Exponential Stability of Linear Discrete Systems with Multiple Delays

Abstract

The paper investigates the exponential stability and exponential estimate of the norms of solutions to a linear system of difference equations with multiple delays , , where , and are square matrices, and . 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.

1. Preliminaries

The investigation of the stability of linear difference systems with delay is a constant priority of research. We refer, for example, to [1–14] and to the references therein.

The paper considers the exponential stability of linear discrete systems with multiple delayswhere , and are matrices, and . For (1) exponential-type stability and exponential estimate of the rate of convergence of solutions are derived.

Set . The initial Cauchy problem for system (1) is as follows: For a vector , we define . Let be the spectral radius of the matrix . Denote by and the maximum and the minimum eigenvalues, respectively, of a symmetric matrix and define . For a given matrix , we use the norm defined by . In the paper, assume .

The trivial solution , of (1) is called Lyapunov exponentially stable if there exist constants and such that, for an arbitrary solution of (1), where 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 . 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 , then the Lyapunov matrix equationhas 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.

2. Exponential Stability

Let be a parameter. Define auxiliary numbers

Theorem 1. Let , be a fixed positive definite symmetric matrix, let matrix solve the equation (5), and, for a fixed , let , , . Then, system (1) is exponentially stable and, for an arbitrary solution , the estimateholds.

Proof. For the Lyapunov function , inequalitieshold. Let where is given. Let a solution of (1) satisfy . Then, for ,i.e.,Below, we prove that (10) is valid for , too. Assume, on the contrary, that (10) is not always valid. Then, an integer exists such that, for , (10) holds, and, for ,Inequality (11) implies that, for ,andNow computeRearranging this computation, we deriveWe estimate the first difference and use the assumption that the matrix is a solution of equation (5); therefore,andNow we apply inequality (13) to getandInequality can be deduced from the assumption . Therefore, utilizing (8),Since , we getThis inequality can be rewritten as or aswhere Now we prove thatInequality (26) is equivalent with an inequalityAfter some simplification, we getwhich is equivalent with the inequality . Then (24), (26), and (10) imply This inequality contradicts (11). Then, inequality (11) is impossible and (10) holds for every . Moreover, (8) and (10) imply i.e., the inequality equivalent with (7).

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 ; 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 above-mentioned independency of results.

Example 2. Let and let system (1) be of the formwhere and and are constants. We show that Theorem 1 is applicable if and are sufficiently small. We have Lyapunov equation (5) is satisfied, e.g., for Then, , , , and . Simple computations result in and Theorem 1 is applicable if and are sufficiently small since this implies , , and, if the expression is positive, provided that ; that is, if then as well. In such a case, for an arbitrary solution of system (32), (33), the estimate holds.

Since 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 , , and , defined by replacing the entries in the previously given matrices , , and by their absolute values, leads to a matrix whose eigenvalues are , and, obviously, .

Finally, we compare the results published in [4–7] with Theorem 1. The assumptions of Theorem 1 are, for the reduced case 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 ) 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, , , , , and are the same as in the paper)where if , is a fixed positive definite matrix, matrix solves the corresponding Lyapunov matrix equation (5), and Assuming that , , we deduce that for (44) to hold, the following is necessary:the limiting value of is and (42) can approximately be written asConsidering the same limiting process as above, for the validity of (7), an analysis of , implies that inequality (45) must hold in addition to inequalityderived from the assumption . Inequality (48), together with the assumption , yields and (7) can be approximatively written asObviously, 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

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The first, third, and fourth authors have been supported by the Czech Science Foundation under Project 16-08549S. Their work has been realized in CEITEC-Central European Institute of Technology with research infrastructure supported by Project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund. The second author has been supported by the Grant FEKT-S-17-4225 of Faculty of Electrical Engineering and Communication, Brno University of Technology. An earlier presentation of preliminary results was introduced on Thursday (May 19, 2016) at Faculty of Physics and Mathematics, University of Latvia.