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

Authors: Hong Zhou; Deqing Huang; Wu-Sheng Wang; Jian-Xin Xu
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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:
(1.1)u(n)≤u0+∑s=n0n-1‍f(s)[u(s)+h(s)]+∑s=n0n-1‍f(s)(∑τ=n0s-1‍g(τ)u(τ)),∀n∈N0.
In 2006, Cheung and Ren [18] studied
(1.2)up(m,n)≤c+∑s=m0m-1‍∑t=n0n-1‍a(s,t)uq(s,t)+∑s=m0m-1‍∑t=n0n-1‍b(s,t)uq(s,t)w(u(s,t)).
Later, Zheng et al. [24] discussed the following discrete inequality:
(1.3)u(n)≤a(n)+∑i=1k‍∑s=0n-1‍fi(n,s)wi(u(s)).
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:(1.4)u(n)≤a(n)+∑s=n0n-1‍f1(n,s)w(u(s))+∑s=n0n-1‍f1(n,s)w(u(s))∑τ=n0s-1‍f2(s,τ)w(u(τ))+∑s=n0n-1‍f1(n,s)w(u(s))∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)w(u(ξ)),(1.5)u(n)≤a(n)+∑s=n0n-1‍f1(n,s)w1(u(s))+∑s=n0n-1‍f1(n,s)w1(u(s))∑τ=n0s-1‍f2(s,τ)w2(u(τ))+∑s=n0n-1‍f1(n,s)w1(u(s))∑τ=n0s-1‍f2(s,τ)w2(u(τ))∑ξ=n0τ-1‍f3(τ,ξ)w3(u(ξ)),
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.
Letu(n) and a(n) be nonnegative functions defined on N0 with a(n) nondecreasing on N0. Moreover, let fi(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
(2.1)u(n)≤W1-1[W2-1(U1(n))],∀n∈[n0,M1)∩N,
where
(2.2)U1(n)=W2(W1(a(n))+∑s=n0n-1‍f1(n,s))+∑s=n0n-1‍f1(n,s)(∑τ=n0s-1‍f2(s,τ)+∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)),(2.3)W2(u)=∫1u‍dsw(W1-1(s)),u>0,(2.4)W1(u)=∫1u‍dsw(s),u>0,W1-1, W2-1 are the inverse functions of W1, W2, respectively, and M1 is the largest natural number such that
(2.5)U1(M1)∈Dom(W2-1),W2-1(U1(M1))∈Dom(W1-1).Proof.
FixM∈NM1=[n0,M1)∩N, where M is chosen arbitrarily and M1 is defined by (2.5). For n∈NM=[n0,M]∩N, from (1.4), we have
(2.6)u(n)≤a(M)+∑s=n0n-1‍f1(M,s)w(u(s))+∑s=n0n-1‍f1(M,s)w(u(s))∑τ=n0s-1‍f2(s,τ)w(u(τ))+∑s=n0n-1‍f1(M,s)w(u(s))∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)w(u(ξ)).
Denote the right-hand side of (2.6) by z1(n), which is a positive and nondecreasing function on NM with z1(n0)=a(M). Then, (2.6) is equivalent to
(2.7)u(n)≤z1(n),∀n∈NM.
From (2.6) and (2.7), we observe that
(2.8)Δz1(n):=z1(n+1)-z1(n)≤f1(M,n)w(z1(n))+f1(M,n)w(z1(n))∑τ=n0n-1‍f2(n,τ)w(z1(τ))+f1(M,n)w(z1(n))∑τ=n0n-1‍f2(n,τ)∑ξ=n0τ-1‍f3(τ,ξ)w(z1(ξ))=f1(M,n)w(z1(n))[∑ξ=n0τ-1‍1+∑τ=n0n-1‍f2(n,τ)w(z1(τ))+∑τ=n0n-1‍f2(n,τ)∑ξ=n0τ-1‍f3(τ,ξ)w(z1(ξ))],∀n∈NM.
Furthermore, it follows from (2.8) that(2.9)Δz1(n)w(z1(n))≤f1(M,n)[∑ξ=n0τ-1‍1+∑τ=n0n-1f2(n,τ)w(z1(τ))+∑τ=n0n-1‍f2(n,τ)∑ξ=n0τ-1‍f3(τ,ξ)w(z1(ξ))],∀n∈NM.
