New Characterization of regional controllability and controllability of deterministic cellular automata via topological and symbolic dynamics notions

Abstract: This article presents a new characterization of controllability and regional controllability of Deterministic Cellular Automata (CA for short). It focuses on analyzing these problems within the framework of control theory, which have been extensively studied for continuous systems modeled by partial differential equations (PDEs). In the analysis of linear systems, the Kalman rank condition is ubiquitous and has been used to obtain the main results characterizing controllability. The aim of this paper is to highlight new ways to prove the regional controllability and controllability of CA using concepts from symbolic dynamics instead of using the Kalman condition. Necessary and sufficient conditions are given using the notions of chain transitive, chain mixing, trace approximation, transitive SFT and mixing SFT. Finally, we demonstrate that the presence of visibly blocking words implies that the cellular automaton is not controllable.