# Globally Exponential Stability of Impulsive Neural Networks with Given Convergence Rate

Authors: Chengyan Liu; Xiaodi Li; Xilin Fu
Journal: Advances in Artificial Neural Systems
 (2013)

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## Abstract

This paper deals with the stability problem for a class of impulsive neural networks. Some sufficient conditions which can guarantee the globally exponential stability of the addressed models with given convergence rate are derived by using Lyapunov function and impulsive analysis techniques. Finally, an example is given to show the effectiveness of the obtained results.


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## Body

## 1. Introduction

Recently, special interest has been devoted to the dynamics analysis of neural networks due to their potential applications in different areas of science. Particularly, there has been a significant development in the theory of neural networks with impulsive effects [1–9], since such neural networks with impulsive effect can be used as an appropriate description of the phenomena of abrupt qualitative dynamical changes of essential continuous time systems. Based on the theory of impulsive differential equations [10–17], some sufficient conditions guaranteeing the exponential stability are derived [18–24]. For example, in [8], the author has obtained a criterion of exponential stability for a Hopfield neural network with periodic coefficients; in [18], by constructing the extended impulsive delayed Halanay inequality and Lyapunov functional methods, authors have got some sufficient conditions ensuring exponential stability of the unique equilibrium point of impulsive Hopfield neural networks with time delays. They all have obtained exponential stability for some kinds of neural networks through different methods. However, most of the existing results about the exponential stability of impulsive neural networks have a common feature that the exponential convergence rate cannot be derived, or derived but not the given one [8, 18, 23, 24]. The purpose of this paper is to establish some criteria which can guarantee the globally exponential stability of impulsive neural networks with the given convergence rate by using Lyapunov function and impulsive analysis techniques. This work is organized as follows. In Section 2, we introduce some basic definitions and notations. In Section 3, the main results are presented. In Section 4, an example is discussed to illustrate the results.


