# Feasible Pairings for Decentralized Integral Controllability of Non-Square Systems

arXiv: https://arxiv.org/abs/2603.01076 · HTML: https://mlsanity.com/paper/2603.01076/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 27 references that are in the map.
- First posted on arXiv 1 Mar 2026.

## Builds on (most cited first, 2 of 2)

- [Sufficient D-Stability Conditions for Non-Square Matrices](https://mlsanity.com/paper/2406.15440/) (2024)
- [A gallery of diagonal stability conditions with structured matrices (and review papers)](https://mlsanity.com/paper/2301.01272/) (2023)

## Abstract (arXiv)

This paper investigates the determination of feasible input-output pairings for the decentralized integral controllability of non-square systems. The relevance of this problem extends beyond traditional industrial processes into modern AI research, particularly Multi-Agent Reinforcement Learning (MARL), where environments frequently act as strongly non-square mappings that evaluate high-dimensional joint action spaces via comparatively low-dimensional global rewards. To address the stability of these complex distributed architectures, we extend the concept of D-stability to non-square matrices, providing a crucial mathematical foundation. We formally define D-stability for non-square matrices as a direct generalization of the square case. By introducing the concept of ``Squared Matrices'', which are derived from specific column selections of the non-square formulation and directly correspond to candidate control pairings, we establish a fundamental link between the stability of these square sub-components and the original non-square system. Ultimately, we propose sufficient conditions under which the individual Volterra-Lyapunov stability of these squared components guarantees the extended D-stability of the non-square matrix, thereby providing a rigorous method to identify feasible pairings that ensure robust decentralized control across both classical and data-driven applications.