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Convex Quadratic Equation

Author

Listed:
  • Li-Gang Lin

    (National Central University)

  • Yew-Wen Liang

    (National Chiao Tung University)

  • Wen-Yuan Hsieh

    (Sonix Technology Co., Ltd)

Abstract

Two main results (A) and (B) are presented in algebraic closed forms. (A) Regarding the convex quadratic equation, an analytical equivalent solvability condition and parameterization of all solutions are formulated, for the first time in the literature and in a unified framework. The philosophy is based on the matrix algebra, while facilitated by a novel equivalence/coordinate transformation (with respect to the much more challenging case of rank-deficient Hessian matrix). In addition, the parameter-solution bijection is verified. From the perspective via (A), a major application is re-examined that accounts for the other main result (B), which deals with both the infinite and finite-time horizon nonlinear optimal control. By virtue of (A), the underlying convex quadratic equations associated with the Hamilton–Jacobi equation, Hamilton–Jacobi inequality, and Hamilton–Jacobi–Bellman equation are explicitly solved, respectively. Therefore, the long quest for the constituent of the optimal controller, gradient of the associated value function, can be captured in each solution set. Moving forward, a preliminary to exactly locate the optimality using the state-dependent (resp., differential) Riccati equation scheme is prepared for the remaining symmetry condition.

Suggested Citation

  • Li-Gang Lin & Yew-Wen Liang & Wen-Yuan Hsieh, 2020. "Convex Quadratic Equation," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 1006-1028, September.
    • Handle: RePEc:spr:joptap:v:186:y:2020:i:3:d:10.1007_s10957-020-01727-5
    DOI: 10.1007/s10957-020-01727-5
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    References listed on IDEAS

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    1. Abolvafaei, Mahnaz & Ganjefar, Soheil, 2020. "Maximum power extraction from wind energy system using homotopy singular perturbation and fast terminal sliding mode method," Renewable Energy, Elsevier, vol. 148(C), pages 611-626.
    2. David G. Luenberger & Yinyu Ye, 2016. "Linear and Nonlinear Programming," International Series in Operations Research and Management Science, Springer, edition 4, number 978-3-319-18842-3, March.
    3. S. C. Beeler & H. T. Tran & H. T. Banks, 2000. "Feedback Control Methodologies for Nonlinear Systems," Journal of Optimization Theory and Applications, Springer, vol. 107(1), pages 1-33, October.
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    Cited by:

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