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

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: 𝑢(𝑛)≤𝑢₀+^𝑛−1_𝑠=𝑛0𝑓(𝑠)[𝑢(𝑠)+ℎ(𝑠)]+^𝑛−1_𝑠=𝑛0𝑓(𝑠)^𝑠−1_𝜏=𝑛0𝑔(𝜏)𝑢(𝜏),∀𝑛∈𝑁₀. (1.1) In 2006, Cheung and Ren [18] studied 𝑢^𝑝(𝑚,𝑛)≤𝑐+^𝑚−1_𝑠=𝑚0^𝑛−1_𝑡=𝑛0𝑎(𝑠,𝑡)𝑢^𝑞(𝑠,𝑡)+^𝑚−1_𝑠=𝑚0^𝑛−1_𝑡=n0𝑏(𝑠,𝑡)𝑢^𝑞(𝑠,𝑡)𝑤(𝑢(𝑠,𝑡)). (1.2) Later, Zheng et al. [24] discussed the following discrete inequality: 𝑢(𝑛)≤𝑎(𝑛)+^𝑘_𝑖=1^𝑛−1_𝑠=0𝑓_𝑖(𝑛,𝑠)𝑤_𝑖(𝑢(𝑠)). (1.3) 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:𝑢(𝑛)≤𝑎(𝑛)+^𝑛−1_𝑠=𝑛0𝑓₁(𝑛,𝑠)𝑤(𝑢(𝑠))+^𝑛−1_𝑠=𝑛0𝑓₁(𝑛,𝑠)𝑤(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑓₂()𝑤(𝑢(𝜏+𝑠,𝜏))_{𝑛−1𝑠=𝑛0}𝑓₁(𝑛,𝑠)𝑤(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃()𝑤(𝑢(𝜉,𝑢(𝑛)(𝑛)+𝜏,𝜉))≤𝑎^𝑛−1_𝑠=𝑛0𝑓₁(𝑛,𝑠)𝑤₁(𝑢(𝑠))+^𝑛−1_𝑠=𝑛0𝑓₁(𝑛,𝑠)𝑤₁(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)𝑤₂(𝑢(𝜏+))_{𝑛−1𝑠=𝑛0}𝑓₁(𝑛,𝑠)𝑤₁(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)𝑤₂(𝑢(𝜏))^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤₃(𝑢(𝜉)), (1.4)(1.5) for all 𝑛∈𝑁₀. 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 𝐍 denote the set of all natural numbers, and 𝑁₀=[𝑛₀,𝐾)∩𝐍 where 𝑛₀ and 𝐾 are two constants, satisfying 𝐾>𝑛₀.

The following theorem summarizes the result on the inequality (1.4).

Theorem 2.1. Let 𝑢(𝑛) and 𝑎(𝑛) be nonnegative functions defined on 𝑁₀ with 𝑎(𝑛) nondecreasing on 𝑁₀. Moreover, let 𝑓_𝑖(𝑛,𝑠), 𝑖=1,2,3 be nonnegative functions for 𝑛₀≤𝑠≤𝑛≤𝐾 and nondecreasing in 𝑛 for fixed 𝑠∈𝑁₀. Suppose that 𝑤(𝑢) is a nondecreasing function on [0,∞) with 𝑤(𝑢)>0 for 𝑢>0. Then, the discrete inequality (1.4) gives 𝑢(𝑛)≤𝑊₁⁻¹𝑊₂⁻¹𝑈₁(𝑛),∀𝑛∈𝑛₀,𝑀₁∩𝐍, (2.1) where 𝑈₁(𝑛)=𝑊₂𝑊₁(𝑎(𝑛))+^𝑛−1_𝑠=𝑛0𝑓₁(+𝑛,𝑠)_{𝑛−1𝑠=𝑛0}𝑓₁(𝑛,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)+^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(,𝑊𝜏,𝜉)₂(𝑢)=^𝑢₁𝑑𝑠𝑤𝑊₁⁻¹(𝑠)𝑊,𝑢>0,₁(𝑢)=^𝑢₁𝑑𝑠𝑤(𝑠),𝑢>0, (2.2)(2.3)(2.4)𝑊₁⁻¹, 𝑊₂⁻¹ are the inverse functions of 𝑊₁, 𝑊₂, respectively, and 𝑀₁ is the largest natural number such that 𝑈₁𝑀₁∈Dom𝑊₂⁻¹,𝑊₂⁻¹𝑈₁𝑀₁∈Dom𝑊₁⁻¹. (2.5)