## 2. Preliminaries

Letℝ denote the set of real numbers, ℝ+ denote the set of nonnegative real numbers, ℤ+ denote the set of positive integers and ℝn denote the n-dimensional real space equipped with the Euclidean norm ∥·∥.Consider the following impulsive neural networks:(1)x˙i(t)=-ai(t)xi(t)+∑j=1n‍bij(t)fj(xj(t))+Ii(t),t≥t0,t≠tk,Δxi|t=tk=xi(tk)-xi(tk-),k∈ℤ+,i∈Λ,
where Λ={1,2,…,n}. n≥2 corresponds to the number of units in a neural network; the impulse times tk satisfy 0≤t0<t1<⋯<tk<⋯, limk→∞tk=∞; xi corresponds to the state of the neurons, fj denotes the measures of response to its incoming potentials of the unit j at time t; Ii(t) is the input of the unit i at time t. PC[I,ℝ]≜{φ:I→ℝ∣φ(t+)=φ(t) for t∈I, φ(t-) exists for t∈I, φ(t-)=φ(t) for all but points tk∈I}, where I⊂ℝ is an interval, φ(t+) and φ(t-) denote the left limit and right limit of function φ(t), respectively. ai(t)> 0, bij(t),Ii(t)∈PC[[t0,+∞),ℝ]. For given t0,x(t0)=x0=(x10,x20,…,xn0)∈ℝn, we denote by x(t) the solution of system (1) with initial value (t0,x(t0)).In this paper, we assume that some conditions are satisfied so that the equilibrium point of system (1) does exist, see [16, 17]. Assume that x⋆=(x1⋆,x2⋆,…,xn⋆)T is an equilibrium point of system (1). Impulsive operator is viewed as perturbation of the point x⋆ of such system without impulsive effects. We assume that the following impulsive condition holds.(H0) Δxi|t=tk=xi(tk)-xi(tk-)=σik(xi(tk-)-xi⋆), σik∈ℝ, i∈Λ, k∈ℤ+.Furthermore, we will assume that the response functionfi satisfies the following condition.(H1) fi is globally Lipschizian with Lipschitz constant li>0, that is, |fi(s1)-fi(s2)|≤li|s1-s2|,  for all  s1,s2∈ℝ, i∈Λ.Note thatx⋆ is an equilibrium point of system (1), one can derive from system (1) that the transformation zi=xi-xi⋆, i∈Λ transforms such system into the following system:
(2)z˙i(t)=-ai(t)zi(t)+∑j=1n‍bij(t)Fj(zj(t)),t≥t0,t≠tk,zi(tk)=(1+σik)zi(tk-),i∈Λ,k∈ℤ+,
where Fj(zj(t))=fj(xj⋆+zj(t))-fj(xj⋆), and from the condition (H1), it holds that ∥Fj(zj(t))∥≤lj∥zj(t)∥, j∈Λ.Furthermore, letyi(t)=zi(t)eα(t-t0), i∈Λ, then system (2) becomes as follows:
(3)y˙i(t)=(α-ai(t))yi(t)+eα(t-t0)∑j=1n‍bij(t)Fj(yj(t)e-α(t-t0)),t≥t0,t≠tk,yi(tk)=(1+σik)yi(tk-),i∈Λ,k∈ℤ+.To prove the stability ofx⋆ of system (1), it is equal to prove the stability of the zero solution of system (2), and also equal to the boundedness of system (3).In the following, the notionAT means the transpose of a square matrix A. We will use the notation A>0 (or A<0, A≥0, A≤0) to denote that the matrix A is a positive definite (negtive definite, positive semidefinite, an negative semidefinite) marix.Lety(t)=(y1(t),y2(t),…,yn(t))T, A(t)=diag[a1(t),a2(t),…,an(t)], B(t)=(bij(t))n×n, L=diag[l1,l2,…,ln], I=diag[1,1,…,1], Dk=diag[1+σ1k,1+σ2k,…,1+σnk], F(y)=(F1(y1),F2(y2),…,andFn(yn))T then system (3) with initial condition becomes as follows:
(4)y˙(t)=(αI-A(t))y(t)+e-α(t-t0)B(t)F(y(t)e-α(t-t0)),t≥t0,t≠tk,y(tk)=Dky(tk-),i∈Λ,k∈ℤ+.We introduce a definition as follows.Definition 1.
Assumex⋆=(x1⋆,x2⋆,…,xn⋆)T∈ℝn is the equilibrium point of system (1), then the equilibrium point x⋆ of system (1) is said to be globally exponential stable with given convergence rate α>0. If for any initial data x(t0)=x0∈ℝ, there exists a constant M⩾1, such that
(5)∥x(t,t0,x0)-x⋆∥⩽∥x0-x⋆∥Me-α(t-t0),t⩾t0.
From the transformationzi=xi-xi⋆, i∈Λ, and z(t0)=z0=x0-x⋆, the globally exponential stability of the equilibrium point x⋆ of system (1) can be transformed into the globally exponential stability of trivial solution of system (2), so (5) can be rewritten as follows:
(6)∥z(t,t0,z0)∥≤∥z(t0)∥Me-α(t-t0),t⩾t0.
Furthermore, form the transformationy(t)=z(t,t0,z0)eα(t-t0), i∈Λ, the globally exponential stability of trivial solution of system (2) can be transformed into the boundedness of the solution of system (4) and it can be rewritten as follows:
(7)∥y(t)∥≤∥y(t0)∥M,t⩾t0.


