General control of linear cellular automata
It provides a foundational framework for controlling discrete dynamical systems like cellular automata, which previously lacked a general theory.
The paper develops a general control theory for linear and affine cellular automata, introducing a controllability matrix and proving that controllability holds if and only if this matrix is invertible. The theory is applied to one- and two-dimensional Boolean cellular automata.
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.