Proof. Fix 𝑀∈𝑁_𝑀1=[𝑛₀,𝑀₁)∩𝐍, where 𝑀 is chosen arbitrarily and 𝑀₁ is defined by (2.5). For 𝑛∈𝑁_𝑀=[𝑛₀,𝑀]∩𝐍, from (1.4), we have 𝑢(𝑛)≤𝑎(𝑀)+^𝑛−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)𝑤(𝑢(𝑠))+^𝑛−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)𝑤(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑓₂()𝑤(𝑢(𝜏+𝑠,𝜏))_{𝑛−1𝑠=𝑛0}𝑓₁(𝑀,𝑠)𝑤(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤(𝑢(𝜉)). (2.6) Denote the right-hand side of (2.6) by 𝑧₁(𝑛), which is a positive and nondecreasing function on 𝑁_𝑀 with 𝑧₁(𝑛₀)=𝑎(𝑀). Then, (2.6) is equivalent to 𝑢(𝑛)≤𝑧₁(𝑛),∀𝑛∈𝑁_𝑀. (2.7) From (2.6) and (2.7), we observe that Δ𝑧₁(𝑛)∶=𝑧₁(𝑛+1)−𝑧₁(𝑛)≤𝑓₁(𝑀,𝑛)𝑤𝑧₁(𝑛)+𝑓₁(𝑀,𝑛)𝑤𝑧₁(𝑛)^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤𝑧₁(𝜏)+𝑓₁(𝑀,𝑛)𝑤𝑧₁(𝑛)^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤𝑧₁(𝜉)=𝑓₁(𝑀,𝑛)𝑤𝑧₁(𝑛)1+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤𝑧₁(𝜏+_{𝑛−1𝜏=𝑛0})𝑓₂(𝑛,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤𝑧₁(𝜉),∀𝑛∈𝑁_𝑀. (2.8) Furthermore, it follows from (2.8) thatΔ𝑧₁(𝑛)𝑤𝑧₁(𝑛)≤𝑓₁(𝑀,𝑛)1+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤𝑧₁(𝜏)+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤𝑧₁(𝜉),∀𝑛∈𝑁_𝑀. (2.9) On the other hand, by the mean-value theorem for integrals, for arbitrarily given integers 𝑛,𝑛+1∈𝑁_𝑀, there exists 𝜂 in the open interval (𝑧₁(𝑛),𝑧₁(𝑛+1)) such that 𝑊₁𝑧₁(𝑛+1)−𝑊₁𝑧₁(𝑛)=^𝑧1_𝑧^(𝑛+1)_1(𝑛)𝑑𝑠𝑤𝑧₁(𝑠=)Δ𝑧₁(𝑛)𝑤𝑧₁𝜂≤Δ𝑧₁(𝑛)𝑤𝑧₁(𝑛)≤𝑓₁(𝑀,𝑛)1+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤𝑧₁(𝜏)+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤𝑧₁(𝜉),∀𝑛∈𝑁_𝑀, (2.10) where 𝑊₁ is defined by (2.4). By setting 𝑛=𝑠 in (2.10) and substituting 𝑠=𝑛₀,𝑛₀+1,𝑛₀+2,…,𝑛−1 successively, we obtain 𝑊₁𝑧₁(𝑛)≤𝑊₁𝑧₁𝑛₀+^𝑀−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)+^𝑛−1_𝑠=𝑛0𝑓₁()×𝑀,𝑠_{𝑠−1𝜏=𝑛0}𝑓₂(𝑠,𝜏)𝑤𝑧₁(𝜏)+^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤𝑧₁(𝜉),∀𝑛∈𝑁_𝑀. (2.11) Let 𝑣₁(𝑛) denote the right-hand side of (2.11), which is a positive and nondecreasing function on 𝑁_𝑀 with 