Stability of a delay difference system

Abstract

We consider the stability problem for the difference system x n = Ax n−1 + Bx n−k , where A, B are real matrixes and the delay k is a positive integer. In the case A = −I, the equation is asymptotically stable if and only if all eigenvalues of the matrix B lie inside a special stability oval in the complex plane. If k is odd, then the oval is in the right half-plane, otherwise, in the left half-plane. If A + B < 1, then the equation is asymptotically stable. We derive explicit sufficient stability conditions for A I and A −I.

Exponential stability of slowly varying discrete systems with multiple state delays

“…Substituting in , we obtain where Applying the aforementioned reasoning in Equation , according to inequality , it follows that x α ( k ) is a bounded function. Consequently, relation implies the global exponential stability of the zero solution of systems .□ Remark Kipnis & Komissarova , using the characteristic equation, obtained a set of sufficient stability conditions to the autonomous case of system . Computing the quantities β 0 and S 0 of Theorem defined by …”

“…By contrast, many methods different from Lyapunov functions have been successfully applied to establish stability results for discrete-time systems. (e.g., [10][11][12][13]).…”

“…Furthermore, the problem of exponential stability become more complicated than that of a system without time delay and/or uncertainties. (e.g., [6,11,14]).…”

Exact stability sets for a linear difference system with diagonal delay

“…To prove the main theorems, we will investigate the distribution of the roots of F (λ) = 0 given by (6). Notice that…”

“…Recently, there have been many results on the asymptotic stability of linear difference systems with delay; see, e.g., [2,[4][5][6][10][11][12][13]15] and references therein.…”