Exponential Stability of Linear Discrete Systems with Multiple Delays

The paper investigates the exponential stability and exponential estimate of the norms of solutions to a linear system of difference equations with multiple delays x(k + 1) = Ax(k) + ∑si=1 Bix(k − mi), k = 0, 1, . . ., where s ∈ N, A and Bi are square matrices, and mi ∈ N. New criterion for exponential stability is proved by the Lyapunov method. An estimate of the norm of solutions is given as well and relations to the well-known results are discussed.

For the foundations of stability theory to difference equations, we refer, e.g., to [15,16].
As it is customary, the asymptotic stability of (1) can be investigated by analyzing the roots of the related characteristic equation.The characteristic equation relevant to (1) is a polynomial equation of degree ( + 1).For large  and , it is impossible, in a general case, to solve such a problem.For example, the Schur-Cohn criterion [16,17] is not applied because the computer calculation is too time-consuming.
Below, the exponential stability of (1) is analyzed by the second Lyapunov method and the following well-known result is utilized: if () < 1, then the Lyapunov matrix equation has a unique solution, a positive definite symmetric matrix  for an arbitrary positive definite symmetric × matrix  (we refer, for example, to [16]).
In Section 2, the exponential stability of system (1) and exponential estimates of solutions are investigated.Concluding remarks and relations to the well-known results are included in Section 3.

Concluding Remarks
Based on the investigations on exponential stability published previously, the present paper brings in Theorem 1 new results.The exponential rate of convergence of solutions is studied in [1] assuming that det  ̸ = 0; therefore, the results are independent.Let us discuss the independence of the results of other sources listed in the references.The criteria for the exponential stability of nonlinear difference systems, for example, are proved in [11,14].The nonlinearities are estimated by some linear terms with matrices having nonnegative entries with the sums of such matrices being, for example, a constant nonnegative matrix with a spectrum less than 1.In general, an attempt to estimate the right-hand sides of the systems by a nonnegative matrix does not provide a matrix with a spectrum less than 1 and the results are independent.For special classes of equations, sharp criteria (depending on delay) for detecting asymptotic stability are proved in [2,3].The following example illustrates the abovementioned independency of results.
Since det  = 0 in the above example, the results of the paper [1] are not applicable to system (32), (33).Moreover, an attempt to apply results of [11,14] is not successful since the sum of matrices  * ,  * 1 , and  * 2 , defined by replacing the entries in the previously given matrices ,  1 , and  2 by their absolute values, leads to a matrix 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].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.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 . . ., ,  and  are the same as in the paper) ,  ≥ 1. (47) Considering the same limiting process as above, for the validity of ( 7), an analysis of   ,  = 1, 2, 3 implies that inequality (45) must hold in addition to inequality (50) 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.Among others, it is proved [6, Theorem 3] that inequality (50) holds for nondelayed linear systems  ( + 1) =  () ,  = 0, 1,, . . . .