Journal of Applied Mathematics

Volume 2012 (2012), Article ID 214609, 14 pages

doi:10.1155/2012/214609

Research Article

Some New Difference Inequalities and an Application to Discrete-Time Control Systems

Hong Zhou,¹ Deqing Huang,² Wu-Sheng Wang,¹ and Jian-Xin Xu²

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.

Abstract.

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.

1. Introduction

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 [17] considered the following discrete inequality:

In 2006, Cheung and Ren [18] studied

Later, Zheng et al. [24] 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:

Implying

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.

3. Applications

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 [17]

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.

Based on Corollary 3.1 and one additional assumption, the next corollary gives the stability result of the control system (3.1) and (3.2).

Corollary 3.2. Under the assumptions of Corollary 3.1, if there exists a positive constant such that

then the perturbed system (3.1) and (3.2) is exponentially asymptotically stable.

Proof. Under condition (3.15), (3.7) can be further estimated as follows:

The exponentially asymptotic stability of system (3.1) and (3.2) is directly implied.

Acknowledgments

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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