Time-autonomy and time-controllability of discrete multidimensional behaviours

Predecessors and starting point of the present paper on multidimensional discrete time-autonomy and time-controllability were three papers from 2002-2004 by Sasane, Controneo, Thomas and Willems on the continuous case, a paper from 2011 by Napp, Rapisarda and Rocha on time-relevant 2D-behaviors and a submitted paper by Bisiacco and Valcher on 2D-time-controllability and dead-beat control. We also acknowledge interesting talks on this subject by Bisiacco and Valcher in Innsbruck. We develop a general framework for discrete multidimensional behaviors in which the discrete time variable in N plays a special part among all independent variables. Following the models of our predecessors we define time-autonomy and time-controllability of a behavior by properties of its trajectories. Goal of the paper is the characterization of these properties by constructive algebraic conditions which is fully achieved for time-autonomy and for timecontrollability in dimension two and partially for time-controllability in higher dimensions than two. AMS-classification: 93B05, 93B25, 93B99, 93C05, 93C25