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

Abstract

We generalize the important paper of Napp et al. (Automatica 47, 2373–2382, 2011), to discrete time-autonomous (ta) (=time-relevant), but not necessarily square-autonomous behaviors in arbitrary dimensions. This paper and therefore also the present one were essentially influenced by the papers of Wood et al. (SIAM J Control Optim 43, 1493–1520 2005), and Willems (Proceedings of the International Conference on Multidimensional (nD) Systems, Aveiro, 2007). In the present paper 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 a spectral condition on its characteristic variety. The behavior is time autonomous if each trajectory is determined by a fixed number of its initial values. Under a weak additional condition, a discrete stable and time-autonomous behavior is asymptotically stable in the sense that under suitable initial conditions 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 papers. Stability and stabilization of multidimensional discrete behaviors were previously discussed by various colleagues, for instance by Bisiacco, Bose, Fornasini, Lin, Marchesini, Pillai, Quadrat, Rogers, Shankar, Sule, Valcher, Wood, but only partly from the analytic point of view.