Interconnections between µ-Value and D-Stable, D(α)-Stable Matrices from Economic Models

Rehman, Mutti-Ur; Akram, Saima; Farxod, Ixtiyarov; Ochilov, Narzillo; Gafurova, Shahlokhon; Fuzaylova, Nigina; Umidjon, Tukhtayev; Atoeva, Makhliyo

doi:10.28924/2291-8639-23-2025-77

Research Article

Interconnections between µ-Value and D-Stable, D(α)-Stable Matrices from Economic Models

Mutti-Ur Rehman*
Saima Akram
Ixtiyarov Farxod
Narzillo Ochilov
Shahlokhon Gafurova
Nigina Fuzaylova
Tukhtayev Umidjon
Makhliyo Atoeva

* Corresponding author

Abstract

In this paper, we review a number of well established methods to study the interconnection between D-stability and µ-values. The D-stability in economic, and dynamic systems plays a crucial role for maintaining equilibrium under proportional changes in parameters, for instance, prices, production levels, or financial flows. The computation of structured singular value a.k.a µ-value is a well-known mathematical tool for analysis of systems appearing in robust control. The µ-value provides the quantitative measure of linear systems stability subject to structured uncertainties. The approximation of an upper bounds of µ-value plays a critical role for ensuring robust stability and performance which guarantees in practical linear control systems. This article also presents the state-of-the-art mathematical methods for approximating upper bounds of µ-values. The µ-value is deeply interconnected with D-stability theory of economic models. The key methods includes the computation of upper bounds of µ-values for mixed real and complex uncertainties, optimization based methods, linear matrix inequalities (LMI)-based techniques.

How to Cite

M.-U. Rehman, S. Akram, I. Farxod, N. Ochilov, S. Gafurova, N. Fuzaylova, T. Umidjon, M. Atoeva, Interconnections between µ-Value and D-Stable, D(α)-Stable Matrices from Economic Models, Int. J. Anal. Appl. 23 (2025), 77. https://doi.org/10.28924/2291-8639-23-2025-77

BibTeX

@article{Rehman2025,
  author  = {Rehman, Mutti-Ur and Akram, Saima and Farxod, Ixtiyarov and Ochilov, Narzillo and Gafurova, Shahlokhon and Fuzaylova, Nigina and Umidjon, Tukhtayev and Atoeva, Makhliyo},
  title   = {Interconnections between µ-Value and D-Stable, D(α)-Stable Matrices from Economic Models},
  journal = {Int. J. Anal. Appl.},
  volume  = {23},
  year    = {2025},
  pages   = {77},
  doi     = {10.28924/2291-8639-23-2025-77}
}

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