## 3. Main Results

Theorem 2.
Given constantα>0. The equilibrium point of the system (1) is globally exponentially stable with the given convergence rate α, if the conditions (H0) and (H1) are fulfilled; moreover, suppose that (i)
αI-A(t)+B(t)L≤0, for all t>t0,(ii)
∏k=1∞‍maxi∈Λ(1+σik)<∞, and σik≥0, i∈Λ, k∈ℤ+.Proof.
We only need to provey(t)=z(t)eα(t-t0) is bounded when t≥t0, where z(t)=z(t,t0,z0) is a solution of (3) through (t0,z0).
Consider the Lyapunov function as follows:(8)V(t)=y(t)Ty(t)=∑i=1n‍yi2(t).
Particularly, V(t0)=∑i=1n‍yi2(t0).
Then from conditions (H0)-(H1) and (i), we get the upper right-hand derivative of V(t) along the solutions of system (3), for t∈[tk,tk+1), k∈ℤ+(9)D+V(t)=2∑i=1n‍yi(t)y˙i(t)=2∑i=1n‍zi(t)e2α(t-t0)[z˙i(t)+αzi(t)]=2∑i=1n‍zi(t)e2α(t-t0)×[-ai(t)zi(t)+∑j=1n‍bij(t)Fj(zj(t))+αzi(t)]=2∑i=1n‍e2α(t-t0)×[(α-ai(t))zi2(t)+zi(t)∑j=1n‍bij(t)Fj(zj(t))]≤2∑i=1n‍e2α(t-t0)×[(α-ai(t))zi(t)2+zi(t)∑j=1n‍bij(t)lj∥zj(t)∥]=2∑i=1n‍e2α(t-t0)zi2(t)(α-ai(t))+2∑i=1n‍e2α(t-t0)zi(t)∑j=1n‍bij(t)lj∥zj(t)∥=2y(t)T[αI-A(t)+B(t)L]y(t)≤0,
which implies the functional V(t) is nonincreasing for t∈[tk,tk+1), k∈ℤ+. By condition (ii), it holds that
(10)V(tk)=∑i=1n‍yi2(tk)=∑i=1n‍zi2(tk)e2α(tk-t0)=∑i=1n‍e2α(t-t0)[zi(tk-)+Ji(zi(tk-))]2=∑i=1n‍e2α(t-t0)(1+σik)2zi2(tk-)≤maxi∈Λ(1+σik)2V(tk-).
For any t∈[t0,t1), since V(t) is nonincreasing, it holds that V(t)≤V(t0); moreover,
(11)V(t1)≤maxi∈Λ(1+σi1)2V(t1-)≤maxi∈Λ(1+σi1)2V(t0).
Similarly, for any t∈[t1,t2), it holds that V(t)≤V(t1)≤maxi∈Λ(1+σi1)2V(t0), and
(12)V(t2)≤maxi∈Λ(1+σi2)2V(t2-)≤maxi∈Λ(1+σi2)2maxi∈Λ(1+σi1)2V(t0).
Thus, it can be deduced that for t∈[tk,tk+1),  k∈ℤ+,
(13)V(t)≤∏j=1k‍maxi∈Λ(1+σij)2V(t0).
Hence, we obtain that for any t≥t0,
(14)V(t)≤∏j=1∞‍maxi∈Λ(1+σij)2V(t0),
which implies that
(15)∥y(t)∥≤M∥y(t0)∥,t≥t0,
where M=∏k=1∞‍maxi∈Λ(1+σik)<∞.
The proof of Theorem2 is complete.Remark 3.
Most of the existing results about the exponential stability of impulsive neural networks cannot effectively control the convergence rate. It is interesting to see that Theorem2 can guarantee the globally exponential stability of impulsive neural networks with the given convergence rate.Remark 4.
In particular, ifA(t)≡A,B(t)≡B in Theorem 2, where A, B are constant matrices, then condition αI-A+BL<0 can be easily checked via Matlab.Theorem 5.
Given constantα>0. The equilibrium point of the system (1) is globally exponentially stable with the given convergence rate α, if the conditions (H0)-(H1) are fulfilled; moreover, suppose that (i)
there exists a constantλ>0, such that (α+λ)I-A(t)+B(t)L<0, for all t>t0,(ii)
τ≜mink∈ℤ+{tk+1-tk}, maxi∈Λ(1+σik)≤Mkeλτ, where 1≤Mk<∞ and ∏k=1∞‍Mk<∞, i∈Λ, k∈ℤ+.Proof.
We only need to prove thaty(t)=z(t)eα(t-t0) is bounded when t≥t0, where z(t)=z(t,t0,z0) is a solution of (3) through (t0,z0).
Consider the Lyapunov function as follows:(16)V(t)=y(t)Ty(t)=∑i=1n‍yi2(t).
In particular, V(t0)=∑i=1n‍yi2(t0).
Then from conditions (H0)-(H1) and (i), we get the upper right-hand derivative of V(t) along the solutions of system (3), for t∈[tk,tk+1), k∈ℤ+(17)D+V(t)=2∑i=1n‍yi(t)y˙i(t)≤2y(t)T(αI-A(t)+B(t)L)y(t)<-2λV(t).
Thus,
(18)V(t)<V(tk)e-2λ(t-tk)<V(tk)e-2λτ,t∈[tk,tk+1),k∈ℤ+.
By condition (ii), it holds that
(19)V(tk)=∑i=1n‍yi2(tk)=∑i=1n‍e2α(t-t0)(1+σik)2zi2(tk-)≤Mk2e2λτ∑i=1n‍e2α(t-t0)zi2(tk-)≤Mk2e2λτV(tk-).
For any t∈[t0,t1), it holds that V(t)≤V(t0)e-2λτ, moreover
(20)V(t1)≤M12e2λτV(t1-)≤M12V(t0).
Similarly, for any t∈[t1,t2), it holds that V(t)≤V(t1)e-2λτ≤M12V(t0)e-2λτ, and
(21)V(t2)≤M2e2λτV(t2-)≤M22M12V(t0).
Without loss of generality, when t∈[tk,tk+1), k∈ℤ+, it can be deduced that
(22)V(t)≤∏j=1k‍Mj2e-2λτV(t0).
Hence, we obtain that for any t≥t0,
(23)V(t)≤∏j=1∞‍Mj2V(t0),
which implies that
(24)∥y(t)∥≤M∥y(t0)∥,t≥t0,
where M=(∏j=1∞‍Mj)<∞.
The proof of Theorem5 is complete.Remark 6.
Although Theorem5 enhances the restriction on condition (i), the impulsive restriction in (ii) is weaker; that is, σik is not necessary to converge to 0 as k is large enough, provided that the impulsive intervals are not too small.Theorem 7.
Given constantα>0. The equilibrium point of the system (1) is globally exponentially stable with the given exponential convergence rate α, if the conditions (H0)-(H1) are fulfilled; moreover, suppose that (i)
there exists a constantλ>0, such that (α-λ)I-A(t)+B(t)L<0,  for all t>t0,(ii)
τ≜maxk∈ℤ+{tk+1-tk}  and βkeλτ≤1, where βk=maxi∈Λ(1+σik), -1≤σik≤0, i∈Λ, k∈ℤ+.Proof.
We only need to prove thaty(t)=z(t)eα(t-t0) is bounded when t≥t0, where z(t)=z(t,t0,z0) is a solution of (3) through (t0,z0).
Consider the Lyapunov functional as follows:(25)V(t)=y(t)Ty(t)=∑i=1n‍yi2(t).
Particularly, V(t0)=∑i=1n‍yi2(t0).
Then from conditions (H0)-(H1) and (i), we get the upper right-hand derivative of  V(t) along the solutions of system (1), for t∈[tk,tk+1), k∈ℤ+(26)D+V(t)=2∑i=1n‍yi(t)y˙i(t)=2y(t)T(αI-A(t)+B(t)L)y(t)<2λV(t).
Thus,
(27)V(t)<V(tk)e2λ(t-tk)<V(tk)e2λτ,t∈[tk,tk+1),k∈ℤ+.
By condition (ii), it holds that
(28)V(tk)=∑i=1n‍yi2(tk)≤maxi∈Λ(1+σik)2V(tk-)=βk2V(tk-),k∈ℤ+.
By simple induction, we can prove that for any t∈[tk,tk+1), k∈ℤ+,
(29)V(t)≤∏j=1k‍βj2e2(k+1)λτV(t0)≤e2λτV(t0),
which implies that
(30)∥y(t)∥≤M∥y(t0)∥,t≥t0,
where M=eλτ.
The proof of Theorem7 is complete.


