Sheaf-theoretic framework for optimal network control

TL;DR

This paper introduces a sheaf-theoretic framework for modeling and solving optimal network control problems, providing theoretical insights and error bounds for discretized Boolean models, with improved bounds for affine dynamics.

Contribution

It develops a novel sheaf-based approach to optimal control, linking sheaf theory with control problems and discretization error analysis.

Findings

Sheaf-theoretic modeling of network control problems.

Equivalence between control solutions and sheaf assignments with minimal consistency radius.

Error bounds for Boolean discretizations, improved for affine dynamics.

Abstract

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.