Exponential Stability Criteria for Nonautonomous Difference Systems

Abstract

The aim of this paper is to characterize the exponential stability of linear systems of difference equations with slowly varying coefficients. Our approach is based on the generalization of the freezing method for difference equations combined with new estimates for the norm of bounded linear operators. The main novelty of this work is that we use estimates for the absolute values of entries of a matrix-valued function, instead of bounds on its eigenvalues. By this method, new explicit stability criteria for linear nonautonomous systems are derived.

1. Introduction

In the theory of difference equations it is well known that the placement of eigenvalues in the complex plane of a time-invariant linear system is a necessary and sufficient condition to ensure the stability or exponential stability. However, in time-varying systems, the stability and exponential stability are not characterized by the spectrum of transition matrices (see, e.g., [1, 2]). Desoer [1] illustrated the same instability characteristic of a class of discrete-time-varying systems, but remedied the situation considering bounded and sufficiently slowly varying coefficients. More explicitly, Desoer considered the system in (the Euclidean -dimensional space): where and for all ( denotes the class of -matrices with real elements), and his assumptions are as follows:(a)there is a finite such that (b)for some , (c) is sufficiently small.

Under this set of conditions it is proven that system (1) is exponentially stable. Actually, Desoer uses (a) and (b) to generate a bound of the form where and depends on , and but is independent of , and then uses a Lyapunov argument to show that system (1) is exponentially stable if is small enough.

Without the restriction on the rate of variation on , the system (1) may have exponentially increasing solutions. Thus, there must be an additional condition on in order to get stability.

It is well known that the Lyapunov function method serves as a main technique to reduce a given complicated system into a relatively simpler system, and it provides useful applications to control theory, but finding Lyapunov functions is still a difficult task (see, e.g., [1, 3–6]). By contrast, many methods different from Lyapunov functions have been successfully applied to the stability analysis of discrete-time systems (see, e.g., [1, 2, 5, 7–10]).

Recently, Gil and Medina [11, 12] and Medina [13–15] begun the study of stability and stabilizability theory for discrete-time systems by means of new estimates for the powers of matrix-valued and operator-valued functions.

In the last decades many investigations have dealt with exponential stability conditions for discrete-time-varying systems [6–8, 16–19] or continuous time-varying systems [2, 7–9].

Our aim is to relax (b) by using estimates for the absolute values of entries of a matrix-valued function, instead of bound on its eigenvalues. Our proof technique is based on the generalization of the “freezing” method for difference equations (see, e.g., Gil and Medina [11]) combined with new estimates for the norm of powers of variable matrices.

If is periodic, then the discrete Floquet theory [18] provides necessary and sufficient conditions for stability. However, the Floquet approach requires calculation of the Monodromy matrix, which is generally a difficult task. Thus, our results are of relevance even in the periodic case.

To the best of our knowledge, this work is the first using the above mentioned approach to develop a theory concerning the exponential stability of linear time-varying discrete systems. Besides our results being explicit, they are easy to verify. Moreover, the solutions’ estimates give us the possibility to investigate linear and nonlinear perturbations of system (1). Indeed, this approach has little overlapping with the existing literature, mainly because our results do not require solving the characteristic polynomial associated with the variable matrices , which in turn is a difficult task for higher dimensional systems.

The structure of this paper is as follows. In Section 2, we introduce some notations and the fundamental results concerning estimates for the absolute values of entries of matrix-valued functions of finite matrices. In Section 3, exponential stability results and its consequences are established for nonautonomous discrete-time systems. In Section 4, as an application of the previous results, we characterize the exponential stability of nonlinear perturbations of system (1). Finally, Section 5 is devoted to the discussion of our results: we highlight the main conclusions and state some directions for future research.

2. Preliminaries and Problem Statement

Let be a complex Euclidean space with the scalar product , as well as the unit matrix . Let be the spectrum of a linear operator (a matrix) and the resolvent of . For a scalar valued function , holomorphic on the spectrum of , the matrix-valued function is defined by where is a closed contour surrounding .

Let be a fixed orthonormal basis in and the entries of a matrix in this basis.

We put ; that is, is a matrix whose entries are absolute values of in . If , then .

Clearly, , where is the diagonal of and is the off-diagonal part of ; that is, the entries of are (if ) and .

Denote by the spectral radius of an operator . Clearly, Thanks to the well-known inequality for the spectral radius, we have where Denote by the closed convex hull of the diagonal entries .

Theorem 1 (see [10, 20]). Let be the off-diagonal part of an -matrix . Let be holomorphic on a neighborhood of the circle Then, with the notation the matrix inequality holds, provided the series in (12) converges.
Note that, according to (9), we have the inequality

Definition 2. A norm in is said to be monotone if for all , we have ; and , provided .

Corollary 3 (see [10, 20]). Let be holomorphic on a neighborhood of . Let be an arbitrary monotone norm in . Then provided the series in (14) converges.
Important contributions to the theory of matrix-valued functions can be found in Verde-Star [21].

Definition 4. The zero solution of (1) is said to be exponentially stable if there exist , , and such that with.
Let , with being the set of nonnegative integers, and let be the evolution operator associated with the system (1); that is,

Definition 5. The zero solution of system (1) is uniformly exponentially stable if there exist positive constants and such that
Now, we are in a position to establish the main results of this work.

