# Global Asymptotic Stability of Impulsive CNNs with Proportional Delays and Partially Lipschitz Activation Functions

Authors: Xueli Song; Jigen Peng
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper researches global asymptotic stability of impulsive cellular neural networks with proportional delays and partially Lipschitz activation functions. Firstly, by means of the transformationvi(t)=ui(et), the impulsive cellular neural networks with proportional delays are transformed into impulsive cellular neural networks with the variable coefficients and constant delays. Secondly, we provide novel criteria for the uniqueness and exponential stability of the equilibrium point of the latter by relative nonlinear measure and prove that the exponential stability of equilibrium point of the latter implies the asymptotic stability of one of the former. We furthermore obtain a sufficient condition to the uniqueness and global asymptotic stability of the equilibrium point of the former. Our method does not require conventional assumptions on global Lipschitz continuity, boundedness, and monotonicity of activation functions. Our results are generalizations and improvements of some existing ones. Finally, an example and its simulations are provided to illustrate the correctness of our analysis.


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## Body

## 1. Introduction

Cellular neural networks (CNNs) introduced by Chua and Yang [1, 2] have found many important applications in biology, the solving of optimization problem, image processing, and pattern recognition [3]. In fact, CNNs can be characterized by an array of identical nonlinear dynamical systems (called cells) locally interconnected in the paper [4] which presented a set of sufficient conditions ensuring the existence of at least one stable equilibrium point in terms of the template elements. As we know, time delays are inevitable in electronic implementation of CNNs [5]. However, time delays may destroy stability of the networks and even lead to the oscillation behaviors. Hence, it is necessary to study the stability of CNNs with different types of delays. Time delays may be proportional delays; that is to say, the delay function τ(t)=(1-q)t is a monotonically increasing function with respect to t>0, where q is a constant and satisfies 0<q<1. The type of proportional delays is usually required in Web quality of service routing decision and one may be convenient to control the network’s running time according to the network allowed delays. Moreover, one can refer to the paper [6] about more information on the proportional delay engineering. Proportional delays [7–10] are unbounded time-varying ones different from constant delays [11], bounded time-varying delays [12–18], and unbounded distributed delay [19–23]. It is relatively difficult to deal with this class of the unbounded time-varying delays because none of any other assumptions are imposed on it compared with other unbounded time-varying delays, such as, unbounded distributed delays often require that the delay kernel functions kij:R+→R+ satisfy ∫0∞‍kij(s)ds=1,∫0∞‍skij(s)ds<∞, or there exists a positive number μ such that ∫0∞‍kij(s)eμsds<∞ [20–23]. Several stability criteria of CNNs with proportional delays have been obtained [7]. Moreover, the abrupt changes in the voltages produced by faulty circuit elements are exemplary of impulse phenomena which can affect the transient behavior of the network [24]. Hence, it is significant to discuss the stability of the CNNs with impulses and proportional delays. However, to the best of the authors’ knowledge, few authors have handled the stability of CNNs with impulses and proportional delays.Among the existing research results about neural networks, some activation functions are assumed to be globally Lipschitz continuous [25–30], bounded and monotonic [31], and bounded [24, 32]. However, these assumptions make these existing results unapplicable to some important engineering problems. For example, when the neural networks are used to solve optimization problems with the presence of constraints (linear, quadratic, or more general programming problems), unbounded (or nonmonotonic, non-globally Lipspchitz continuous) activations modeled by diode-like exponential-type functions are needed such that constraints are satisfied [33]. Motivated by this, we attempt to abandon these assumptions and only require activation functions to be partially Lipschitz continuous. Moreover, the relative nonlinear measure is more efficient than the nonlinear measure for exponential stability analysis of different classes of neural networks without delays where the equilibrium points are given [20, 34].According to the foregoing analysis, this paper is devoted to analyzing stability of impulsive CNNs with proportional delays and Lipschitz continuous activation functions by relative nonlinear measure. The remainder of this paper is arranged as follows. Section2 describes the model of proportion-delayed impulsive CNNs with partial Lipschitz continuous activation functions and provides its equivalent form by some transformation. Being preliminaries, Section 3 is devoted to uniqueness and exponential stability of equilibrium point of a nonlinear impulsive functional differential equation with variable coefficients and constant delays by means of relative nonlinear measure. In Section 4, a sufficient condition is obtained for global asymptotic stability of equilibrium point of impulsive proportion-delayed CNNs with partially Lipschitz continuous activation functions by results derived in Section 3. Furthermore, an example and its simulations are presented to illustrate that our method is valid and that our derived results are new and correct. Conclusions are given in Section 5.


