Stability Results for a Class of Difference Systems with Delay

Eva Kaslik

doi:10.1155/2009/938492

Abstract

Considering the linear delay difference system , where , is a real matrix, and is a positive integer, the stability domain of the null solution is completely characterized in terms of the eigenvalues of the matrix . It is also shown that the stability domain becomes smaller as the delay increases. These results may be successfully applied in the stability analysis of a large class of nonlinear difference systems, including discrete-time Hopfield neural networks.

Highlights

IntroductionWe will characterize the stability region of the null solution for the following class of linear delay difference systems:
In this paper, we will characterize the stability region of the null solution for the following class of linear delay difference systems:x n 1 ax n Bx n − k ∀n ≥ k, 1.1 where a ∈ 0, 1, B is a p × p real matrix, and k is a positive integer
Considering the linear delay difference system x n 1 ax n Bx n − k, where a ∈ 0, 1, B is a p × p real matrix, and k is a positive integer, the stability domain of the null solution is completely characterized in terms of the eigenvalues of the matrix B

Summary

Introduction

We will characterize the stability region of the null solution for the following class of linear delay difference systems:. Due to the properties of the curve Γ stated in Remark 2.3, we obtain that the polynomial Pλ z has a unique root on the unit circle if and only if λ ∈ Γ \ {ca,k θa1,k , ca,k θa2,k , . The step of the proof is to show that when the complex parameter λ λ1 iλ[2] leaves the domain Δja,k by crossing its boundary Γja,k at a value λ ca,k θ isa,k θ , with θ ∈ −θaj ,k, −θaj−,k1 ∪ θaj−,k1, θaj ,k , the root z z λ of the polynomial Pλ z , which is equal to eiθ when λ λ , crosses the unit circle. As zk 1 − azk λ1 iλ[2] and zk 1 − azk λ1 − iλ[2], differentiating with respect to λ1 and with respect to λ2, we obtain

Re zPλ z Pλ z 2

Re w1 eiθ Pλ

Characterization of the Stability Domain

Dependence of the Stability Domain on the Delay

Some Particular Cases

Conclusions and Future Directions