A necessary and sufficient condition of asymptotic stability for a class of Fornasini-Marchesini models

In this paper, we study the asymptotic stability of a particular class of linear 2D discrete Fornasini-Marchesini models. The solutions of the model can be expressed in terms of doubly indexed sequences in a simple way only when the matrices describing the model commute. In this situation, we are able to analyse directly the limit of all the trajectories. By doing so, we propose the first necessary and sufficient condition for asymptotic stability of Fornasini-Marchesini matrix models. This condition is not computationally challenging as it is ultimately based on the eigenvalues of the matrices describing the model. We support our result with numerical simulations.