Dead-beat control of two-dimensional behaviors

In this paper two-dimensional (2D) discrete behaviors, defined on the grid ℤ₊ × 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 sum decompositions. Finally, the dead-beat control (DBC) problem and the concept of admissible DBC are introduced and related to the zero-time-controllability property.