Distributed Computation for Solving the Sylvester Equation Based on Optimization

This letter solves the Sylvester equation in the form of $AX+XB=C $ in a distributed way, and proposes a distributed continuous-time algorithm when there is at least one exact solution to the equation. 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 design a distributed algorithm based on the saddle-point dynamics and derivative feedback. Finally, we prove the exponential convergence of the proposed algorithm to an optimal solution with help of the convex optimization and semi-stability theory.