An inverse-free fixed-time stable dynamical system and its forward-Euler discretization for solving generalized absolute value equations

TL;DR

This paper tackles solving the generalized absolute value equation $Ax - B|x|=c$ by constructing an inverse-free dynamical system that achieves fixed-time convergence to the unique GAVE solution under the condition $\\sigma_{\\min}(A) > \\|B\\|$. The continuous-time flow $\\dot{x} = -\\rho(x) g(\\gamma,x)$ with $g(\\gamma,x)=\\gamma A^T( Ax - B|x| - c)$ and a carefully chosen $\\rho(x)$ yields a Lyapunov decrease $\\dot{V}(x) \\le -c_1 V(x)^{(\\lambda_1+1)/2} - c_2 V(x)^{(\\lambda_2+1)/2}$, guaranteeing global fixed-time stability with an explicit upper bound $T_{\\max}$ that depends on design parameters. The method is inverse-free (no matrix inversions) and the paper further shows that its forward-Euler discretization $x^{(k+1)} = x^{(k)} - \\eta \,\\rho(x^{(k)}) g(\\gamma,x^{(k)})$ yields an explicit $(T,\\epsilon)$-close discrete-time approximation, with a provable range of step sizes $\\eta$ ensuring convergence to an $\\epsilon$-neighborhood of the unique solution within a finite number of iterations. Numerical experiments compare the discrete scheme against several existing methods, demonstrating that the inverse-free Euler discretization is faster and reliably achieves high accuracy. Overall, the work advances stable, fast, and inversion-free strategies for solving GAVE with rigorous fixed-time guarantees and practical discretization schemes.

Abstract

An inverse-free dynamical system is proposed to solve the generalized absolute value equation (GAVE) with a fixed time convergence, where the time of convergence is finite and is uniformly bounded for all initial points. Moreover, an iterative method obtained by using the forward-Euler discretization of the proposed dynamic model is developed and sufficient conditions which guarantee that the discrete iteration globally converge to an arbitrarily small neighborhood of the unique solution of GAVE within a finite number of iterative steps are given. Numerical results illustrate the effectiveness of the proposed methods.

Figures (1)

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Theorems & Definitions (24)

• Lemma 2.1: wuli2020
• Theorem 2.1
• proof
• Corollary 2.1
• Theorem 2.2
• proof
• Remark 2.1
• Definition 2.1
• Lemma 2.2: poly2012
• Lemma 2.3: gbgb2022