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Solution Bounds and Numerical Methods of the Unified Algebraic Lyapunov Equation

Author

Listed:
  • Juan Zhang

    (Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan 411105, China)

  • Shifeng Li

    (Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China)

  • Xiangyang Gan

    (Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan 411105, China)

Abstract

In this paper, applying some properties of matrix inequality and Schur complement, we give new upper and lower bounds of the solution for the unified algebraic Lyapunov equation that generalize the forms of discrete and continuous Lyapunov matrix equations. We show that its positive definite solution exists and is unique under certain conditions. Meanwhile, we present three numerical algorithms, including fixed point iterative method, the acceleration fixed point method and the alternating direction implicit method, to solve the unified algebraic Lyapunov equation. The convergence analysis of these algorithms is discussed. Finally, some numerical examples are presented to verify the feasibility of the derived upper and lower bounds, and numerical algorithms.

Suggested Citation

  • Juan Zhang & Shifeng Li & Xiangyang Gan, 2022. "Solution Bounds and Numerical Methods of the Unified Algebraic Lyapunov Equation," Mathematics, MDPI, vol. 10(16), pages 1-19, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:16:p:2858-:d:885346
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    References listed on IDEAS

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    1. C. H. Lee, 2004. "Solution Bounds of the Continuous and Discrete Lyapunov Matrix Equations," Journal of Optimization Theory and Applications, Springer, vol. 120(3), pages 559-578, March.
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