Boundary Constraints for Minimum Cost Control of Directed Networks

Controlling directed networks with minimum cost has become an emerging branch in the areas of complex networks and control recently. In this paper, we focus on this minimum cost control problem subject to two types of boundary constraints, namely, trace boundary constraint and orthonormal boundary constraint on the input matrices. First, the minimum cost control problem is formulated as an optimization model for each type of boundary constraint. Next, two iterative algorithms, named as trace-constraint-based projected gradient method and orthonormal-constraint-based projected gradient method, are proposed to solve the optimal problem, respectively. Then, convergence properties of both algorithms are established. Finally, extensive simulation results show the effectiveness of our methods based on detailed comparisons between the two boundary conditions. We believe the results reveal some interesting physical insights for the optimal control of directed networks.