## 2. Model Description and Its Equivalent Form

We consider the following CNNs with impulses and multiproportional delays:(1)u˙i(t)=-diui(t)+∑j=1n‍[aijfj(uj(t))+bijgj(uj(pjt))+cijhj(uj(qjt))]+Ii,t≥1,t≠tk,Δui(tk)=Ii,k(ui(tk)),k∈N,ui(s)=ϕi(s),r≤s≤1,
for i=1,2,…,n, where n≥2 is the number of cells in the networks; ui(t) denotes the potential of the ith cell at time t; di>0 represents the rate with which the ith cell resets its potential to the resting state when isolated from other cells and inputs at time t; aij,bij, and cij denote the strengths of connectivity between the jth and the ith cells at time t, pjt, and qjt, respectively; pj and qj are proportional delay factors and satisfy 0<pj,  qj<1,  r=min⁡1≤j≤n{pj,qj} and pjt=t-(1-pj)t,  qjt=t-(1-qj)t, in which (1-pj)t,(1-qj)t correspond to the time delays required in processing and transmitting a signal from the jth cell to the ith cell, and (1-pj)t→+∞,  (1-qj)t→+∞ as t→+∞; Δui(tk)=ui(tk+)-ui(tk-) is the impulse at moments tk and 1=t1<t2<⋯ is a strictly increasing sequences such that lim⁡k→+∞tk=+∞; fj, gj, and hj are the nonlinear activation functions; Ii>0 denotes the ith component of an external input source introduced from outside the network to the ith cell at time t.To discuss stability of the networks (1), we only assume the following.(H)
Activation functionsfj, gj, and hj are partially Lipschitz continuous on R for j=1,2,…,n.In what follows, we plan to transform model (1) into a model what we can directly deal with. Motivated by this paper [7], we define the transformation by
(2)vi(t)=ui(et),i=1,2,…,n.(I) Whenet≥1 and et≠tk, then t≥0,  t≠ln⁡tk and v˙i(t)=u˙i(et)et; that is,
(3)u˙(et)=v˙(t)e-t.
Taking h=et and then h≥1, then the transformation (2) is written as
(4)u˙i(h)=h-1v˙i(t).
From (1) and (4), we derive
(5)v˙i(t)h-1=-diui(h)+∑j=1n‍[(uj(qjh))aijfj(uj(h))+bijgj(uj(pjh))+cijhj(uj(qjh))]+Ii;
that is,
(6)u˙i(et)=-diui(et)+∑j=1n‍[aijfj(uj(et))+bijgj(uj(pjet))+cijhj(uj(qjet))]+Ii.