𝑣₁(𝑛₀)=𝑊₁(𝑧₁(𝑛₀))+∑^𝑀−1_𝑠=𝑛0𝑓₁(𝑀,𝑠). Then, (2.11) is equivalent to 𝑧₁(𝑛)≤𝑊₁⁻¹𝑣₁(𝑛),∀𝑛∈𝑁_𝑀. (2.12) By the definition of 𝑣₁, we obtain Δ𝑣₁(𝑛)∶=𝑣₁(𝑛+1)−𝑣₁(𝑛)=𝑓₁(𝑀,𝑛)^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤𝑧₁(𝜏)+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤𝑧₁(𝜉),∀𝑛∈𝑁_𝑀. (2.13) Considering (2.12), (2.13) and the monotonicity properties of 𝑤, 𝑊₁⁻¹, and 𝑧₁, we get Δ𝑣₁(𝑛)𝑤𝑊₁⁻¹𝑣₁(𝑛)≤𝑓₁(𝑀,𝑛)^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉), (2.14) for all 𝑛∈𝑁_𝑀. Once again, performing the same procedure as in (2.10) and (2.11), (2.14) gives 𝑊₂𝑣₁(𝑛)≤𝑊₂𝑣₁𝑛₀+^𝑛−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)+^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉), (2.15) for all 𝑛∈𝑁_𝑀, where 𝑊₂ is defined in (2.3). In the sequel, (2.7), (2.12), and (2.15) render to 𝑢(𝑛)≤𝑧₁(𝑛)≤𝑊₁⁻¹𝑣₁(𝑛)=𝑊₁⁻¹𝑊₂⁻¹𝑊₂𝑊₁(𝑎(𝑀))+^𝑀−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)+^𝑛−1_𝑠=𝑛0𝑓₁()×𝑀,𝑠_{𝑠−1𝜏=𝑛0}𝑓₂(𝑠,𝜏)+^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉),∀𝑛∈𝑁_𝑀. (2.16) Let 𝑛=𝑀 in (2.16), then, we have𝑢(𝑛)≤𝑊₁⁻¹𝑊₂⁻¹𝑊₂𝑊₁(𝑎(𝑀))+^𝑀−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)+^𝑀−1_𝑠=𝑛0𝑓₁()×𝑀,𝑠_{𝑠−1𝜏=𝑛0}𝑓₂(𝑠,𝜏)+^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉). (2.17) Noticing that 𝑀 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 𝑢(𝑛), 𝑎(𝑛), 𝑓_𝑖(𝑛,𝑠), 𝑖=1,2,3, and 𝜑(𝑢) be the same as in Theorem 2.1. Suppose that 𝑤_𝑖(𝑢), 𝑖=1,2,3 are nondecreasing functions on [0,∞) with 𝑤_𝑖(𝑢)>0 for 𝑢>0. If 𝑢(𝑛) satisfies the discrete inequality (1.5), then 𝑢(𝑛)≤Φ₁⁻¹Φ₂⁻¹Φ₃⁻¹𝑈₂(𝑛),∀𝑛∈𝑁_𝑀3=𝑛₀,𝑀₃∩𝐍, (2.18) where 𝑈₂(𝑛)=Φ₃Φ₂Φ₁(𝑎(𝑛))+^𝑛−1_𝑠=𝑛0𝑓₁(𝑛,𝑠)+^𝑛−1_𝑠=𝑛0𝑓₁(𝑛,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂(+𝑠,𝜏)_{𝑛−1𝑠=𝑛0}𝑓₁(𝑛,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(),Φ𝜏,𝜉₁(𝑢)=^𝑢₁𝑑𝑠𝑤₁(𝑠)Φ,𝑢>0,₂(𝑢)=^𝑢₁𝑑𝑠𝑤₂Φ₁⁻¹(𝑠Φ),𝑢>0,₃(𝑢)=^𝑢₁𝑑𝑠𝑤₃Φ₁⁻¹Φ₂⁻¹(𝑠),𝑢>0, (2.19)(2.20)(2.21)(2.22)Φ_𝑖⁻¹, 𝑖=1,2,3 are the inverse functions of Φ_𝑖, 𝑖=1,2,3, respectively, and 𝑀₂ is the largest natural number such that 𝑈₂𝑀₂∈DomΦ₃⁻¹,Φ₃⁻¹𝑈₂𝑀₂∈DomΦ₂⁻¹,Φ₂⁻¹Φ₃⁻¹𝑈₂𝑀₂∈DomΦ₁⁻¹. (2.23)

