Exponential Stability Criteria for Nonautonomous Difference Systems

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.


Introduction
In the theory of difference equations it is well known that the placement of eigenvalues in the complex plane of a timeinvariant 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):  ( + 1) =  ()  () ,  = 0, 1, . . ., 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 sup (c) sup ≥0 ‖( + 1) − ()‖ 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       ()     ≤   ; ∀,∀ ≥ 1, where  = 1 −  and  depends on ,  and   but is independent of , and then uses a Lyapunov argument to show that system (1) is exponentially stable if sup ≥0 ‖(+1)−()‖ 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][4][5][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][8][9][10]).
Recently, Gil and Medina [11,12] and Medina [13][14][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.
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.

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 Γ is a closed contour surrounding ().
We put Clearly,  =  + , where  = diag{ 11 , . . .,   } is the diagonal of  and  =  −  is the off-diagonal part of ; that is, the entries V  of  are V  =   (if  ̸ = ) and V  = 0. Denote by   () the spectral radius of an operator .Clearly, Thanks to the well-known inequality for the spectral radius, we have where Denote by  0 () the closed convex hull of the diagonal entries  11 , . . .,   .

Main Results
Consider in   the system with the initial condition with a given  0 ∈   and a fixed  ≥ 0.
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 timeinvariant 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.
Remark 12.In the particular case  0 < 1, the system ( 1) is exponentially stable because in this situation we have and no further conditions are needed.
Proof.Given a fixed integer  ≥ 0, we can write (48) in the form The variation of constants formula yields Taking  = , we have Hence, From this relation, we obtain max But the right-hand side of this inequality does not depend on .Thus, it follows that sup =1,2,...
Bound (52) proves the Lyapunov stability with ‖(0)‖ small enough.To establish the exponential stability of the zero solution of system (48), we take the new variable with  > 0 small enough.Here () is the solution of ( 48)-(50).
Applying the above reasoning to (62), according to (52), it follows that   () is a bounded function for  > 0 small enough.Consequently, relation (61) implies the exponential stability of the zero solution of (48).
Hence, the function () = 0.5 − satisfies the conditions of Theorem 14; that is, Consequently, by Theorem 14, the zero solution of system (68) is exponentially stable.

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.