General control of linear cellular automata

TL;DR

This paper develops a general controllability theory for linear cellular automata, extending control concepts from continuous systems to discrete cellular automata, with practical criteria based on a controllability matrix.

Contribution

It introduces a controllability matrix for linear cellular automata and establishes invertibility as a criterion for controllability, filling a gap in control theory for discrete systems.

Findings

Controllability of linear cellular automata is characterized by the invertibility of the controllability matrix.

The paper applies the theory to 1D and 2D Boolean cellular automata examples.

A new framework extends control theory to discrete, cellular automaton systems.

Abstract

In mathematics and engineering, control theory is concerned with the analysis of dynamical systems through the application of suitable control inputs. One of the prominent problems in control theory is controllability which concerns the ability to determine whether there exists a control input that can steer a dynamical system from an initial state to a desired final state within a finite time horizon. There is a general theory for controlling linear or linearizable system, but it cannot be applied to discrete systems like cellular automata, which is the problem of that we address in this paper. We develop a general theory for linear (and affine) cellular automata, and apply it to examples of one-dimensional and two-dimensional Boolean cases. We introduce the concept of controllability matrix and show that controllability holds if and only if the controllability matrix is invertible.