From transformation (2), we obtain
(7)uj(pjet)=uj(et+ln⁡pj)=vj(t+ln⁡pj)=vj(t-τj),uj(qjet)=uj(et+ln⁡qj)=vj(t+ln⁡qj)=vj(t-ςj),
where τj=-ln⁡pj, ςj=-ln⁡qj.By (2), (6), and (7), we enjoy
(8)v˙i(t)=et{∑j=1n‍-divi(t)+∑j=1n‍[aijfj(vj(t))+bijgj(vj(t-τj))+cijhj(vj(t-ςj))]+Ii∑j=1n‍}.(II) Whenet≥1 and et=tk, then t≥0 and t=ln⁡tk. By transformation (2), we have
(9)Δvi(t)=vi(t+)-vi(t)=ui(et+)-ui(et)=ui(tk+)-ui(tk)=Ii,k(ui(tk))=Ii,k(vi(t)).(III) Whenet∈[r,1], from (1) we have
(10)ui(et)=ϕi(et),t∈[-τ,0],
where τ=max⁡1≤j≤n{τj,ςj}. Hence, the initial functions associated with (8) are given by
(11)vi(s)=ψi(s)=ϕi(es),-τ≤s≤0,i=1,2,…,n.Conversely, letτj=-ln⁡pj, ςj=-ln⁡qj in (8); by transformation (2), then (8) can be written as (1) for t≥1 and t≠tk, Δ(ui(t))=ui(t+)-ui(t)=vi(ln⁡t+)-vi(ln⁡t) for t≥1 and t=tk, and for t∈[r,1], from (10) and (11), the initial function associated with (1) is given by ui(s)=ϕi(s), s∈[r,1].In conclusion, in the sense of solutions, the CNNs with impulses and multiproportional delays (1) is equivalent to the following CNNs with constant delays and variable coefficients
(12)v˙i(t)=et{∑j=1n‍-divi(t)+∑j=1n‍[aijfj(vj(t))+bijgj(vj(t-τj))+cijhj(vj(t-ςj))]+Ii∑j=1n‍},t≠ln⁡tk,Δvi(t)=Ii,k(vi(t)),t=ln⁡tk,k∈N,vi(s)=ψi(s),s∈[-τ,0],
for t≥0, where τ=max⁡1≤j≤n{τj,ςj}, τj=-ln⁡pj>0, ςj=-ln⁡qj>0, ψi∈C([-τ,0],R) denoting the space of all continuous functions from [-τ,0] to R for i=1,2,…,n and ψ=(ψ1,ψ2,…,ψn)T.


## 3. Preliminaries

Letn-dimensional real vector space Rn be endowed with 1-norm ∥·∥1 defined by
(13)∥x∥1=∑i=1n‍|xi|foreveryx=(x1,x2,…,xn)T∈Rn,
where the superscript T denotes the transpose. Let 〈·,·〉 denote the inner product in Rn and sign⁡(x)=(sign⁡(x1),sign⁡(x2),…,sign⁡(xn))T the sign vector of x∈Rn, where sign⁡(r) represents the sign function of r∈R. Obviously, the relations
(14)∥x∥1=〈x,sign⁡(x)〉,∥x∥1≥〈x,sign⁡(y)〉
hold for all x,y∈Rn.In order to discuss the stability of the neural networks (1), we firstly consider exponential stability of the following differential equation with variable coefficients, delays, and impulses
(15)dz(t)dt=et[F(z(t))+G(zt(s))],t≥0,t≠tk,Δz(tk)=z(tk+)-z(tk-)=Ik(z(tk)),k∈N,z0=ϕ∈C([-τ,0],Ω),
where τ>0,  C([-τ,0],Ω) denotes the space of all continuous functions from [-τ,0] into the open subset Ω of Rn; zt∈C([-τ,0],Ω) is defined by zt(s)=z(t+s) for all s∈[-τ,0] and ∥zt∥C=sup⁡-τ≤s≤0∥z(t+s)∥1; F and G:Ω→Rn are nonlinear operators; 0=t0<t1<t2<⋯ is a strictly increasing sequence such that lim⁡k→+∞tk=+∞; zt(s) is defined as follows:
(16)zt(s)=(z1(t-s),z2(t-s),…,zn(t-s))T.
The nonlinear operators F and G are defined, respectively, by
(17)F(u)=(F1(u),F2(u),…,Fn(u))T,G(u)=(G1(u),G2(u),…,Gn(u))T.Definition 1 (see [20]).
(
1
) A nonlinear operator T:Ω→Rn is called be Lipschitz continuous on Ω if there exists a nonnegative constant M such that
(18)∥T(x)-T(y)∥1≤M∥x-y∥1∀x,y∈Ω,
where M is called the Lipschitz constant of T on Ω. The constant
(19)LΩ(T)=sup⁡x,y∈Ω,y≠x∥T(y)-T(x)∥1∥y-x∥1
is called the minimal Lipschitz constant (MLC) of T on Ω. Furthermore, the operator T is called globally Lipschitz continuous if Ω=Rn.
