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

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DOI: https://doi.org/10.28924/2291-8639-23-2025-77
Published: Mar 31, 2025
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Mutti-Ur Rehman, Saima Akram, Ixtiyarov Farxod, Narzillo Ochilov, Shahlokhon Gafurova, Nigina Fuzaylova, Tukhtayev Umidjon, Makhliyo Atoeva, Interconnections between µ-Value and D-Stable, D(α)-Stable Matrices from Economic Models, Int. J. Anal. Appl., 23 (2025), 77.

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

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.

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