Distributed Computation for Solving the Sylvester Equation Based on Optimization

This paper solves the Sylvester equation in the form of AX+XB=C in a distributed way, and proposes three distributed continuous-time algorithms for three cases. We start with the basic algorithm for solving a least squares solution of the equation, and then give a simplified algorithm for the case when there is an exact solution to the equation, followed by an algorithm with regularization case. Based on local information and appropriate communication among neighbor agents, we solve the distributed computation problem of the Sylvester equation from the optimization viewpoint, and we prove the convergence of proposed algorithms to an optimal solution in three different cases, with help of the convex optimization and semi-stability.