Exponential Stability of Linear Discrete Systems with Variable Delays via Lyapunov Second Method

Abstract

The paper investigates the exponential stability of a linear system of difference equations with variable delays ( + 1) = ( ) + ∑ =1 ( ) ( − ( )), = 0, 1, . . ., where ∈ N, is a constant square matrix, ( ) are square matrices, ( ) ∈ N ∪ {0}, and ( ) ≤ for an ∈ N. New criteria for exponential stability are derived using the method of Lyapunov functions and formulated in terms of the norms of matrices of linear terms and matrices solving an auxiliary Lyapunov equation. An exponential-type estimate of the norm of solutions is given as well. The efficiency of the derived criteria is numerically demonstrated by examples and their relations to the well-known results are discussed.

Exponential Stability of Linear Discrete Systems with Multiple Delays

“…Although the results are independent, a limiting process (for → 1 + ) indicates that the conditions of the main result in [7] are, in general, more restrictive. Now we will demonstrate that, with respect to the derived estimates of the norms of solutions, the situation is just the opposite and that the estimation (7) is, in general, better than that in [4,Theorem 2]. The last estimation mentioned says that (below, , , , = 1 .…”

“…Finally, we compare the results published in [4][5][6][7] with Theorem 1. The assumptions of Theorem 1 are, for the reduced case = 1 of a single delay, weaker than those of Theorem 2 in [7].…”

“…Obviously, estimation (50) is (due to the absence of the maximal delay ) better than estimation (47). We finish this part with a remark that the results of [5] are generalized in [4]. Results of [6] are on the exponential stability of linear perturbed systems with a single delay.…”

“…The assumptions of Theorem 1 are, for the reduced case = 1 of a single delay, weaker than those of Theorem 2 in [7]. In [4] an analysis of Theorem 2 is carried out. Although the results are independent, a limiting process (for → 1 + ) indicates that the conditions of the main result in [7] are, in general, more restrictive.…”

“…The investigation of the stability of linear difference systems with delay is a constant priority of research. We refer, for example, to [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and to the references therein.…”