Distributed Minimum Weighted Norm Solution To Linear Equations Associated With Weighted Inner Product

In this paper, we propose a distributed method to find the solution of the linear equations Ax = b with minimum energy, i.e. the minimum weighted norm associated with the weighted inner product. We first prove that for a special case when the norm is two-norm, the algorithm can make multiple agents reach the minimum two-norm solution of the global linear equations Ax - b if the agents are initialized at the minimum two-norm solutions of their local equations. We then prove that if there are bounded initialization errors, the final convergence of the algorithm is also bounded away from the minimum two-norm solution of the global linear equations. Next, we prove the case with the two-norm replaced with a weighted norm associated with the weighted inner product. Finally, simulation examples are presented to show the effectiveness of the results in this paper.