Stability of Difference Equations and Applications to Robustness Problems
© Bogdan Sasu. 2010
Received: 3 November 2009
Accepted: 23 February 2010
Published: 28 February 2010
The aim of this paper is to obtain new necessary and sufficient conditions for the uniform exponential stability of variational difference equations with applications to robustness problems. We prove characterizations for exponential stability of variational difference equations using translation invariant sequence spaces and emphasize the importance of each hypothesis. We introduce a new concept of stability radius for a variational system of difference equations with respect to a perturbation structure and deduce a very general estimate for the lower bound of . All the results are obtained without any restriction concerning the coefficients, being applicable for any system of variational difference equations.
In the last decades an increasing interest was focused on the asymptotic properties of the most general class of evolution equations—the variational systems and a number of open questions were answered, increasing the applicability area not only to partial differential equations but also to systems arising from the linearization of nonlinear equations (see [1–10] and the references therein). In this context a special attention was devoted to the general case of variational systems of difference equations of the form:
where is a family of bounded linear operators on a Banach space and is a flow on a metric space . The interest is motivated by several notable advantages related with the system (A): the obtained results are applicable to a large class of systems; since no measurability or continuity conditions are needed, this system often models the families of equations proceeding from the linearization of nonlinear equations and also extends the nonautonomous case in infinite-dimensional spaces (see [1, 2, 4–10] and the references therein).
In recent years a notable progress was made in the study of the qualitative properties of various classes of difference equations (see [4–8, 11–32]). The input-output methods or the so-called "theorems of Perron type" have proved to be important tools in the study of the asymptotic behavior of difference equations like stability (see [27, 29, 30]), expansiveness (see ), dichotomy (see [1, 5, 8, 15, 17, 19, 20, 32]), and trichotomy as well (see [7, 18, 20]). A distinct method for the study of exponential stability relies on the convergence of some associated series and this was used in  for difference equations and in  for variational difference equations. The exponential stability of difference equations with several variable delays and variable coefficients was studied in , where the authors obtained interesting conditions for global exponential stability using new computational formulas with respect to the coefficients. The uniform asymptotic stability of positive Volterra difference equations was recently studied in , the authors proving the equivalence between the uniform asymptotic stability of the zero solution, the summability of the fundamental solution, and the invertibility of an associated operator outside the unit disk. A very efficient method in the study of the stability of difference equations is represented by the so-called "freezing technique," which was used in  for the study of absolute stability of discrete-time systems with delay and also in  in order to deduce explicit conditions for global feedback exponential stabilizability of discrete-time control systems with multiple state delays. In the study of the asymptotic behavior of discrete-time systems, there is an increasing interest in finding methods arising from control theory (see [5–8, 21–27, 29–32]). This is motivated by the fact that besides their large applicability area, the control-type techniques can be also applied to the analysis of the robustness of diverse properties in the presence of perturbations (see [5, 25, 26, 30, 32, 33]). In this context it is natural to extend the study to the variational case. Thus, two main questions arise: which is the most general framework for the study of the stability of variational difference equations using control type methods and how one may apply the new techniques in order to determine the behavior of the initial system in the presence of perturbations. In what follows, our attention will focus in order to provide complete answers to these open questions.
In this paper we propose a new study concerning the stability of variational difference equations and the robustness of this property. We associate with the system (A) a family of control systems and we attack the subject from the perspective of the solvability of between two Banach sequence spaces invariant under translations. We split the class of Banach sequence spaces which are invariant under translations into two central subclasses and deduce necessary and sufficient conditions for uniform and exponential stability with respect to the solvability of the control system ( ) when the input sequences belong to a space from a subclass or the solution lies in a space from the other subclass (see Theorem 3.8). By an example we show that the stability result is the most general in the topic and that the assumptions on the underlying sequence spaces cannot be removed. As particular cases of the stability results we deduce many interesting situations; among them we mention some direct generalizations at the variational case of the theorems from [27, 29, 30]. We also mention that the associated control system is distinct compared with those considered in the study of dichotomy and trichotomy (see [7, 8, 32, 34]), the input-output conditions are different, the Banach sequence norm is more flexible, and the underlying classes of sequence spaces are the largest and with more permissive properties.
Next, we apply the stability results and we propose a new approach for stability robustness of variational difference equations. We study the stable behavior of the system (A) in the presence of a general perturbation structure with , , by means of the stability radius (see Definition 4.3). Our target is to obtain a lower bound for the stability radius of variational systems of difference equations as well as to determine the largest class of Banach sequence spaces within the robustness properties hold. With this purpose we associate with the system (A) an input-output control system and consider the general class of all Banach sequence spaces with the property that if there is such that , for all , then and . For every Banach sequence space , we introduce the index , where is the family of input-output operators associated with the system ( ) and we obtain a lower bound for in terms of . Thus, we point out an interesting connection between the family of input-output operators and the size of the smallest perturbation in the presence of which the perturbed system loses its exponential stability. The variational case requires a special analysis and the methods are substantially more complicated compared to those used in the nonautonomous case (see [30, 33]). We note that the study is done without any restriction or assumption on the coefficients, the obtained results being applicable for any system of variational difference equations.