On the other hand, by the mean-value theorem for integrals, for arbitrarily given integers n,n+1∈NM, there exists η in the open interval (z1(n),z1(n+1)) such that
(2.10)W1(z1(n+1))-W1(z1(n))=∫z1(n)z1(n+1)‍dsw(z1(s))=Δz1(n)w(z1(η))≤Δz1(n)w(z1(n))≤f1(M,n)[∑ξ=n0τ-1‍1+∑τ=n0n-1‍f2(n,τ)w(z1(τ))+∑τ=n0n-1‍f2(n,τ)∑ξ=n0τ-1‍f3(τ,ξ)w(z1(ξ))],∀n∈NM,
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
(2.11)W1(z1(n))≤W1(z1(n0))+∑s=n0M-1‍f1(M,s)+∑s=n0n-1‍f1(M,s)×[∑τ=n0s-1‍f2(s,τ)w(z1(τ))+∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)w(z1(ξ))],∀n∈NM.
Let v1(n) denote the right-hand side of (2.11), which is a positive and nondecreasing function on NM with v1(n0)=W1(z1(n0))+∑s=n0M-1‍f1(M,s). Then, (2.11) is equivalent to
(2.12)z1(n)≤W1-1(v1(n)),∀n∈NM.
By the definition of v1, we obtain
(2.13)Δv1(n):=v1(n+1)-v1(n)=f1(M,n)[∑τ=n0n-1‍f2(n,τ)w(z1(τ))+∑τ=n0n-1‍f2(n,τ)∑ξ=n0τ-1‍f3(τ,ξ)w(z1(ξ))],∀n∈NM.
Considering (2.12), (2.13) and the monotonicity properties of w, W1-1, and z1, we get
(2.14)Δv1(n)w(W1-1(v1(n)))≤f1(M,n)[∑τ=n0n-1‍f2(n,τ)+∑τ=n0n-1‍f2(n,τ)∑ξ=n0τ-1‍f3(τ,ξ)],
for all n∈NM. Once again, performing the same procedure as in (2.10) and (2.11), (2.14) gives
(2.15)W2(v1(n))≤W2(v1(n0))+∑s=n0n-1‍f1(M,s)[∑τ=n0s-1‍f2(s,τ)+∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)],
for all n∈NM, where W2 is defined in (2.3). In the sequel, (2.7), (2.12), and (2.15) render to
(2.16)u(n)≤z1(n)≤W1-1(v1(n))=W1-1[W2-1(W2(∑W1(a(M))+∑s=n0M-1‍f1(M,s))+∑s=n0n-1‍f1(M,s)×(∑τ=n0s-1‍f2(s,τ)+∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)∑))],∀n∈NM.
Let n=M in (2.16), then, we have(2.17)u(n)≤W1-1[∑ξ=n0τ-1‍W2-1(∑W2(W1(a(M))+∑s=n0M-1‍f1(M,s))+∑s=n0M-1‍f1(M,s)×(∑τ=n0s-1‍f2(s,τ)+∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)∑))].
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 functionsu(n), a(n), fi(n,s), i=1,2,3, and φ(u) be the same as in Theorem 2.1. Suppose that wi(u), i=1,2,3 are nondecreasing functions on [0,∞) with wi(u)>0 for u>0. If u(n) satisfies the discrete inequality (1.5), then
(2.18)u(n)≤Φ1-1[Φ2-1(Φ3-1(U2(n)))],∀n∈NM3=[n0,M3)∩N,
where
(2.19)U2(n)=Φ3(Φ2(Φ1(a(n))+∑s=n0n-1‍f1(n,s))+∑s=n0n-1‍f1(n,s)∑τ=n0s-1‍f2(s,τ))+∑s=n0n-1‍f1(n,s)∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ),(2.20)Φ1(u)=∫1u‍dsw1(s),u>0,(2.21)Φ2(u)=∫1u‍dsw2(Φ1-1(s)),u>0,(2.22)Φ3(u)=∫1u‍dsw3(Φ1-1(Φ2-1(s))),u>0,Φi-1, i=1,2,3 are the inverse functions of Φi, i=1,2,3, respectively, and M2 is the largest natural number such that
(2.23)U2(M2)∈Dom(Φ3-1),Φ3-1(U2(M2))∈Dom(Φ2-1),Φ2-1(Φ3-1(U2(M2)))∈Dom(Φ1-1).Proof.