## 4. Applications

The following illustrative example will demonstrate the effectiveness of our results.Example 8.
Consider the following impulsive neural networks:(31)x˙1(t)=-a1(t)x1(t)+b11(t)f1(x1(t))+b12(t)f2(x2(t))+I1(t),x˙2(t)=-a2(t)x2(t)+b21(t)f1(x1(t))+b22(t)f2(x2(t))+I2(t),x1(tk)=(1+σ1k)x1(tk-)-σ1k,k∈ℤ+,x2(tk)=(1+σ2k)x2(tk-)-σ2k,k∈ℤ+,
where f1(u)=f2(u)=(|u+1|-|u-1|)/2.
It is easy to see thatfj, j=1,2 satisfying hypothesis (H1) with li=l2=1. We have a1(t)=3, a2(t)=2-(3/2)cos(2t), b11(t)=|sint|/2, b12(t)=1/2, b21(t)=2, b22(t)=-1-(cos(2t)/2), I1=(5/2)-(|sint|/2), I2=1-cos(2t), σ1k=1+(1/5k2)-1, σ2k=1+(1/6k2)-1.
Letα=3. It can be deduced that αI-A(t)+B(t)L≤0  and ∏k=1∞‍maxi=1,2|1+σik|<∞. Hence, all the conditions of Theorem 2 are satisfied; then the equilibrium point x⋆=(1,1) of the above system (31) is globally exponentially stable with the given convergence rate α=3.



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