2. Banach Sequence Spaces
In this section, for the sake of clarity, we will recall some basic definitions and properties of Banach sequences spaces. Let denote the set of the integers, let denote the set of all non negative integers, let denote the set of all real numbers, and let be the linear space of all sequences . Let . For every set we denote by the characteristic function of the set . For every we consider the sequence defined by and , for all .
Example 2.3 (Orlicz sequence spaces).
(see, e.g., [30, Lemma 2.1]);
Necessity. If , then . Let . Then there is a strictly increasing sequence such that , for all and all . Setting we deduce that , for all and all . It follows that is a Cauchy sequence in ; so this is convergent. Let be such that in . According to Lemma 2.7 we deduce that , and so . This implies that .
Suppose that ; so . Then , for all . This implies that . Let and let . Then , for every . Since is nondecreasing, this yields , for every , so . It follows that ; so . Hence and using Remark 2.5(iii) we obtain the conclusion.
Let be a real or complex Banach space. For every Banach sequence space we denote by the space of all sequences with the property that the mapping belongs to . is a Banach space with respect to the norm .
3. Stability of Variational Difference Equations
Let . We consider the variational system of variational difference equations (A). We note that in the particular case when and we obtain the case of difference equations. There are several distinct directions of generalizing the case of difference equations. One of the most interesting methods is to consider them in the general framework of dynamic equations on time scales (see [35, 36]), having a wide potential for applications in the study of population dynamics. Another method is to consider them as particular cases of variational difference equations, which often proceed from the linearization of nonlinear equations (see [2, 10] and the references therein). It is also interesting to note that the exponential stability of a variational equation is equivalent with the exponential stability of the variational difference equation associated with it. Therefore, concerning the stability of variational equations it is recommended to study the discrete-time case, because no measurability or continuity conditions are required.
The discrete cocycle associated with the system (A) is
Our first result provides a sufficient condition for uniform stability and is given by the following.
Necessity is immediate via Definition 3.4 and sufficiency follows from Theorem 3.5.
The first main result of this section is the following.
From relations (3.19) and (3.23) it follows that . Taking into account that does not depend on or we obtain that , for all and all . Using similar arguments with those from Case 1, we deduce that the system (A) is uniformly exponentially stable.
The second main result of this section is the following.
This follows from Theorem 3.7.
Sufficiency. This follows from (i).
In what follows, we prove that the result obtained in Theorem 3.8 is the most general in this topic. Precisely, we will show that if and , then the -stability of the system does not assure the uniform exponential stability of the system (A).
Taking into account that does not depend on or it follows that the system is -stable. But, for all that, it is easy to verify that there are not such that , for all , so the system (A) is not uniformly exponentially stable.
As consequences of Theorem 3.8 we obtain the following.
From the above results it follows that in the study of exponential stability of variational difference equations using input-output techniques one may work with Banach sequence spaces which are invariant under translations, such that either the input space contains at least a sequence whose series is divergent or the output space has unbounded fundamental function. Moreover, according to Corollary 3.11 we deduce that when the input space and the output space coincide, then there is no other requirement on the underlying sequence spaces and it is sufficient to consider any Banach sequence space which is invariant under translations and contains at least a characteristic function of a singleton.
4. Applications to Robustness of Exponential Stability
In this section, by applying our main results we will study the persistence of the exponential stability in the presence of variational structured perturbations.
In what follows we suppose that the system (A) is uniformly exponentially stable. The main question is how large may be the norm of the perturbation such that the perturbed system (4.2) remains uniformly exponentially stable. With this purpose we introduce in the following.
is called the stability radius of the system (A).
We note that in the existent literature there is not an explicit computational formula for the stability radius of systems, only in some sporadic special cases. For linear retarded systems of differential equations on , some interesting formulas were obtained by Ngoc and Son in [26, Theorems 3.5 and 3.10], but the estimates are still complicated. For the case of positive linear retarded systems which are Hurwitz stable, the authors succeeded to deduce a formula for the stability radius corresponding to multiaffine perturbations (see Theorem 4.7 and the following example). In  Murakami and Nagabuchi obtained an explicit formula for the stability radius of uniformly asymptotically stable positive Volterra difference equations on Banach lattices (see , Theorem 4.3). Generally, in order to analyze the persistence of the exponential stability in the presence of perturbations, it is interesting to find a lower bound for the stability radius of systems (see [3, 25, 26, 30, 33]) because in this manner we estimate the possible size of the disturbance operator under which the (additively) perturbed system remains exponentially stable.
In what follows, using the results obtained in the previous section we will estimate a lower bound for the stability radius of the system (A). We will provide a detailed study, when the system (A) is subject to a very general perturbation structure.
has the property
has the property
The main result of this section is the following.
From Theorem 4.8 we deduce that the property of uniform exponential stability of a system of variational difference equations is preserved in the presence of structured perturbations, provided that the norm of the perturbation factor is less than and this estimate holds for any sequence space in the general class . The study points out an interesting connection between the family of the input-output operators associated with the control system and the size of the "largest" perturbation in the presence of which the perturbed system still remains uniformly exponentially stable.
The central result of this section is the following.
As a consequence, we deduce the following.
The author would like to thank the referees for carefully reading the manuscript and for helpful comments, which led to the improvement of the paper. The work was supported by CNCSIS – UEFISCSU, project number PN II – IDEI 1081/2008 and project number PN II – IDEI 1080/2008.
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