  In this paper, we investigate optimal control of network-coupled subsystems
where the dynamics and the cost couplings depend on an underlying undirected
weighted graph. The graph coupling matrix in the dynamics may be the adjacency
matrix, the Laplacian matrix, or any other symmetric matrix corresponding to
the underlying graph. The cost couplings can be any polynomial function of the
underlying coupling matrix. We use the spectral decomposition of the graph
coupling matrix to decompose the overall system into (L+1) systems with
decoupled dynamics and cost, where L is the rank of the coupling matrix.
Furthermore, the optimal control input at each subsystem can be computed by
solving (Ldist + 1) decoupled Riccati equations where Ldist (Ldist \leq L) is
the number of distinct non-zero eigenvalues of the coupling matrix. A salient
feature of the result is that the solution complexity depends on the number of
distinct eigenvalues of the coupling matrix rather than the size of the
network. Therefore, the proposed solution framework provides a scalable method
for synthesizing and implementing optimal control laws for large-scale
network-coupled subsystems.
