Stability of Multidimensional Linear Discrete Systems

We study linear multidimensional discrete systems of the form \[ x_α=\sum_{k=1}^m A_kx_{α-e_k}, \qquad α\in\mathbb N^m, \] with uniformly bounded boundary data prescribed on the codimension-one coordinate faces and attractivity understood in the sense of Pringsheim convergence. We first derive an explicit solution representation in terms of sums of ordered matrix products and introduce a shell decomposition whose associated sums satisfy a one-dimensional affine recursion. Our main result is a generalized Fornasini--Marchesini theorem in arbitrary dimension: asymptotic stability is equivalent to the condition \[ ρ\!\left(z_1A_1 + \cdots + z_m A_m \right)<1 \qquad \text{for every } |z_1| = \cdots = |z_m|=1. \] We then study the critical spectral boundary, where the maximal toral spectral radius equals one. We prove that a nonsemisimple unimodular eigenvalue of a toral pencil forces instability, yielding necessary spectral conditions for Lyapunov stability. In the simultaneously diagonalizable case, we obtain a complete classification of asymptotically stable, neutrally stable, and unstable systems in terms of the common eigenvalues. Finally, we show that bounds involving only the individual spectral radii can fail outside this setting: even when all coefficient matrices are nilpotent and their spectral radii sum to zero, a critical system may still be unstable.