Matrix Nearness Problems, <i>D</i> -Stability and <i>μ</i> -Values in Economic Models

In this paper, we investigate the interconnection between <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i>-stability and the structured singular value (μ-values) to analyze linear time invariant dynamical systems subject to structured uncertainties. The new results on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i>-stability ensures that the spectrum remains enclosed into a specified domain of the complex plane subject to admissible perturbations which are critical for stability. The computation of structured singular values allow to measure the robustness, performance and stability under structured uncertainty. The results on relationship between <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i>-stability and μ-analysis are developed by exploring how <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i>-stable regions can be characterized within the μ-framework. Numerical testing demonstrate the the behavior of singular values and structured singular values under the proposed framework.

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