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Neurodynamic optimization approaches with finite/fixed-time convergence for absolute value equations.

Xingxing Ju1, Xinsong Yang1, Gang Feng2

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|July 16, 2023
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This study introduces three novel neurodynamic methods for solving absolute value equations (AVEs). Two methods offer finite-time convergence, while the third provides fixed-time convergence, robust against perturbations.

Keywords:
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Area of Science:

  • Computational Mathematics
  • Dynamical Systems Theory
  • Numerical Analysis

Background:

  • Absolute Value Equations (AVEs) present significant challenges in various scientific and engineering fields.
  • Existing methods for solving AVEs often suffer from slow convergence or sensitivity to initial conditions.
  • Neurodynamic approaches offer a promising alternative for real-time solutions to complex equations.

Purpose of the Study:

  • To develop novel, accelerated, inverse-free neurodynamic approaches for solving Absolute Value Equations (AVEs).
  • To introduce both finite-time and fixed-time converging algorithms for AVEs.
  • To analyze the convergence properties and robustness of the proposed methods.

Main Methods:

  • Design and implementation of three distinct inverse-free neurodynamic models.
  • Development of finite-time converging algorithms for AVEs.
  • Development of a fixed-time converging algorithm with uniformly bounded settling time.

Main Results:

  • The proposed finite-time converging approaches demonstrate convergence to AVE solutions within a finite time.
  • The fixed-time converging approach achieves convergence in a fixed time, independent of initial conditions.
  • All proposed neurodynamic approaches exhibit robustness against bounded vanishing perturbations.

Conclusions:

  • The novel neurodynamic approaches provide efficient and robust solutions for Absolute Value Equations.
  • The fixed-time convergence offers predictable performance across various initial states.
  • The methods are validated through numerical examples and applications in boundary value problems.