(
2
) A nonlinear operator T:Ω→Rn is said to be partially Lipschitz continuous on Ω if, for any x∈Ω, there exists a constant Lx>0 such that
(20)∥T(y)-T(x)∥1≤Lx∥y-x∥1,∀y∈Ω.
The constant
(21)LΩp(T,x)=sup⁡x,y∈Ω,y≠x∥T(y)-T(x)∥1∥y-x∥1
is called minimal partial Lipschitz constant (MPLC) of T on Ω with respect to x. Furthermore, the operator T is called partially Lipschitz continuous if Ω=Rn.From the paper [20] we conclude that every Lipschitz continuous operator on Ω is partially Lipschitz continuous on Ω and LΩp(T,x)≤LΩ(T) for any Lipschitz continuous operator T and x∈Ω.Definition 2 (see [34]).
Assume thatΩ is an open subset of Rn, F is a nonlinear operator from Ω into Rn, and x0∈Ω is any vector. The constant
(22)mΩ(F,x0)=sup⁡x∈Ω,x≠x0〈F(x)-F(x0),sign⁡(x-x0)〉∥x-x0∥1
is called relative nonlinear measure of F at x0.Definition 3.
z
* is said to be an equilibrium point of (15) if (F+G)z*=0 and Ik(z*)=0 for all k∈N.Definition 4.
Letz* be an equilibrium point of (15) and Ω an open neighborhood of z*. z* is exponentially stable on Ω if there exist two positive constants σ and M such that
(23)∥z(t)-z*∥1≤Me-σtmax⁡-τ≤s≤0∥ϕ(s)-z*∥1
holds for t≥0, where z(t) is the unique solution of (15) initiated from the function ϕ∈C([-τ,0],Ω).
Particularly, ifΩ=Rn holds, then z* is the unique equilibrium point and (15) is said to be globally exponentially stable.Lemma 5 (see [35]).
Ifa>c≥0, for every nonnegative real number b, the equation
(24)0=λ-a+ceλb
has a unique positive solution.Lemma 6 (see [36]).
Letv(t)>0 for t∈R and t0∈R. Suppose that
(25)v′(t)≤-av(t)+b[sup⁡-∞<s≤tv(s)]fort≥t0.
If a>b>0, there exist constants γ>0 and k>0 such that
(26)v(t)≤ke-γ(t-t0)sup⁡-∞<s≤t0v(s)
holds for t≥t0.Theorem 7.
LetΩ be an open neighborhood of the equilibrium point z* of (15). Equation (15) has no other equilibrium point in Ω different from z* if mΩ(F+G,z*)<0.Proof.
Assume thatz~∈Ω is any equilibrium point of (15) different from z*; that is,
(27)F(z*)+G(z*)=F(z~)+G(z~)=0,Ik(z*)=Ik(z~)=0.
Then, we derive
(28)mΩ(F+G,z*)=sup⁡x∈Ω,x≠z*((∥x-z*∥1)-1(〈F(x)+G(x)-(F(z*)-G(z*)),sign⁡(x-z*)(F(z*)-G(z*))〉)×(∥x-z*∥1)-1)≥〈F(z~)+G(z~)-(F(z*)+G(z*)),sign⁡(z~-z*)〉∥z~-z*∥1=0,
which contradicts mΩ(F+G,z*)<0.Theorem 8.
LetΩ be a neighborhood of the equilibrium z* of (15). Assume F and G to be partially Lipschitz continuous on Ω with respect to z* and
(29)Ik(z(tk))=-γk(z(tk)-z*),0≤γk≤2,k∈N.
If there exists some diagonal matrix A=diag⁡(a1,a2,…,an) with ai>0 such that the inequality
(30)mA-1(Ω)(FA,z*)+LA-1(Ω)p(GA,z*)<0
holds, then z* is exponentially stable on Ω. Particularly, the solution z(t) of (15) initiated from ϕ∈C([-τ,0],Ω) decays by
(31)∥z(t)-z*∥1≤e-σt·sup⁡-τ≤s≤0∥ϕ(s)-z*∥1∀t≥0,
where σ is the unique positive solution of the equation
(32)0=σ·min⁡1≤i≤nai+mA-1(Ω)(FA,z*)+LA-1(Ω)p(GA,z*)·eσ.Proof.