FixM∈NM2=[n0,M2)∩N, where M is chosen arbitrarily and M2 is given in (2.23). For n∈NM, from (1.5), we have
(2.24)u(n)≤a(M)+∑s=n0n-1‍f1(M,s)w1(u(s))+∑s=n0n-1‍f1(M,s)w1(u(s))∑τ=n0s-1‍f2(s,τ)w2(u(τ))+∑s=n0n-1‍f1(M,s)w1(u(s))∑τ=n0s-1‍f2(s,τ)w2(u(s))∑ξ=n0τ-1‍f3(τ,ξ)w3(u(ξ)).
Let z2(n) represent the right-hand side of (2.24), which is a positive and nondecreasing function on NM2 with z2(n0)=a(M). Then, (2.24) is equivalent to
(2.25)u(n)≤z2(n),∀n∈NM.
Using (2.24) and (2.25), Δz2(n):=z2(n+1)-z2(n) can be estimated as follows:
(2.26)Δz2(n)≤f1(M,n)w1(z2(n))+f1(M,n)w1(z2(n))∑τ=n0n-1‍f2(n,τ)w2(z2(τ))+f1(M,n)w1(z2(n))∑τ=n0n-1‍f2(n,τ)w2(z2(n))∑ξ=n0τ-1‍f3(τ,ξ)w3(z2(ξ))=f1(M,n)w1(z2(n))[∑ξ=n0τ-1‍1+∑τ=n0n-1‍f2(n,τ)w2(z2(τ))+∑τ=n0n-1‍f2(n,τ)w2(z2(τ))∑ξ=n0τ-1‍f3(τ,ξ)w3(z2(ξ))],∀n∈NM,
Implying
(2.27)Δz2(n)w1(z2(n))≤f1(M,n)[1+∑τ=n0n-1‍f2(n,τ)w2(z2(τ))+∑τ=n0n-1‍f2(n,τ)w2(z2(τ))∑ξ=n0τ-1‍f3(τ,ξ)w3(z2(ξ))],
for all n∈NM. Performing the same derivation as in (2.10) and (2.11), we obtain from (2.27) that
(2.28)Φ1(z2(n))≤Φ1(z2(n0))+∑s=n0M-1‍f1(M,s)+∑s=n0n-1‍f1(M,s)×[∑ξ=n0τ-1‍∑τ=n0s-1‍f2(s,τ)w2(z2(τ))+∑τ=n0s-1‍f2(s,τ)w2(z2(τ))∑ξ=n0τ-1‍f3(τ,ξ)w3(z2(ξ))],∀n∈NM,
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 NM2 with v2(n0)=Φ1(z2(n0))+∑s=n0M-1‍f1(M,s)=Φ1(a(M))+∑s=n0M-1‍f1(M,s). Then, (2.28) is equivalent to
(2.29)z2(n)≤Φ1-1(v2(n)),∀n∈NM.
By the definition of v2, we obtain
(2.30)Δv2(n):=v2(n+1)-v2(n)=f1(M,n)[∑τ=n0n-1‍f2(n,τ)w2(z2(τ))+∑τ=n0n-1‍f2(n,τ)w2(z2(τ))∑ξ=n0τ-1‍f3(τ,ξ)w3(z2(ξ))],∀n∈NM.
From (2.29), (2.30) and the monotonicity of w2,Φ1-1, and z2, we get
(2.31)Δv2(n)w2(Φ1-1(v2(n)))≤f1(M,n)[∑τ=n0n-1‍f2(n,τ)+∑τ=n0n-1‍f2(n,τ)∑ξ=n0τ-1‍f3(τ,ξ)w3(Φ1-1(v2(ξ)))],
for all n∈NM. Similarly to (2.28), it follows from (2.31) that
(2.32)Φ2(v2(n))≤Φ2(v2(n0))+∑s=n0M-1‍f1(M,s)∑τ=n0s-1‍f2(s,τ)+∑s=n0n-1‍f1(M,s)∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)w3(Φ1-1(v2(n))),
for all n∈NM, 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 NM2 with
(2.33)v3(n0)=Φ2(v2(n0))+∑s=n0M-1‍f1(M,s)∑τ=n0s-1‍f2(s,τ)=Φ2(Φ1(a(M))+∑s=n0M-1‍f1(M,s))+∑s=n0M-1‍f1(M,s)∑τ=n0s-1‍f2(s,τ).
