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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 214609, 14 pages
Some New Difference Inequalities and an Application to Discrete-Time Control Systems
1Department of Mathematics, Hechi University, Guangxi, Yizhou 546300, China
2Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117583
Received 1 July 2012; Accepted 17 September 2012
Academic Editor: Jong Hae Kim
Copyright © 2012 Hong Zhou et al. 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.
Two new nonlinear difference inequalities are considered, where the inequalities consist of multiple iterated sums, and composite function of nonlinear function and unknown function may be involved in each layer. Under several practical assumptions, the inequalities are solved through rigorous analysis, and explicit bounds for the unknown functions are given clearly. Further, the derived results are applied to the stability problem of a class of linear control systems with nonlinear perturbations.
Being an important tool in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall inequalities [1, 2] and their applications have attracted great interests of many mathematicians [3–5]. Some recent works can be found in [6–16] and references therein. Along with the development of the theory of integral inequalities and the theory of difference equations, more and more attentions are paid to discrete versions of Gronwall type inequalities [17–24]. For instance, Pachpatte  considered the following discrete inequality: In 2006, Cheung and Ren  studied Later, Zheng et al.  discussed the following discrete inequality: However, the above results are not applicable to inequalities that consist of multiple iterated sums, in particular those in which composite function of nonlinear function and unknown function is involved in each layer of iterated sums. Hence, it is desirable to consider more general difference inequalities of these extended types. They can be used in the study of certain classes of difference equations or applied in many practical engineering problems.
Motivated by the results given in [7, 8, 11, 16–19, 21], in this paper we discuss the following two types of inequalities: for all . All the assumptions on (1.4) and (1.5) are given in the next sections. The inequalities (1.5) consist of multiple iterated sums, and composite function of nonlinear functions and unknown function may be involved in each layer. Under several practical assumptions, the inequalities are solved through rigorous analysis, and explicit bounds for the unknown functions are given clearly. Further, the derived results are applied to the stability problem of a class of linear control systems with nonlinear perturbations.
2. Main Result
In this section, we proceed to solving the difference inequalities (1.4) and (1.5) and present explicit bounds on the embedded unknown functions. Throughout this paper, let denote the set of all natural numbers, and where and are two constants, satisfying .
The following theorem summarizes the result on the inequality (1.4).
Theorem 2.1. Let and be nonnegative functions defined on with nondecreasing on . Moreover, let , be nonnegative functions for and nondecreasing in for fixed . Suppose that is a nondecreasing function on with for . Then, the discrete inequality (1.4) gives where , are the inverse functions of , , respectively, and is the largest natural number such that
Proof. Fix , where is chosen arbitrarily and is defined by (2.5). For , from (1.4), we have Denote the right-hand side of (2.6) by , which is a positive and nondecreasing function on with . Then, (2.6) is equivalent to From (2.6) and (2.7), we observe that Furthermore, it follows from (2.8) that On the other hand, by the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that where is defined by (2.4). By setting in (2.10) and substituting successively, we obtain Let denote the right-hand side of (2.11), which is a positive and nondecreasing function on with . Then, (2.11) is equivalent to By the definition of , we obtain Considering (2.12), (2.13) and the monotonicity properties of , , and , we get for all . Once again, performing the same procedure as in (2.10) and (2.11), (2.14) gives for all , where is defined in (2.3). In the sequel, (2.7), (2.12), and (2.15) render to Let in (2.16), then, we have Noticing that is chosen arbitrarily, (2.1) is directly induced by (2.17). The proof of Theorem 2.1 is complete.
Now, we are in the position of solving the inequality (1.5).
Theorem 2.2. Let the functions , , , , and be the same as in Theorem 2.1. Suppose that , are nondecreasing functions on with for . If satisfies the discrete inequality (1.5), then where , are the inverse functions of , , respectively, and is the largest natural number such that
Proof. Fix , where is chosen arbitrarily and is given in (2.23). For , from (1.5), we have
Let represent the right-hand side of (2.24), which is a positive and nondecreasing function on with . Then, (2.24) is equivalent to
Using (2.24) and (2.25), can be estimated as follows:
for all . Performing the same derivation as in (2.10) and (2.11), we obtain from (2.27) that
where is defined in (2.20). Denote by the right-hand side of (2.28), which is a positive and nondecreasing function on with ++. Then, (2.28) is equivalent to
By the definition of , we obtain
From (2.29), (2.30) and the monotonicity of , and , we get
for all . Similarly to (2.28), it follows from (2.31) that
for all , where is defined in (2.21). Let denote the right-hand side of (2.32), which is a positive and nondecreasing function on with
Then, (2.32) is equivalent to
By the definition of ,
In consequence, (2.34), (2.35) and the monotonicity properties of , and lead to
Similarly to (2.28) and (2.32), we obtain from (2.36) that
where is defined in (2.22).
Summarizing the results in (2.25), (2.29), (2.34), and (2.37), we can conclude that for all . As , (2.38) yields Since is chosen arbitrarily in (2.39), the inequality (2.18) is derived. This completes the proof of Theorem 2.2.
In this section, the result of Theorem 2.2 is applied to explore the asymptotic stability behavior of a class of discrete-time control systems  where Control system (3.1) can be regarded as the perturbation counterpart of the following closed-loop system: The functions , , , are defined on , the -dimensional vector space, is an matrix with , and the functions and are defined on and , respectively. Moreover, and are supposed to meet the following constraints: where is a constant, , are nonnegative real-valued functions defined on and , respectively, and are nondecreasing in for fixed , and , are positive and continuous functions defined on . The symbol denotes norm on as well as a corresponding consistent matrix norm.
Corollary 3.1. Consider the discrete-time control systems (3.1) and (3.2), where the perturbation-related functions and satisfy the conditions (3.4) and (3.5). Assume that the fundamental solution matrix of the linear system (3.3) satisfies where is a constant. Then, any solutions of the control systems (3.1) and (3.2), denoted by , can be estimated by where , are the inverse functions of , , respectively, and is the largest natural number such that
Proof. By using the variation of constants formula, any solution of (3.1) and (3.2) can be represented by for all . Using the conditions (3.4) and (3.6) in (3.10), we have Further, using the relationships (3.2), (3.5), and (3.11), we derive for all . Let , then, (3.12) can be rewritten as Let , , , and , then (3.13) can be further estimated as follows: for all . Notice that, by our assumption, all functions in (3.14) satisfy the conditions of Theorem 2.2. Applying Theorem 2.2 to the inequality (3.14), (3.7) is immediately derived, where the relationship is adopted. This completes the proof of Corollary 3.1.
This research was supported by National Natural Science Foundation of China (Project no. 11161018), the SERC Research Grant (Project no. 092 101 00558), Scientific Research Foundation of the Education Department of Guangxi Province of China (Project no. 201106LX599), and the Key Discipline of Applied Mathematics of Hechi University of China (200725).
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