Letx(t)=z(t)-z* for all t≥0. From the relations (14) we derive that
(33)∥x(t)∥1-∥x(t-s)∥1s≤1s〈x(t)-x(t-s),sign⁡(x(t))〉
holds for all s>0. Consequently, the function t↦∥x(t)∥1 is absolutely continuous in (0,+∞), which implies that derivatives of ∥x(t)∥1 exist almost everywhere in (0,+∞). Furthermore, from (15) we conclude that derivatives of ∥x(t)∥1 satisfy
(34)d∥x(t)∥1dt≤〈dx(t)dt,sign⁡(x(t))〉=〈et[F(z(t))+G(zt(s))],sign⁡(x(t))〉=〈[et(F(z(t))+G(zt(s)))-(F(z*)+G(z*))et(zt(s))],sign⁡(x(t))〉=et[〈F(z(t))-F(z*),sign⁡(x(t))〉+〈G(zt(s))-G(z*),sign⁡(x(t))〉]≤et[∥G(zt(s))-G(z*)∥1〈F(z(t))-F(z*),sign⁡(A-1x(t))〉+∥G(zt(s))-G(z*)∥1〈F(z(t))-F(z*),sign⁡(A-1x(t))〉]≤et[LA-1(Ω)p〈F(z(t))-F(z*),sign⁡(A-1x(t))〉+LA-1(Ω)p(GA,z*)∥A-1zt(s)-A-1z*∥1]≤et[sup⁡-τ≤s≤0∥A-1(z(t+s)-y(t+s))∥1mA-1(Ω)(FA,z*)∥A-1x(t)∥1+LA-1(Ω)p(GA,z*)×sup⁡-τ≤s≤0∥A-1(z(t+s)-y(t+s))∥1]≤et[sup⁡-τ≤s≤t∥x(s)∥1mA-1(Ω)(FA,z*)∥x(t)∥1+LA-1(Ω)p(GA,z*)×sup⁡-τ≤s≤t∥x(s)∥1](min⁡1≤i≤nai)-1≤etk[sup⁡-τ≤s≤t∥x(s)∥1mA-1(Ω)(FA,z*)∥x(t)∥1+LA-1(Ω)p(GA,z*)×sup⁡-τ≤s≤t∥x(s)∥1](min⁡1≤i≤nai)-1,t∈(tk-1,tk).
The combination of condition (30) and Lemmas 6 and 5 implies that
(35)∥x(t)∥1≤e-σtktsup⁡-τ≤s≤0∥x(s)∥1
holds for all t∈(tk-1,tk), where σtk is the unique positive solution of the equation
(36)0=e-tk·σ·min⁡1≤i≤nai+mA-1(Ω)(FA,z*)+LA-1(Ω)p(GA,z*)·eσ.
It needs to point out that the positive solution σtk of (36) is strictly monotonically increasing with respect to tk. In fact, let σtk+1 be the positive solution of the equation
(37)0=e-tk+1·σ·min⁡1≤i≤nai+mA-1(Ω)(FA,z*)+LA-1(Ω)p(GA,z*)·eσ.
By subtracting (37) from (36), we derive
(38)(e-tkσtk-e-tk+1σtk+1)min⁡1≤i≤nai+LA-1(Ω)p(GA,z*)(eσtk-eσtk+1)=0.
Furthermore, we have
(39)eσtk+1-eσtk=min⁡1≤i≤naiLA-1(Ω)p(GA,z*)(e-tkσtk-e-tk+1σtk+1).
Since both of min⁡1≤i≤nai and LA-1(Ω)(GA,z*) are positive, we have
(40)(eσtk+1-eσtk)(e-tkσtk-e-tk+1σtk+1)>0.
It is obvious that σtk is not equal to σtk+1, that is, σtk+1<σtk or σtk+1>σtk. If σtk+1<σtk, then eσtk+1<eσtk and e-tkσtk-e-tk+1σtk+1<0 from inequality (40), that is,
(41)σtk+1σtk>e(tk+1-tk).
