# A Graph Theory Approach for Regional Controllability of Boolean Cellular Automata

## TL;DR

This paper introduces a graph theory-based method to analyze and determine regional controllability in Boolean cellular automata, focusing on boundary control and leveraging Hamiltonian circuits and strongly connected components.

## Contribution

It presents a novel graph-theoretic framework for regional controllability in cellular automata, providing necessary and sufficient conditions and a preimage-based control method.

## Key findings

- Conditions for controllability are characterized using Hamiltonian circuits.
- Control strategies are derived using a preimage approach.
- The method applies to boundary control in spatially extended systems.

## Abstract

Controllability is one of the central concepts of modern control theory that allows a good understanding of a system's behaviour. It consists in constraining a system to reach the desired state from an initial state within a given time interval. When the desired objective affects only a sub-region of the domain, the control is said to be regional. The purpose of this paper is to study a particular case of regional control using cellular automata models since they are spatially extended systems where spatial properties can be easily defined thanks to their intrinsic locality. We investigate the case of boundary controls on the target region using an original approach based on graph theory. Necessary and sufficient conditions are given based on the Hamiltonian Circuit and strongly connected component. The controls are obtained using a preimage approach.