# Sheaf-theoretic framework for optimal network control

arXiv: https://arxiv.org/abs/2012.00120 · HTML: https://mlsanity.com/paper/2012.00120/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Nov 2020.

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

- [Prospects for inconsistency detection using large language models and sheaves](https://mlsanity.com/paper/2401.16713/) (2024)

## Abstract (arXiv)

In this paper, we use tools from sheaf theory to model and analyze optimal network control problems and their associated discrete relaxations. We consider a general problem setting in which pieces of equipment and their causal relations are represented as a directed network, and the state of this equipment evolves over time according to known dynamics and the presence or absence of control actions. First, we provide a brief introduction to key concepts in the theory of sheaves on partial orders. This foundation is used to construct a series of sheaves that build upon each other to model the problem of optimal control, culminating in a result that proves that solving our optimal control problem is equivalent to finding an assignment to a sheaf that has minimum consistency radius and restricts to a global section on a particular subsheaf. The framework thus built is applied to the specific case where a model is discretized to one in which the state and control variables are Boolean in nature, and we provide a general bound for the error incurred by such a discretization process. We conclude by presenting an application of these theoretical tools that demonstrates that this bound is improved when the system dynamics are affine.