Since tk+1>tk, σtk+1/σtk>e(tk+1-tk)>1 contradict the assumption σtk+1<σtk. This means that σtk+1>σtk holds for tk+1>tk; that is, the positive solution σtk of (36) is strictly monotonically increasing with respect to tk. Hence, σ=σt0<σt1<σt2<⋯, where σt0 is the unique positive solution of (37) at t0=0; that is,
(42)0=σ·min⁡1≤i≤nai+mA-1(Ω)(FA,z*)+LA-1(Ω)p(GA,z*)·eσ.
Since e-σtkt<e-σt0t=e-σt for all t∈(tk-1,tk), k∈N, inequality (35) means that
(43)∥x(t)∥1≤e-σtsup⁡-τ≤s≤0∥x(s)∥1
holds for all t∈(tk-1,tk), k∈N, where σ is the unique positive solution of (32). Inequality (43) is accordingly changed into the following form:
(44)∥z(t)-z*∥1≤e-σtsup⁡-τ≤s≤0∥ϕ(s)-z*∥1
which holds for all t∈(tk-1,tk),  k∈N. According to condition (29), we enjoy
(45)x(tk+)=z(tk+)-z*=z(tk)+Ik(z(tk))-z*=(1-γk)x(tk),k∈N.
This implies
(46)∥x(tk+)∥1=∥(1-γk)x(tk)∥1≤∥x(tk)∥1,k∈N.
From (35) and (46) we derive
(47)∥z(tk+)-z*∥1=∥x(tk+)∥1≤∥x(tk)∥1=∥x(tk-)∥1≤e-σtsup⁡-τ≤s≤0∥x(s)∥1=e-σtsup⁡-τ≤s≤0∥ϕ(s)-z*∥1,k∈N.
In conclusion, we obtain inequality (31).Remark 9.
Our proof idea mainly comes from Theorem 2 of [20] investigating the exponential stability of the special case of (15) (i.e., (15) with constant coefficients). However, they are essentially different because Theorem 8 in this paper has to deal with time-varying coefficients. Consequently, Theorem 8 in this paper is a generalization of Theorem 2 in [20]. Moreover, it needs to point out that the exponential stability criterion (30) and exponential decay index σ in (31) are independent of time t although the abstract equation (15) enjoys time-varying coefficients, which means that our method is essential to qualitatively and quantitatively characterize exponential stability of (15). Moreover, Theorem 8 is not only generalization and improvement of Theorem 1 in [35] because there indeed exists a nonlinear Lipschitz continuous map T on Ω such that LΩp(T,x) is strictly less than LΩ(T) for any x∈Ω and (15) enjoys time-varying coefficients.
It is obvious that the CNNs model (12) can be changed into the form of (15). By Theorem 8, we can obtain the exponential stable criterion of equilibrium point of the CNNs model (12). Since the model (1) is equivalent to the model (12) in the sense of solution, models (12) and (1) enjoy the same equilibrium point v*=u*, where v*=(v1*,v2*,…,vn*)T and u*=(u1*,u2*,…,un*)T are the equilibrium point of models (12) and (1), respectively. What qualitative property of the model (1) can be derived from the global exponential stability of the model (12)? The next theorem can answer this problem.Theorem 10.
Suppose that the equilibrium pointu* of the model (12) is globally exponentially stable, that is, that exist two positive constants M and σ such that
(48)∥v(t)-u*∥1≤Meσtmax⁡-τ≤s≤0∥ψ(s)-u*∥1
holds for t≥0, where v(t) is the unique solution of the model (12) initiated from ψ∈C([-τ,0],Rn). Then u* of the model (1) is globally asymptotic stable. Particularly, the inequality
(49)∥u(t)-u*∥1≤Mt-σmax⁡r≤s≤1∥ϕ(s)-u*∥1
holds for t≥1, where u(t) is the unique solution of the model (1) initiated from ϕ∈C([r,1],Rn), r=min⁡1≤j≤n{pj,qj}.Proof.
