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A Comprehensive Review of Matrix Equations in Dynamical Systems and Control Theory

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  • Chacha Stephen Chacha

Abstract

Matrix equations are of foundational importance in the modeling, investigation, and control of dynamical systems. This review discusses various classes of matrix equations, their solutions, and their relevance in control theory and dynamical systems. Key topics include Lyapunov and Riccati equations, iterative techniques such as Krylov subspace approaches and the alternating direction implicit technique, and the influence of condition numbers on numerical robustness. In addition, model‐order reduction methods and reduced‐order models are examined for their impact on transient behavior and computational efficiency. Finally, emerging directions such as data‐driven methodologies, the Koopman operator perspective, and quantum computing are explored, alongside outstanding challenges and prospects for future research.

Suggested Citation

  • Chacha Stephen Chacha, 2025. "A Comprehensive Review of Matrix Equations in Dynamical Systems and Control Theory," Journal of Applied Mathematics, John Wiley & Sons, vol. 2025(1).
    • Handle: RePEc:wly:jnljam:v:2025:y:2025:i:1:n:3745816
    DOI: 10.1155/jama/3745816
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    References listed on IDEAS

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    1. Dehghan, Mehdi & Shirilord, Akbar, 2019. "A generalized modified Hermitian and skew-Hermitian splitting (GMHSS) method for solving complex Sylvester matrix equation," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 632-651.
    2. Long-Sheng Liu & Qing-Wen Wang & Mahmoud Saad Mehany, 2022. "A Sylvester-Type Matrix Equation over the Hamilton Quaternions with an Application," Mathematics, MDPI, vol. 10(10), pages 1-20, May.
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