Distributed Control Design for Solving Linear Algebraic Equations via Adjustable Domains

This article aims to develop a general and designable distributed algorithm for solving linear algebraic equations (LAEs), which departs from the design framework based on orthogonal projection. The concept of adjustable domains for the parameter matrix is introduced, enabling the algorithm to derive flexible and variable updating rules for agents. By leveraging adjustable domains in control design, all agents can exponentially converge to a common (least squares) solution of (un)solvable LAEs under arbitrary initialization conditions, regardless of whether the LAEs admit a unique solution or multiple solutions. Moreover, two novel distributed algorithms for obtaining the least squares solution are proposed within both row and column partitioning frameworks. A simulation example is provided to demonstrate the effectiveness of the proposed distributed algorithms.