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

Copyright © 2017 Josef Dibĺık. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The paper investigates the exponential stability of a linear system of difference equations with variable delays x(k + 1) = Ax(k) + ∑si=1 Bi(k)x(k − mi(k)), k = 0, 1, . . ., where s ∈ N, A is a constant square matrix, Bi(k) are square matrices, mi(k) ∈ N ∪ {0}, and mi(k) ≤ m for anm ∈ N. New criteria for exponential stability are derived using the method of Lyapunov functions and formulated in terms of the norms ofmatrices of linear terms andmatrices 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.


Introduction
Many mathematical real-life models are described by nonlinear systems of difference equations with delay.To derive local exponential stability results for such systems, it is reasonable to investigate the corresponding linear difference systems first.Such systems are the subject of the paper.
Simultaneously, we give an exponential estimate of the norms of solutions.The results are compared with the results published previously.
It is obvious that system (1) can be transformed into a system with constant delays where  *  (),  = 0, . . ., , are suitable  ×  matrices and  *  are natural numbers such that defines the same solution to system (4) as for system (1).
It should, however, be emphasized that such a transformation does not simplify the problem itself because the matrices  *  (),  = 1, . . ., , remain variable and, for example, the method based on the study of the corresponding characteristic equation is not applicable.
The paper studies the exponential stability of (1) by the second Lyapunov method.Recall the well-known fact, necessary for this method to be applied, that when the stability of linear discrete equations is investigated by Lyapunov function () =    with an × positive definite symmetric matrix , an important role is played by what is called the Lyapunov equation where  is a given  ×  matrix and  is an  ×  matrix.The linear system ( + 1) = (),  = 0, 1, . .., is exponentially stable (i.e., () < 1, where  is the spectral radius of  defined by () = max{||:  ∈ ()}, () fl { ∈ C: det( − ) = 0} is the set of all eigenvalues of , and  is the unit matrix), if and only if, for an arbitrary positive definite symmetric  ×  matrix , the matrix equation ( 5) has a unique solution-a positive definite symmetric matrix  ( [12,13]).
The remaining part of the paper is organized as follows.In Section 2, exponential stability of system (1) and exponential estimates are investigated.The case of constant matrices is treated in Section 3. Section 4 contains a concluding discussion, comparisons, and examples demonstrating the results obtained and their independence of the well-known results.

Exponential Stability in the Case of Constant Matrices
The result of Theorem 2 can be improved if matrices   (),  = 1, . . ., , in (1) are constant; that is,   () ≡   = const.In such a case, the numbers (, ),   (, ),  = 1, . . ., , do not depend on  and are constant as well.We redefine them as follows: In such a case, the function Θ(, ),  ≥ 0, defined by (11), does not depend on  either and, in the following, we use a constant Theorem 3 is an analog of Theorem 2 for this "constant" case.
The difference (except for the above-mentioned changes) is that inequality (9) in Theorem 2 is improved to inequality (31), where no number  ∈ (0, 1) is necessary.
Proof.The proof can be done along the lines of the proof of Theorem 2. The inequality 0 < Θ() < 1 can be proved in much the same way as the inequality 0 < Θ(, ) ≤ ,  = 0, 1, . ...Then, the proof can be repeated with  fl Θ().

A Discussion of the Results Obtained, Comparisons, and Examples
Theorems 2 and 3 provide sufficient conditions for exponential stability.The novelty of our approach is based on a new method of estimating the full difference of Lyapunov function via auxiliary functions (, ),   (, ),  = 1, . . ., , or via auxiliary constants (),   (),  = 1, . . ., .Theorem 3 generalizes Theorem 1 in [4].Some results can detect asymptotic stability but do not provide estimates of solutions, often necessary for computational purposes (in spite of the fact that, for special classes of equations, they can provide criteria on asymptotic stability, depending on delay, of the type "if on only if").We refer, for example, to the papers [3,6,14].Thus, an advantage over these results is the explicitly expressed estimation of the norm of an arbitrary solution.
The results are also independent of those in other sources mentioned in the list of references.For example, some new criteria for exponential stability of nonlinear difference systems with time-varying delay were recently proved in [9] where nonlinearities are estimated by linear terms whose matrices are nonnegative and their sum can, for example, be estimated by a constant nonnegative matrix with a spectrum less than 1.For more details, we refer to [9, Theorem 2.2].Examples 5 and 6 consider linear systems.Unfortunately, an attempt to estimate the right-hand sides of the systems by a nonnegative matrix does not provide a matrix with a spectrum less that 1 and, moreover, none of the cases described by the theorem is applicable.The results in [11] can be considered similarly in the case of absence of stochastic terms.
The exponential stability of linear systems is analyzed in [15] as well, where det ̸ = 0 is assumed.In Examples 5 and 6, det = 0. So, the results of [15] are not applicable either.
Discussing how realistic conditions of Theorems 2 and 3 are, we can conclude that, assuming that the norms of matrices   are small enough, condition (30) of Theorem 3 will be valid if Condition ( 31) is equivalent with and is true as well if the above norms are small enough.Thus, crucial for the applicability of Theorem 3 is inequality (33).In such a case, Theorem 3 is applicable in principle.A similar discussion applies with respect to the assumptions of Theorem 2.
In establishing the stability or exponential stability of linear systems with constant coefficients and a single delay, [5] utilizes a different set of sufficient conditions (independent of our results).The main result [5,Theorem 2] has the form of the following theorem.Theorem 4. Let () < 1,  be a fixed positive definite symmetric  ×  matrix, let matrix  solve the corresponding Lyapunov matrix equation (5), and, for a fixed  > 1, let where (36) Then, the system with delay is exponentially stable and, for an arbitrary solution  = (), the estimate We compare this result with our Theorem 3 adapted for this case (in its formulation, we set  = 1,  1 () ≡ ,  1 ≡ ).
and, evidently, is not equivalent with (39).It is more restrictive and can be regarded as equivalent with (39) only if () = 1 and  → 1 + .In addition, in Theorem 3, an inequality similar to inequality  * 2 (, , ) > 0 is not necessary.We demonstrate the results obtained by the following examples.
Example 5. Let  = 2 and let system (1) be of the form where  ≥ 0. Using Theorem and give an exponential estimate of the norm of solutions.We have and () = 0. Lyapunov equation ( 5) is satisfied, for example, for  = ( 1 1.99 For an arbitrary solution (), the estimate ≐ 24, 99126 (0.0079928) /2(+1) ‖ (0)‖  ,  ≥ 0, Example 6.Let  = 2 and let system (1) be of the form where  ≥ 0. Using Theorem 2, we prove that the system is exponentially stable and, for an arbitrary solution (), the estimate it is possible to use some previous computations with the same matrices  and .We obtain and is valid, for example, for  = 0.99984.From (10), we derive inequality (52).
Remark 7. As noted above, the results of the recent paper [9] are not applicable to Example 5 or 6.We will explain this using Example 6.To apply, for example, Theorem 2.3 in [9] we develop the best possible estimations of the right-hand sides of (51) (below, the notation of the above-mentioned paper is used).We get  (57) The eigenvalues of the matrix  0 +  1 are  1 ≈ −0.00012,  2 ≈ 4.00012.The inequality ( 0 +  1 ) < 1, necessary for exponential stability, does not hold and Theorem 2.3 is not applicable.Similarly, we can show that neither Theorem 2.2 nor Theorem 2.4 is applicable and, therefore, such results are independent of the presented results.Similarly, the results of the papers [1,2,7,8,10] are not applicable either since, in an analysis in [9], the authors show that their results are more general.Then, it is obvious that our results are independent of these results as well.