Abstract
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.
Highlights
Stability and convergence properties are very important topics when dealing with both continuousand discrete-time controlled dynamic systems
This paper relies on partitioned Hermitian matrices and Cauchy’s interlacing theorem and the associated stability results
Based on the fact that convergent matrices are a discrete counterpart of stability matrices, the results presented above are extended to sequences of convergent matrices
Summary
Stability and convergence properties are very important topics when dealing with both continuousand discrete-time controlled dynamic systems. Our main objective is to adapt the interlacing theorem in order to use it to derive stability or convergence conditions of the sequence of matrices, and to use the results for the stability of a large-scale discrete aggregation-type dynamic system [9,10,11,12,13,14]. The subsequent result relies on some conditions which guarantee the boundedness of the determinant and eigenvalues of a recursive sequence of Hermitian matrices which were obtained and supported by Lemma 1 and Cauchy’s interlacing theorem. Lemma 5 holds “mutatis-mutandis” if A(n0 ) ∈ Cn0 ×n0 is antistable
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