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

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.