The Stability Cone for a Matrix Delay Difference Equation

Abstract

We construct a stability cone, which allows us to analyze the stability of the matrix delay difference equation 𝑥𝑛=𝐴𝑥𝑛−1+𝐵𝑥𝑛−𝑘. We assume that 𝐴 and 𝐵 are 𝑚×𝑚 simultaneously triangularizable matrices. We construct 𝑚 points in ℝ3 which are functions of eigenvalues of matrices 𝐴, 𝐵 such that the equation is asymptotically stable if and only if all the points lie inside the stability cone.