The asymptotic stability of stable and time-autonomous discrete multidimensional behaviors

We generalize the important paper D. Napp-Avelli, P. Rapisarda, P. Rocha, 'Time-relevant stability of 2D systems', Automatica 47(2011), 2373-2382, to discrete time-autonomous (ta) (=time-relevant), but not necessarily square-autonomous behaviors in arbitrary dimensions. The discrete domain of the independent variables is the lattice of vectors of integers of arbitrary (but fixed) length whose first component is a natural number and interpreted as a discrete time instant. The stability of an autonomous behavior is defined by algebraic conditions on its characteristic variety. Under suitable additional conditions a discrete stable and time-autonomous behavior is asymptotically stable in the sense that its trajectories converge to zero when the time tends to infinity. We derive algorithms for the constructive verification of the assumptions of most of our results and in particular establish a constructive normal form of ta behaviors in arbitrary dimensions. The Fourier transform on finitely generated free abelian groups plays an important part in the derivations as it already did in the quoted paper for the group of integers. Stability and stabilization of multidimensional discrete behaviors were previously discussed by various colleagues, for instance by Bisiacco, Bose, Fornasini, Lin, Marchesini, Quadrat, Shankar, Sule, Valcher and Willems, but only partly from the analytic point of view.