Proof. Fix 𝑀∈𝑁_𝑀2=[𝑛₀,𝑀₂)∩𝐍, where 𝑀 is chosen arbitrarily and 𝑀₂ is given in (2.23). For 𝑛∈𝑁_𝑀, from (1.5), we have 𝑢(𝑛)≤𝑎(𝑀)+^𝑛−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)𝑤₁(𝑢(𝑠))+^𝑛−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)𝑤₁(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)𝑤₂(𝑢(𝜏+))_{𝑛−1𝑠=𝑛0}𝑓₁(𝑀,𝑠)𝑤₁(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)𝑤₂(𝑢(𝑠))^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤₃(𝑢(𝜉)). (2.24) Let 𝑧₂(𝑛) represent the right-hand side of (2.24), which is a positive and nondecreasing function on 𝑁_𝑀2 with 𝑧₂(𝑛₀)=𝑎(𝑀). Then, (2.24) is equivalent to 𝑢(𝑛)≤𝑧₂(𝑛),∀𝑛∈𝑁_𝑀. (2.25) Using (2.24) and (2.25), Δ𝑧₂(𝑛)∶=𝑧₂(𝑛+1)−𝑧₂(𝑛) can be estimated as follows: Δ𝑧₂(𝑛)≤𝑓₁(M,𝑛)𝑤₁𝑧₂(𝑛)+𝑓₁(𝑀,𝑛)𝑤₁𝑧₂(𝑛)^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤₂𝑧₂(𝜏)+𝑓₁(𝑀,𝑛)𝑤₁𝑧₂(𝑛)^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤₂𝑧₂(𝑛)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤₃𝑧₂(𝜉)=𝑓₁(𝑀,𝑛)𝑤₁𝑧₂(𝑛)1+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤₂𝑧₂(𝜏)+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤₂𝑧₂(𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤₃𝑧₂(𝜉),∀𝑛∈𝑁_𝑀, (2.26) Implying Δ𝑧₂(𝑛)𝑤₁𝑧₂(𝑛)≤𝑓₁(𝑀,𝑛)1+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤₂𝑧₂(𝜏+_{𝑛−1𝜏=𝑛0})𝑓₂(𝑛,𝜏)𝑤₂𝑧₂(𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤₃𝑧₂(𝜉), (2.27) for all 𝑛∈𝑁_𝑀. Performing the same derivation as in (2.10) and (2.11), we obtain from (2.27) that Φ₁𝑧₂(𝑛)≤Φ₁𝑧₂𝑛₀+^𝑀−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)+^𝑛−1_𝑠=𝑛0𝑓₁()×𝑀,𝑠_{𝑠−1𝜏=𝑛0}𝑓₂(𝑠,𝜏)𝑤₂𝑧₂(𝜏)+^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)𝑤₂𝑧₂(𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤₃𝑧₂(𝜉),∀𝑛∈𝑁_𝑀, (2.28) where Φ₁ is defined in (2.20). Denote by 𝑣₂(𝑛) the right-hand side of (2.28), which is a positive and nondecreasing function on 𝑁_𝑀2 with 