# Optimal control of network-coupled subsystems: Spectral decomposition and low-dimensional solutions

arXiv: https://arxiv.org/abs/2009.12367 · HTML: https://mlsanity.com/paper/2009.12367/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Sep 2020.

## Built on by (most cited first, 1 of 1)

- [Scalable regret for learning to control network-coupled subsystems with unknown dynamics](https://mlsanity.com/paper/2108.07970/) (2021)

## Abstract (arXiv)

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 ≤ 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.