# On the Optimal Control of Network LQR with Spatially-Exponential Decaying Structure

arXiv: https://arxiv.org/abs/2209.14376 · HTML: https://mlsanity.com/paper/2209.14376/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2022.

## Built on by (most cited first, 4 of 4)

- [Global Convergence of Localized Policy Iteration in Networked Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2211.17116/) (2022)
- [Scalable Reinforcement Learning for Linear-Quadratic Control of Networks](https://mlsanity.com/paper/2401.16183/) (2024)
- [Distributed Policy Gradient for Linear Quadratic Networked Control with Limited Communication Range](https://mlsanity.com/paper/2403.03055/) (2024)
- [Exploiting inter-agent coupling information for efficient reinforcement learning of cooperative LQR](https://mlsanity.com/paper/2504.20927/) (2025)

## Abstract (arXiv)

This paper studies network LQR problems with system matrices being spatially-exponential decaying (SED) between nodes in the network. The major objective is to study whether the optimal controller also enjoys a SED structure, which is an appealing property for ensuring the optimality of decentralized control over the network. We start with studying the open-loop asymptotically stable system and show that the optimal LQR state feedback gain K is `quasi'-SED in this setting, i.e. |[K]ij|~ Oleft(e^-fraccpolyln(N)dist(i,j)right). The decaying rate c depends on the decaying rate and norms of system matrices and the open-loop exponential stability constants. Then the result is further generalized to unstable systems under a SED stabilizability assumption. Building upon the `quasi'-SED result on K, we give an upper-bound on the performance of kappa-truncated local controllers, suggesting that distributed controllers can achieve near-optimal performance for SED systems. We develop these results via studying the structure of another type of controller, disturbance response control, which has been studied and used in recent online control literature; thus as a side result, we also prove the `quasi'-SED property of the optimal disturbance response control, which serves as a contribution on its own merit.