3. Main Results

Consider in the system with the initial condition with a given and a fixed .

Assume that there exists a nonnegative sequence such that where .

Firstly, we recall a boundedness result for (18)-(19) which is proven in [22, Lemma  1.1]; namely, we recall the following lemma.

Theorem 6 (see [11]). Under (20), let Then, every solution of (18)-(19) satisfies the inequality where

Theorem 7 (see [12]). Assume that there exists such that Then (18)-(19) is uniformly stable. Moreover, any solution of (18) satisfies the inequality

Corollary 8 (see [12]). Let condition (24) hold. In addition, for a constant , let Then (18)-(19) is exponentially stable. Moreover, any solution of (18) satisfies the inequality where .
In fact, due to (25), . Putting in (18), one has Hence, by conditions (24) and (26), one has because of the fact that if , then .

Remark 9. Condition (20), combined with the freezing technique, allows us to reduce the stability analysis of a time-varying system to the analysis of the related time-invariant system, thus exploiting the tools developed for linear autonomous systems. However, this condition is conservative compared with the approach proposed by Jetto and Orsini [2, 8], which does not require slowly varying conditions on the coefficients.

Our main results are made possible by the following bound.

Lemma 10. Let be the off-diagonal part of an -matrix . Then, for an arbitrary monotone norm in ,

Proof. In particular, if in Theorem 1 we consider the matrix-valued function where is an constant matrix, hence, Thus, for every , relation (11) yields where . Thus, by (12), where . Hence, by (14), we have

As a consequence of Lemma 10, we obtain where Assume that Hence

Consequently, where .

Theorem 11. Under (20), let Then (18)-(19) is exponentially stable. Moreover, any solution of (18) satisfies the inequality

Proof. By (43), . Hence, the uniform stability is a consequence of Theorem 1. To establish the exponential stability of (18)-(19), let us define a new variable with small enough, where is a solution of (18). Substituting (45) in (18), we have where .
By (22) and Theorem 6, we obtain the estimate (43). It provides the stability of (46). Besides, the boundedness of , for small enough, implies the exponential stability of .

Remark 12. In the particular case , the system (1) is exponentially stable because in this situation we have and no further conditions are needed.

4. Application to Nonlinear Equations

The previous estimates give us the possibility to investigate the stability of nonlinear perturbations of system (1).

Consider the equation where maps into , with the property where is the set of nonnegative integers. Take the initial condition

Theorem 13. Under conditions (20) and (49), let Then the zero solution of (48)–(50) is exponentially stable. Moreover, a solution of (48), with , satisfies the inequality where .

Proof. Given a fixed integer , we can write (48) in the form The variation of constants formula yields Taking , we have Hence, This inequality yields Thus, From this relation, we obtain But the right-hand side of this inequality does not depend on . Thus, it follows that Bound (52) proves the Lyapunov stability with small enough. To establish the exponential stability of the zero solution of system (48), we take the new variable with small enough. Here is the solution of (48)–(50).
Substituting (61) in (48), we have where The growth condition (49) yields Applying the above reasoning to (62), according to (52), it follows that is a bounded function for small enough. Consequently, relation (61) implies the exponential stability of the zero solution of (48).

The estimates established in Section 3 give us the possibility to improve condition (51) and estimate (52) of Theorem 13.

Theorem 14. Under conditions (20) and (49), let Then, the zero solution of (48) is exponentially stable. Moreover, a solution , with , satisfies the inequality where .

Proof. By (40) and (51), for any monotone norm, we have Consequently, if , then, by Theorem 13, the zero solution of (48) is exponentially stable.

Example 15. Consider the following nonautonomous difference system in : where , and is a given function satisfying the growth condition: there is a constant such that If , where , and , then we have On the other hand, by choosing a suitable matrix norm, we obtain By the mean value theorem we have, for some and , Hence, the function satisfies the conditions of Theorem 14; that is, Consequently, by Theorem 14, the zero solution of system (68) is exponentially stable.

Remark 16. An efficient method to construct difference systems with slowly time-varying matrix functions is using a weaker condition than (20), namely, the uniform Lipschitz condition with small enough (see [16]).

5. Conclusion

The stability analysis of discrete-time-varying systems is harder than the stability analysis of time-invariant systems, because the stability and exponential stability of time-varying systems are not characterized by the spectrum of the transition matrices. Several approaches have been proposed in the literature to reduce its stability analysis to the analysis of related time-invariant systems [1, 2, 7–9, 11, 18, 22]. The “freezing” technique has become well known among these techniques. In particular, it has been used to prove that exponential stability of a discrete-time-invariant system implies the exponential stability of the original (time-variant) system provided that the original linear system varies sufficiently slowly and a correct eigenvalue placement in the complex plane. The main novelty of our work is that we use estimates for the absolute values of entries of matrix-valued functions instead of bounds on its eigenvalues. Thus, we establish new stability results for time-varying systems with nonlinear perturbations which complement the existing literature concerning this subject. Natural directions for future research are the generalizations of our results to time-varying delay systems as well as to the stabilization of discrete control systems using new estimates to the norm of the powers of operators.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the anonymous referees for their careful review and encouraging comments. This research was supported by Fondo Nacional de Ciencia y Tecnologia (Fondecyt), Chile, under Grant no. 1130112.