By the transformation (2) and the inequality (48), we derive
(50)∥u(et)-u*∥1=∥v(t)-u*∥1≤Me-σtmax⁡-τ≤s≤0∥ψ(s)-u*∥1=Me-σtmax⁡-τ≤s≤0∥ϕ(es)-u*∥1,et≥1.
Let et=h, then h≥1 and t=ln⁡h≥0. Let es=ξ, then ξ∈[r,1]. The inequality (50) implies
(51)∥u(h)-u*∥≤Me-σln⁡hmax⁡r≤ξ≤1∥ϕ(ξ)-u*∥1=Mh-σmax⁡r≤ξ≤1∥ϕ(ξ)-u*∥1,h≥1.
Taking t=h, we furthermore derive
(52)∥u(t)-u*∥≤Mt-σmax⁡r≤ξ≤1∥ϕ(ξ)-u*∥1,t≥1.
This implies that the equilibrium point u* of the model (1) is globally asymptotic stable.Remark 11.
It need point out that the paper [7] has obtained not exponential stability, but asymptotic stable criteria of CNNs with multi-proportional delays because it mistakes asymptotic stability as exponential stability, which can be easily seen from the Remark 3.2 in [7] and Theorem 10 in this paper.


## 4. Uniqueness and Global Asymptotic Stability of Equilibrium Point of Model (1)

In this subsection, we firstly prove that model (1) has a unique equilibrium point in Rn. It is enough to prove that model (12) has a unique equilibrium point in Rn because models (12) and (1) enjoy the same equilibrium point. For this, we define that F=(F1,F2,…,Fn)T and G=(G1,G2,…,Gn)T:Rn→Rn are defined, respectively, by
(53)Fi(v)=-divi+∑j=1n‍aijfj(vj),Gi(v)=∑j=1n‍[bijgj(vj)+cijhj(vj)]+Ii.Theorem 12.
Suppose that the assumption(H) holds and u* is an equilibrium point of the model (1). For each set of external inputs, Ii, model (1) has no other equilibrium point in Rn different from u* if there exist positive real numbers ai (i=1,2,…,n) such that
(54)max⁡1≤j≤n1dj∑i=1n[LRjp(fj,uj*)ajai|aij|+LRjp(gj,uj*)ajai|bij|‍+LRjp(hj,uj*)ajai|cij|]<1
holds, where LRjp(fj,uj*), LRjp(gj,uj*), and LRjp(hj,uj*) denote MPLC of fj, gj, and hj on Rj with respect to uj*, respectively.Proof.
Obviously, it is enough to prove that model (12) has no other equilibrium point in Rn different from u* if the inequality (54) holds. Define A=diag⁡(a1,a2,…,an) and we need only prove mA-1(Rn)(A-1(F+G)A,u*)<0 according to Theorem 7. In detail, for v∈A-1(Rn), we enjoy
(55)〈A-1(F+G)(Av)-A-1(F+G)(Au*),sign⁡(v-u*)〉=∑i=1n‍sign⁡(vi-ui*)×{∑j=1n‍-ai-1[di(aivi)-di(aiui*)]+∑j=1n‍[ai-1aij(fj(ajvj)-fj(ajuj*))+ai-1bij(gj(ajvj)-gj(ajuj*))+ai-1cij(hj(ajvj)-hj(ajuj*))]∑j=1n‍}≤∑i=1n‍ai-1{∑j=1n‍-|di(aivi)-di(aiui*)|+∑j=1n‍[|aij||fj(ajvj)-fj(ajuj*)|+|bij||gj(ajvj)-gj(ajuj*)|+|cij||hj(ajvj)-hj(ajuj*)|]∑j=1n‍}≤-∑i=1n‍di|vi-ui*|+∑j=1n‍∑i=1n‍ajai[|aij|LRjp(fj,uj*)+|bij|LRjp(gj,uj*)+|cij|LRjp(hj,uj*)]|vj-uj*|=-∑j=1n‍{dj-∑i=1n‍ajai[|aij|LRjp(fj,uj*)+|bij|LRjp(gj,uj*)+|cij|LRjp(hj,uj*)]∑i=1n‍}|vj-uj*|.