𝑣₂(𝑛₀)=Φ₁(𝑧₂(𝑛₀))+∑^𝑀−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)=Φ₁(𝑎(𝑀))+∑^𝑀−1_𝑠=𝑛0𝑓₁(𝑀,𝑠). Then, (2.28) is equivalent to 𝑧₂(𝑛)≤Φ₁⁻¹𝑣₂(𝑛),∀𝑛∈𝑁_𝑀. (2.29) By the definition of 𝑣₂, we obtain Δ𝑣₂(𝑛)∶=𝑣₂(𝑛+1)−𝑣₂(𝑛)=𝑓₁(𝑀,𝑛)^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤₂𝑧₂(𝜏)+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)𝑤₂𝑧₂(𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤₃𝑧₂(𝜉),∀𝑛∈𝑁_𝑀. (2.30) From (2.29), (2.30) and the monotonicity of 𝑤₂,Φ₁⁻¹, and 𝑧₂, we get Δ𝑣₂(𝑛)𝑤₂Φ₁⁻¹𝑣₂(𝑛)≤𝑓₁(𝑀,𝑛)^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)+^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤₃Φ₁⁻¹𝑣₂(𝜉), (2.31) for all 𝑛∈𝑁_𝑀. Similarly to (2.28), it follows from (2.31) that Φ₂𝑣₂(𝑛)≤Φ₂𝑣₂𝑛₀+^𝑀−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂()+𝑠,𝜏_{𝑛−1𝑠=𝑛0}𝑓₁(𝑀,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤₃Φ₁⁻¹𝑣₂(𝑛), (2.32) for all 𝑛∈𝑁_𝑀, where Φ₂ is defined in (2.21). Let 𝑣₃(𝑛) denote the right-hand side of (2.32), which is a positive and nondecreasing function on 𝑁_𝑀2 with 𝑣₃𝑛₀=Φ₂𝑣₂𝑛₀+^𝑀−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)=Φ₂Φ₁(𝑎(𝑀))+^𝑀−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)+^𝑀−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏). (2.33) Then, (2.32) is equivalent to 𝑣₂(𝑛)≤Φ₂⁻¹𝑣₃(𝑛),∀𝑛∈𝑁_𝑀. (2.34) By the definition of 𝑣₃, Δ𝑣₃(𝑛)∶=𝑣₃(𝑛+1)−𝑣₃(𝑛)=𝑓₁(𝑀,𝑛)^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤₃Φ₁⁻¹𝑣₂(𝜉),∀𝑛∈𝑁_𝑀. (2.35) In consequence, (2.34), (2.35) and the monotonicity properties of 𝑤₃,Φ₁⁻¹, and 𝑣₂ lead to Δ𝑣₃(𝑛)𝑤₃Φ₁⁻¹Φ₂⁻¹𝑣₃(𝑛)≤𝑓₁(𝑀,𝑛)^𝑛−1_𝜏=𝑛0𝑓₂(𝑛,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉),∀𝑛∈𝑁_𝑀. (2.36) Similarly to (2.28) and (2.32), we obtain from (2.36) that Φ₃𝑣₃(𝑛)≤Φ₃𝑣₃𝑛₀+^𝑛−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉),∀𝑛∈𝑁_𝑀, (2.37) where Φ₃ is defined in (2.22).
