Journal of Applied Mathematics

Volume 2012, Article ID 214609, 14 pages

http://dx.doi.org/10.1155/2012/214609

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

^{1}Department of Mathematics, Hechi University, Guangxi, Yizhou 546300, China^{2}Department 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).

#### References

- T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,”
*Annals of Mathematics*, vol. 20, no. 4, pp. 292–296, 1919. View at Publisher · View at Google Scholar - R. Bellman, “The stability of solutions of linear differential equations,”
*Duke Mathematical Journal*, vol. 10, pp. 643–647, 1943. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. S. Mitrinović, J. E. Pečarić, and A. M. Fink,
*Inequalities Involving Functions and Their Integrals and Derivatives*, Kluwer Academic, Dordrecht, The Netherlands, 1991. - D. Baĭnov and P. Simeonov,
*Integral Inequalities and Applications*, vol. 57, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. - B. G. Pachpatte,
*Inequalities for Differential and Integral Equations*, vol. 197, Academic Press, New York, USA, 1998. - W. Zhang and S. Deng, “Projected Gronwall-Bellman's inequality for integrable functions,”
*Mathematical and Computer Modelling*, vol. 34, no. 3-4, pp. 393–402, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,”
*Applied Mathematics and Computation*, vol. 165, no. 3, pp. 599–612, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B.-I. Kim, “On some Gronwall type inequalities for a system integral equation,”
*Bulletin of the Korean Mathematical Society*, vol. 42, no. 4, pp. 789–805, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - O. Lipovan, “Integral inequalities for retarded Volterra equations,”
*Journal of Mathematical Analysis and Applications*, vol. 322, no. 1, pp. 349–358, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W.-S. Cheung, “Some new nonlinear inequalities and applications to boundary value problems,”
*Nonlinear Analysis A*, vol. 64, no. 9, pp. 2112–2128, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. P. Agarwal, C. S. Ryoo, and Y.-H. Kim, “New integral inequalities for iterated integrals with applications,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 24385, 18 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W.-S. Wang, “A generalized retarded Gronwall-like inequality in two variables and applications to BVP,”
*Applied Mathematics and Computation*, vol. 191, no. 1, pp. 144–154, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. P. Agarwal, Y.-H. Kim, and S. K. Sen, “New retarded integral inequalities with applications,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 908784, 15 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W.-S. Wang and C.-X. Shen, “On a generalized retarded integral inequality with two variables,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 518646, 9 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C.-J. Chen, W.-S. Cheung, and D. Zhao, “Gronwall-bellman-type integral inequalities and applications to BVPs,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 258569, 15 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus - A. Abdeldaim and M. Yakout, “On some new integral inequalities of Gronwall-Bellman-Pachpatte type,”
*Applied Mathematics and Computation*, vol. 217, no. 20, pp. 7887–7899, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. G. Pachpatte, “Finite difference inequalities and discrete time control systems,”
*Indian Journal of Pure and Applied Mathematics*, vol. 9, no. 12, pp. 1282–1290, 1978. View at Google Scholar · View at Zentralblatt MATH - W.-S. Cheung and J. Ren, “Discrete non-linear inequalities and applications to boundary value problems,”
*Journal of Mathematical Analysis and Applications*, vol. 319, no. 2, pp. 708–724, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. G. Pachpatte,
*Integral and Finite Difference Inequalities and Applications*, vol. 205 of*North-Holland Mathematics Studies*, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. - Q.-H. Ma and W.-S. Cheung, “Some new nonlinear difference inequalities and their applications,”
*Journal of Computational and Applied Mathematics*, vol. 202, no. 2, pp. 339–351, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W.-S. Wang, “A generalized sum-difference inequality and applications to partial difference equations,”
*Advances in Difference Equations*, vol. 2008, Article ID 695495, 12 pages, 2008. View at Google Scholar · View at Zentralblatt MATH - W.-S. Wang, “Estimation on certain nonlinear discrete inequality and applications to boundary value problem,”
*Advances in Difference Equations*, vol. 2009, Article ID 708587, 8 pages, 2009. View at Google Scholar · View at Zentralblatt MATH - X.-M. Zhang and Q.-L. Han, “Delay-dependent robust H
^{∞}filtering for uncertain discrete-time systems with time-varying delay based on a finite sum inequality,”*IEEE Transactions on Circuits and Systems II*, vol. 53, no. 12, pp. 1466–1470, 2006. View at Publisher · View at Google Scholar · View at Scopus - K. L. Zheng, S. M. Zhong, and M. Ye, “Discrete nonlinear inequalities in time control systems,” in
*Proceedings of the International Conference on Apperceiving Computing and Intelligence Analysis (ICACIA '09)*, pp. 403–406, 2009.