**1. Introduction**

Stability and convergence properties are very important topics when dealing with both continuousand discrete-time controlled dynamic systems. In this context, one of the most important design tools is the closed-loop stabilization of control systems via the appropriate incorporation of stabilizing controllers; see, for instance, [1–4] and references therein. In particular, in [1], and in some references therein, the robust stable adaptive control of tandem of master-slave robotic manipulators using a multi-estimation scheme is discussed. There are several questions of interest in the analysis, such as the fact that the dynamics may be time-varying and imperfectly known, and the fact that a parallel multi-estimation with eventual switching through time is incorporated into the adaptive controller to improve the transient behavior. The speed estimation and stable control of an induction motor based on the use of artificial neural networks is analyzed in [2]. Strategies of decentralized control, including several applications and stabilization tools, are given in [3,4]. In particular, decentralized control is useful when the various subsystems which are integrated in a whole integrated system are located in separate areas, or when the amount of information needed presents difficulties with regards to obtaining completely optimal suitable performance. Thus, the individual controllers associated with the various subsystems ge<sup>t</sup> local information about the corresponding subsystems, and eventually some extra partial information about the remaining ones to achieve stabilization, provided that the neglected coupling dynamics are weak enough. Stabilizing decentralized control designs are described in [3]

for networked composite systems. Some technical aspects and the results of non-negative matrices of usefulness to describe the properties and behavior of positive dynamic systems, the robustness of matrices against numerical parameterization perturbations of their entries, and the properties of linear dynamic systems are discussed in [5–8].

This paper focuses on the study of sequences of Hermitian matrices of increasing order which are built via block partition aggregation at each iteration, in such a way that both the current iteration and the next one are Hermitian matrices. The basic mathematical tool is the use of the interlacing Cauchy's theorem of the matrix eigenvalues of the matrices of the sequence, which orders the sequences of the eigenvalues as the iteration progresses [8]. 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 [9–14]. The paper is organized as follows. Section 2 is devoted to investigating the properties of boundedness and convergence of the sequences of the determinants and the sequences of eigenvalues as the iteration progresses by aggregation of the updated information while maintaining a Hermitian structure. In the particular case when the matrices describing the problem are real, the updated information has a symmetrical structure. The results are used, in particular, to give stability or anti-stability (in the sense that all the matrix eigenvalues of the matrices of the iterative sequence are unstable) conditions to the matrices used in the standard factorization of Hermitian positive definite matrices. Section 3 extends some of the above results to the convergence of sequences of partitioned Hermitian matrices constructed by aggregation of the updated information. Note that the concept of the convergence of matrices is a discrete counterpart of the matrix stability property in the continuous-time domain, since matrices are stability matrices if all their eigenvalues are in the open complex left-half plane. The basic idea that complex square matrices are convergen<sup>t</sup> if their eigenvalues are within the open unit circle centered at zero is taken into account. An example is discussed concerning a SIR epidemic model with contagions between populations of adjacent clusters in Section 4. Section 5 is devoted to developing an application for the stability of an aggregation discrete-time dynamic controlled system whose order increases by successive incorporations of new subsystems as the iteration index progresses, and whose structure keeps a symmetry. Finally, some conclusions are presented at the end the paper. The relevant mathematical proofs are given in the appendix in order to facilitate a direct reading of the manuscript. The system is assumed to be parameterized by real parameters and controlled by linear output-feedback control laws; it is also assumed that the former whole aggregation system and each new aggregated subsystem at each iteration might eventually be coupled.