Then, (2.32) is equivalent to
(2.34)v2(n)≤Φ2-1(v3(n)),∀n∈NM.
By the definition of v3,
(2.35)Δv3(n):=v3(n+1)-v3(n)=f1(M,n)∑τ=n0n-1‍f2(n,τ)∑ξ=n0τ-1‍f3(τ,ξ)w3(Φ1-1(v2(ξ))),∀n∈NM.
In consequence, (2.34), (2.35) and the monotonicity properties of w3,Φ1-1, and v2 lead to
(2.36)Δv3(n)w3(Φ1-1(Φ2-1(v3(n))))≤f1(M,n)∑τ=n0n-1‍f2(n,τ)∑ξ=n0τ-1‍f3(τ,ξ),∀n∈NM.
Similarly to (2.28) and (2.32), we obtain from (2.36) that
(2.37)Φ3(v3(n))≤Φ3(v3(n0))+∑s=n0n-1‍f1(M,s)∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ),∀n∈NM,
where Φ3 is defined in (2.22).
Summarizing the results in (2.25), (2.29), (2.34), and (2.37), we can conclude that
(2.38)u(n)≤z2(n)≤Φ1-1[v2(n)]≤Φ1-1[Φ2-1(v3(n))]≤Φ1-1[Φ2-1(Φ3-1(Φ3(v3(n0))+∑s=n0n-1‍f1(M,s)∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)))]=Φ1-1[Φ2-1(∑ξ=n0τ-1‍Φ3-1(Φ3(Φ2(Φ1(a(M))+∑s=n0M-1‍f1(M,s))+∑s=n0M-1‍f1(M,s)∑τ=n0s-1‍f2(s,τ))+∑s=n0n-1‍f1(M,s)∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)))],
for all n∈NM. As n=M, (2.38) yields
(2.39)u(M)≤Φ1-1[Φ2-1(Φ3-1(Φ3(Φ2(Φ1(a(M))+∑s=n0M-1‍f1(M,s))+∑s=n0M-1‍f1(M,s)∑τ=n0s-1‍f2(s,τ))+∑s=n0M-1‍f1(M,s)∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ)))].
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 Theorem2.2 is applied to explore the asymptotic stability behavior of a class of discrete-time control systems [17]
(3.1)x(n+1)=A(n)x(n)+f(n,x(n),σ(n)),x(n0)=x0,
where
(3.2)σ(n)=θ(n)+∑s=n0n-1‍k(n,s,x(s)).
Control system (3.1) can be regarded as the perturbation counterpart of the following closed-loop system:
(3.3)y(n+1)=A(n)y(n),y(n0)=x0.
The functions x, y, θ, σ are defined on N→Rr, 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×Rr×Rr and N×N×Rr, respectively. Moreover, f and k are supposed to meet the following constraints:
(3.4)|f(n,x(n),σ(n))|≤g1(n)e-αnw1(|x(n)|eαn)(1+|σ(n)|),(3.5)|k(n,s,x(s))|≤g2(n,s)w2(|x(n)|eαn)(1+∑τ=n0s-1‍g3(s,τ)w3(|x(τ)|eατ)),
where α>0 is a constant, gi, 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 wi(u), i=1,2,3 are positive and continuous functions defined on [0,∞). The symbol |·| denotes norm on Rr 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
(3.6)|Y(n)Y-1(s)|≤Cexp(-α(n-s)),0≤s≤n≤∞,
where C>0 is a constant. Then, any solutions of the control systems (3.1) and (3.2), denoted by xσ(n,n0,x0), can be estimated by
(3.7)|xσ(n,n0,x0)|≤exp(-αn){Φ4-1[Φ5-1(Φ6-1(U4(n)))]},∀n∈NM4=[n0,M4)∩N,
where
(3.8)U4(n)=Φ6(Φ5(Φ4(|x0|Cexp(αn0))+∑s=n0n-1‍Ceαg1(s)(1+|θ(s)|))+∑s=n0n-1‍Ceαg1(s)(1+|θ(s)|)∑τ=n0s-1‍f2(s,τ))+∑s=n0n-1‍Ceαg1(s)(1+|θ(s)|)∑τ=n0s-1‍f2(s,τ)∑ξ=n0τ-1‍f3(τ,ξ),Φ4(u)=∫1u‍dsw1(s),u>0,Φ5(u)=∫1u‍dsw2(Φ1-1(s)),u>0,Φ6(u)=∫1u‍dsw3(Φ1-1(Φ2-1(s))),u>0,Φi-1, i=4,5,6 are the inverse functions of Φi, i=4,5,6, respectively, and M4 is the largest natural number such that
(3.9)U4(M4)∈Dom(Φ6-1),Φ6-1(U4(M4))∈Dom(Φ5-1),Φ5-1(Φ6-1(U4(M4)))∈Dom(Φ4-1).Proof.
