but with interpolation conditions on derivatives [24]; one could then try to prove analogs of the present results for this wider class of interpolation problems (this should not be difficult).…”

Section: Implementation Of the Solution Proceduresmentioning

confidence: 93%

A Case of $\mu$-Synthesis as a Quadratic Semidefinite Program

Agler¹,

Lykova²,

Young³

2013

SIAM J. Control Optim.

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Abstract. We analyse a special case of the robust stabilization problem under structured uncertainty. We obtain a new criterion for the solvability of the spectral NevanlinnaPick problem, which is a special case of the µ-synthesis problem of H ∞ control in which µ is the spectral radius. Given n distinct points λ1, . . . , λn in the unit disc and 2 × 2 nonscalar complex matrices W1, . . . , Wn, the problem is to determine whether there is an analytic 2 × 2 matrix function F on the disc such that F (λj) = Wj for each j and the supremum of the spectral radius of F (λ) is less than 1 for λ in the disc. The condition is that the minimum of a quadratic function of pairs of positive 3n-square matrices subject to certain linear matrix inequalities in the data be attained and be zero.

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“…In contrast to the spectral ball, the symmetrized polydisc G n is a taut domain, and should be more accessible with techniques from hyperbolic geometry. Solutions to this lifting problem have been found for dimensions n = 2, 3 by Nikolov, Pflug and Thomas [NPT11] and recently for dimensions n = 4, 5 by Nikolov, Thomas and Tran [NTT14]. They also provide the solution to a localised version of the spectral Nevanlinna-Pick lifting problem:…”

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confidence: 81%