Zero-Time-Controllability and Dead-Beat Control of Two-Dimensional Behaviors

In this paper two-dimensional (2D) discrete behaviors, defined on the grid $\mathbb{Z}_+\times \mathbb{Z}$ and having the time as (first) independent variable, are investigated. For these behaviors, by emphasizing the causality notion that is naturally associated with the time variable, we introduce two new concepts of controllability. Algebraic characterizations of time-controllability and of zero-time-controllability are provided, and it is shown that behaviors endowed with these properties admit special decompositions. Next, the dead-beat control (DBC) problem and the concept of admissible DBC are introduced and related to the zero-time-controllability property. Differently from what happens with one-dimensional behaviors, zero-time-controllability does not ensure the existence of regular DBCs, and stronger algebraic properties need to be imposed on the behavior. Finally, necessary and sufficient conditions for the existence of a DBC that makes the resulting behavior both strongly autonomous and nilpotent are provided.