By using the variation of constants formula, any solutionxσ(n,n0,x0) of (3.1) and (3.2) can be represented by
(3.10)xσ(n,n0,x0)=Y(n)Y-1(n0)x0+∑s=n0n-1‍Y(s)Y-1(s+1)f(s,xσ(s,n0,x0),σ(s)),
for all n∈N0. Using the conditions (3.4) and (3.6) in (3.10), we have
(3.11)|xσ(n,n0,x0)|≤|x0|Cexp(-α(n-n0))+∑s=n0n-1‍Cexp(-α(n-s-1))×g1(s)e-αsw1(|xσ(s,n0,x0)|eαs)(1+|σ(s)|),∀n∈N0.
Further, using the relationships (3.2), (3.5), and (3.11), we derive
(3.12)|xσ(n,n0,x0)|≤|x0|Cexp(-α(n-n0))+∑s=n0n-1‍Cexp(-α(n-1))×g1(s)w1(|xσ(s,n0,x0)|eαs)[∑τ=n0τ-1‍1+|θ(s)|+∑τ=n0s-1‍g2(s,τ)w2(|xσ(τ,n0,x0)|eατ)×(1+∑τ=n0τ-1‍g3(τ,ξ)w3(|xσ(ξ,n0,x0)|eαξ))],
for all n∈N0. Let u(n)=|xσ(n,n0,x0)|exp(αn), then, (3.12) can be rewritten as
(3.13)u(n)≤|x0|Cexp(αn0)+∑s=n0n-1‍Ceαg1(s)(1+|θ(s)|)w1(u(s))+∑s=n0n-1‍Ceαg1(s)(1+|θ(s)|)w1(u(s))∑τ=n0s-1‍g2(s,τ)w2(u(τ))+∑s=n0n-1‍Ceαg1(s)(1+|θ(s)|)w1(u(s))∑τ=n0s-1‍g2(s,τ)×w2(u(τ))∑ξ=n0τ-1‍g3(τ,ξ)w3(u(ξ)),∀n∈N0.
Let a(n)=|x0|Cexp(αn0), f1(n,s)=Cg1(s)eα(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:
(3.14)u(n)≤a(n)+∑s=n0n-1‍f1(n,s)w1(u(s))+∑s=n0n-1‍f1(n,s)w1(u(s))∑τ=n0s-1‍f2(s,τ)w2(u(τ))+∑s=n0n-1‍f1(n,s)w1(u(s))∑τ=n0s-1‍f2(s,τ)w2(u(τ))∑ξ=n0τ-1‍f3(τ,ξ)w3(u(ξ)),
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)=|xσ(n,n0,x0)|exp(αn) is adopted. This completes the proof of Corollary 3.1.Based on Corollary3.1 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 Corollary3.1, if there exists a positive constant B such that
(3.15){Φ4-1[Φ5-1(Φ6-1(U4(n)))]}≤B,∀n∈N,
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:
(3.16)|xσ(n,n0,x0)|≤Bexp(-αn),∀n∈[n0,∞)∩N.
The exponentially asymptotic stability of system (3.1) and (3.2) is directly implied.



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*Source: 214609-2012-10-17.xml*