Continuous-time distributed algorithms for solving linear algebraic equation

In this paper, a multi-agent distributed continuous-time algorithm is proposed to solve a large-scale linear algebraic equation Ax = d
_o
. Unlike many existing results assuming each agent knows a few rows of A, the algorithm proposed in this paper assumes each agent knows a few columns of A. To solve the linear algebraic equation, the problem is first converted to an optimization problem with a linear constraint. Then, a distributed continuous-time algorithm is designed based on the Lagrangian function of the optimization problem. The algorithm is proved to solve the linear algebraic equation with any initial condition via a Lyapunov approach. An example is presented to show the efficacy of the proposed algorithm.