A graph theory approach for regional controllability of Boolean cellular automata

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.

Programmable pattern formation in cellular systems with local signaling

“…S1). The ten distinct rules for linear topology were also independently 22 identified by another study 23,24 .…”

Some Control and Observation Issues in Cellular Automata

“…Theorem 4. A Cellular Automata is regionally controllable iff there exists a t such that the graph associated to the transformation matrix C t contains a Hamiltonian circuit, that is a circuit of a graph G = (V, AR) is a simple directed path of G that includes every vertex exactly once [11].…”

“…It consists in considering objective functions defined on a sub-region of the domain and exerting control actions on the boundary of the target region. This problem has been dealt with several tools: namely Kalman theorem, Markov Chains, graph theory [10,11]. The extension to non-linear CA has also been studied in these works.…”

Synchronization of elementary cellular automata

“…This state estimation problem is widely studied by classical control theory (Kalman 1963;Sarachik and Kreindler 1965) and it follows from the verification of observability, a notion that ensures that the sensors are well placed. This notion of observability can be applied to cellular automata (CA) (El Yacoubi et al 2021;Ple ´net et al 2022;Dridi et al 2019) (and by extension to Boolean networks Zhu et al (2018), which can be seen as a generalization of CA) but its evaluation has proven to be extremely complicated when one deals with non-linear CA Ple ´net et al (2022).…”