Summarizing the results in (2.25), (2.29), (2.34), and (2.37), we can conclude that 𝑢(𝑛)≤𝑧₂(𝑛)≤Φ₁⁻¹𝑣₂(𝑛)≤Φ₁⁻¹Φ₂⁻¹𝑣₃(𝑛)≤Φ₁⁻¹Φ₂⁻¹Φ₃⁻¹Φ₃𝑣₃𝑛₀+^𝑛−1_𝑠=𝑛0𝑓₁(𝑀,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)=Φ₁⁻¹Φ₂⁻¹Φ₃⁻¹Φ₃Φ₂Φ₁(𝑎(𝑀))+^𝑀−1_𝑠=𝑛0𝑓₁()+𝑀,𝑠_{𝑀−1𝑠=𝑛0}𝑓₁(𝑀,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂()+𝑠,𝜏_{𝑛−1𝑠=𝑛0}𝑓₁(𝑀,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉), (2.38) for all 𝑛∈𝑁_𝑀. As 𝑛=𝑀, (2.38) yields 𝑢(𝑀)≤Φ₁⁻¹Φ₂⁻¹Φ₃⁻¹Φ₃Φ₂Φ₁(𝑎(𝑀))+^𝑀−1_𝑠=𝑛0𝑓₁()+𝑀,𝑠_{𝑀−1𝑠=𝑛0}𝑓₁(𝑀,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂()+𝑠,𝜏_{𝑀−1𝑠=𝑛0}𝑓₁(𝑀,𝑠)^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉). (2.39) Since 𝑀 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] 𝑥(𝑛+1)=𝐴(𝑛)𝑥(𝑛)+𝑓(𝑛,𝑥(𝑛),𝜎(𝑛)),𝑥𝑛₀=𝑥₀, (3.1) where 𝜎(𝑛)=𝜃(𝑛)+^𝑛−1_𝑠=𝑛0𝑘(𝑛,𝑠,𝑥(𝑠)). (3.2) Control system (3.1) can be regarded as the perturbation counterpart of the following closed-loop system: 𝑦(𝑛+1)=𝐴(𝑛)𝑦(𝑛),𝑦𝑛₀=𝑥₀. (3.3) The functions 𝑥, 𝑦, 𝜃, 𝜎 are defined on 𝑁→𝐑^𝑟, the 𝑟-dimensional vector space, 𝐴(𝑛) is an 𝑟×𝑟 matrix with det𝐴(𝑛)≠0, and the functions 𝑓 and 𝑘 are defined on 𝑁×𝐑^𝑟×𝐑^𝑟 and 𝑁×𝑁×𝐑^𝑟, respectively. Moreover, 𝑓 and 𝑘 are supposed to meet the following constraints: 𝑓(𝑛,𝑥(𝑛),𝜎(𝑛))≤𝑔₁(𝑛)𝑒^−𝛼𝑛𝑤₁|𝑥(𝑛)|𝑒^𝛼𝑛(|𝜎(𝑛,|𝑘((𝑠1+)|)𝑛,𝑠,𝑥))|≤𝑔₂(𝑛,𝑠)𝑤₂|𝑥(𝑛)|𝑒^𝛼𝑛1+^𝑠−1_𝜏=𝑛0𝑔₃(𝑠,𝜏)𝑤₃|𝑥(𝜏)|𝑒^𝛼𝜏, (3.4)(3.5) where 𝛼>0 is a constant, 𝑔_𝑖, 𝑖=1,2,3 are nonnegative real-valued functions defined on 𝑁₀ and 𝑁₀×𝑁₀, respectively, 𝑔₂(𝑛,𝑠) and 𝑔₃(𝑛,𝑠) are nondecreasing in 𝑛 for fixed 𝑠∈𝑁₀, and 𝑤_𝑖(𝑢), 𝑖=1,2,3 are positive and continuous functions defined on [0,∞). The symbol |⋅| denotes norm on 𝐑^𝑟 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 𝑓 and 𝑘 satisfy the conditions (3.4) and (3.5). Assume that the fundamental solution matrix 𝑌(𝑛) of the linear system (3.3) satisfies 𝑌(𝑛)𝑌⁻¹(𝑠)≤𝐶exp(−𝛼(𝑛−𝑠)),0≤𝑠≤𝑛≤∞, (3.6) where 𝐶>0 is a constant. Then, any solutions of the control systems (3.1) and (3.2), denoted by 𝑥_𝜎(𝑛,𝑛₀,𝑥₀), can be estimated by 𝑥_𝜎𝑛,𝑛₀,𝑥₀≤exp(−𝛼𝑛)Φ₄⁻¹Φ₅⁻¹Φ₆⁻¹𝑈₄(𝑛),∀𝑛∈𝑁_𝑀4=𝑛₀,𝑀₄∩𝐍, (3.7) where 