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Article

On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls

Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, Leioa, PO Box 48940, Bizkaia, Spain
Symmetry 2019, 11(5), 712; https://doi.org/10.3390/sym11050712
Received: 8 May 2019 / Revised: 19 May 2019 / Accepted: 21 May 2019 / Published: 24 May 2019
(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Applications)
This paper links the celebrated Cauchy’s interlacing theorem of eigenvalues for partitioned updated sequences of Hermitian matrices with stability and convergence problems and results of related sequences of matrices. The results are also applied to sequences of factorizations of semidefinite matrices with their complex conjugates ones to obtain sufficiency-type stability results for the factors in those factorizations. Some extensions are given for parallel characterizations of convergent sequences of matrices. In both cases, the updated information has a Hermitian structure, in particular, a symmetric structure occurs if the involved vector and matrices are complex. These results rely on the relation of stable matrices and convergent matrices (those ones being intuitively stable in a discrete context). An epidemic model involving a clustering structure is discussed in light of the given results. Finally, an application is given for a discrete-time aggregation dynamic system where an aggregated subsystem is incorporated into the whole system at each iteration step. The whole aggregation system and the sequence of aggregated subsystems are assumed to be controlled via linear-output feedback. The characterization of the aggregation dynamic system linked to the updating dynamics through the iteration procedure implies that such a system is, generally, time-varying. View Full-Text
Keywords: Aggregation dynamic system; Discrete system; Epidemic model; Cauchy’s interlacing theorem; Output-feedback control; Stability; Antistable/Stable matrix Aggregation dynamic system; Discrete system; Epidemic model; Cauchy’s interlacing theorem; Output-feedback control; Stability; Antistable/Stable matrix
MDPI and ACS Style

De la Sen, M. On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls. Symmetry 2019, 11, 712. https://doi.org/10.3390/sym11050712

AMA Style

De la Sen M. On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls. Symmetry. 2019; 11(5):712. https://doi.org/10.3390/sym11050712

Chicago/Turabian Style

De la Sen, Manuel. 2019. "On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls" Symmetry 11, no. 5: 712. https://doi.org/10.3390/sym11050712

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