𝑈₄(𝑛)=Φ₆Φ₅Φ₄𝑥₀𝐶exp𝛼𝑛₀+^𝑛−1_𝑠=𝑛0𝐶𝑒^𝛼𝑔₁(𝑠|𝜃(𝑠+)(1+)|)_{𝑛−1𝑠=𝑛0}𝐶𝑒^𝛼𝑔₁(𝑠)(1+|𝜃(𝑠)|)^𝑠−1_𝜏=𝑛0𝑓₂()+𝑠,𝜏_{𝑛−1𝑠=𝑛0}𝐶𝑒^𝛼𝑔₁(𝑠)(1+|𝜃(𝑠)|)^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)^𝜏−1_𝜉=𝑛0𝑓₃(),Φ𝜏,𝜉₄(𝑢)=^𝑢₁𝑑𝑠𝑤₁(𝑠)Φ,𝑢>0,₅(𝑢)=^𝑢₁𝑑𝑠𝑤₂Φ₁⁻¹(𝑠)Φ,𝑢>0,₆(𝑢)=^𝑢₁𝑑𝑠𝑤₃Φ₁⁻¹Φ₂⁻¹(𝑠),𝑢>0, (3.8)Φ_𝑖⁻¹, 𝑖=4,5,6 are the inverse functions of Φ_𝑖, 𝑖=4,5,6, respectively, and 𝑀₄ is the largest natural number such that 𝑈₄𝑀₄∈DomΦ₆⁻¹,Φ₆⁻¹𝑈₄𝑀₄∈DomΦ₅⁻¹,Φ₅⁻¹Φ₆⁻¹𝑈₄𝑀₄∈DomΦ₄⁻¹. (3.9)

Proof. By using the variation of constants formula, any solution 𝑥_𝜎(𝑛,𝑛₀,𝑥₀) of (3.1) and (3.2) can be represented by 𝑥_𝜎𝑛,𝑛₀,𝑥₀=𝑌(𝑛)𝑌⁻¹𝑛₀𝑥₀+^𝑛−1_𝑠=𝑛0𝑌(𝑠)𝑌⁻¹(𝑠+1)𝑓𝑠,𝑥_𝜎𝑠,𝑛₀,𝑥₀,𝜎(𝑠), (3.10) for all 𝑛∈𝑁₀. Using the conditions (3.4) and (3.6) in (3.10), we have 𝑥_𝜎𝑛,𝑛₀,𝑥₀≤𝑥₀𝐶exp−𝛼𝑛−𝑛₀+^𝑛−1_𝑠=𝑛0𝐶exp(−𝛼(𝑛−𝑠−1))×𝑔₁(𝑠)𝑒^−𝛼𝑠𝑤₁𝑥_𝜎𝑠,𝑛₀,𝑥₀𝑒^𝛼𝑠(1+|𝜎(𝑠)|),∀𝑛∈𝑁₀. (3.11) Further, using the relationships (3.2), (3.5), and (3.11), we derive 𝑥_𝜎𝑛,𝑛₀,𝑥₀≤𝑥₀𝐶exp−𝛼𝑛−𝑛₀+^𝑛−1_𝑠=𝑛0𝐶exp(−𝛼(𝑛−1))×𝑔₁(𝑠)𝑤₁𝑥_𝜎𝑠,𝑛₀,𝑥₀𝑒^𝛼𝑠1+|𝜃(𝑠)|+^𝑠−1_𝜏=𝑛0𝑔₂(𝑠,𝜏)𝑤₂𝑥_𝜎𝜏,𝑛₀,𝑥₀𝑒^𝛼𝜏×1+^𝜏−1_𝜏=𝑛0𝑔₃(𝜏,𝜉)𝑤₃𝑥_𝜎𝜉,𝑛₀,𝑥₀𝑒^𝛼𝜉, (3.12) for all 𝑛∈𝑁₀. Let 𝑢(𝑛)=|𝑥_𝜎(𝑛,𝑛₀,𝑥₀)|exp(𝛼𝑛), then, (3.12) can be rewritten as 𝑢(𝑛)≤𝑥₀𝐶exp𝛼𝑛₀+^𝑛−1_𝑠=𝑛0𝐶𝑒^𝛼𝑔₁(𝑠)(1+|𝜃(𝑠)|)𝑤₁(𝑢(𝑠+))_{𝑛−1𝑠=𝑛0}𝐶𝑒^𝛼𝑔₁(𝑠)(1+|𝜃(𝑠)|)𝑤₁(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑔₂(𝑠,𝜏)𝑤₂(𝑢(𝜏+))_{𝑛−1𝑠=𝑛0}𝐶𝑒^𝛼𝑔₁(𝑠)(1+|𝜃(𝑠)|)𝑤₁(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑔₂(𝑠,𝜏)×𝑤₂(𝑢(𝜏))^𝜏−1_𝜉=𝑛0𝑔₃(𝜏,𝜉)𝑤₃(𝑢(𝜉)),∀𝑛∈𝑁₀. (3.13) Let 𝑎(𝑛)=|𝑥₀|𝐶exp(𝛼𝑛₀), 𝑓₁(𝑛,𝑠)=𝐶𝑔₁(𝑠)𝑒^𝛼(1+|𝜃(𝑠)|), 𝑓₂(𝑛,𝑠)=𝑔₂(𝑛,𝑠), and 𝑓₃(𝑛,𝑠)=𝑔₃(𝑛,𝑠), then (3.13) can be further estimated as follows: 𝑢(𝑛)≤𝑎(𝑛)+^𝑛−1_𝑠=𝑛0𝑓₁(𝑛,𝑠)𝑤₁(𝑢(𝑠))+^𝑛−1_𝑠=𝑛0𝑓₁(𝑛,𝑠)𝑤₁(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)𝑤₂(𝑢(𝜏+))_{𝑛−1𝑠=𝑛0}𝑓₁(𝑛,𝑠)𝑤₁(𝑢(𝑠))^𝑠−1_𝜏=𝑛0𝑓₂(𝑠,𝜏)𝑤₂(𝑢(𝜏))^𝜏−1_𝜉=𝑛0𝑓₃(𝜏,𝜉)𝑤₃(𝑢(𝜉)), (3.14) for all 𝑛∈𝑁₀. 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 𝑢(𝑛)=|𝑥_𝜎(𝑛,𝑛₀,𝑥₀)|exp(𝛼𝑛) 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 𝐵 such that Φ₄⁻¹Φ₅⁻¹Φ₆⁻¹𝑈₄(𝑛)≤𝐵,∀𝑛∈𝐍, (3.15) 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: 𝑥_𝜎𝑛,𝑛₀,𝑥₀≤𝐵exp(−𝛼𝑛),∀𝑛∈𝑛₀,∞∩𝐍. (3.16) 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).