The combination of (55) and (54) implies that mA-1(Rn)(A-1(F+G)A,u*)<0, which implies that model (12) enjoys no other equilibrium point in Rn different from u*. That is to say, u* is the unique equilibrium point in Rn of model (1).Secondly, we prove that condition (54) also guarantees global asymptotic stability of equilibrium point of model (1) by Theorems 8 and 10.Theorem 13.
Assume that assumption(H) holds, u* is the equilibrium point of the model (1), and Ii,k(ui(tk))=-γi,k(ui(tk)-ui*), 0≤γi,k≤2, for k∈N and i=1,2,…,n. If there exist a set of positive real numbers ai (i=1,2,…,n) such that condition (54) holds, then for each set of external input, Ii, model (1) is globally asymptotic stable. Particularly, if u(t) is the solution of the model (1) initiated from ϕ∈C([r,1],Rn), then the inequality
(56)∥u(t)-u*∥1≤t-σ·max⁡1≤i≤naimin⁡1≤i≤nai·sup⁡r≤s≤1∥ϕ(s)-u*∥1,
holds for t≥1, where σ is the unique positive solution of the equation
(57)σ·min⁡1≤j≤ncj-1-1+keσ=0
with
(58)cj=dj-LRjp(fj,uj*)∑i=1n‍ajai|aij|,k=max⁡1≤j≤n{cj-1∑i=1n‍ajai[|bij|LRjp(gj,uj*)+|cij|LRjp(hj,uj*)]}.Proof.
Obviously,u* is also the equilibrium point of model (12) because model (1) is equivalent to the model (12) in the sense of solution. Firstly, we prove that model (12) is globally exponentially stable by Theorem 8. For this, let A=diag⁡(a1,a2,…,an) and C=diag⁡(c1-1,c2-1,…,cn-1). It immediately follows from the condition (54) that
(59)cj=dj-LRjp(fj,uj*)∑i=1n‍ajai|aij|>0,forj=1,2,…,n.
For all v∈C-1A-1(Rn),
(60)〈A-1F(ACv)-A-1F(ACu*),sign⁡(v-u*)〉≤∑i=1n‍ai-1{∑j=1n‍-|di(aici-1vi)-di(aici-1ui*)|+∑j=1n‍|aij||fj(ajcj-1vj)-fj(ajcj-1uj*)|}≤∑i=1n‍ai-1{∑j=1n‍-aici-1di|vi-ui*|+∑j=1n‍|aij|LRjp(fj,uj*)ajcj-1|vj-uj*|}=-∑j=1n‍cj-1(dj-LRjp(fj,uj*)∑i=1n‍ajai|aij|)|vj-uj*|=-∥v-u*∥1,
which implies that mC-1A-1(Rn)(A-1FAC,u*)≤-1. For all v∈C-1A-1(Rn), we have
(61)∥A-1GACv-A-1GACu*∥1=∑i=1n‍|ai-1∑j=1n‍{bij[gj(ajcj-1vj)-gj(ajcj-1uj*)]+cij[hj(ajcj-1vj)-hj(ajcj-1uj*)]}∑j=1n‍|≤∑j=1n‍{cj-1∑i=1n‍ajai[|bij|LRjp(gj,uj*)+|cij|LRjp(hj,uj*)]∑i=1n‍}|vj-uj*|;
thus,
(62)LC-1A-1(Rn)p(A-1GAC,u*)≤max⁡1≤j≤n{cj-1∑i=1n‍ajai[|bij|LRjp(gj,uj*)+|cij|LRjp(hj,uj*)]}=k.
Consequently, from (54) we conclude that
(63)mC-1A-1(Rn)(A-1FAC,u*)+LC-1A-1(Rn)p(A-1GAC,u*)≤-1+max⁡1≤j≤n{cj-1∑i=1n‍ajai[|bij|LRjp(gj,uj*)+|cij|LRjp(hj,uj*)]∑i=1n‍}=max⁡1≤j≤n{(dj-∑i=1n‍ajai|aij|LRjp(fj,uj*))-1(-dj+∑i=1n‍ajai|aij|LRjp(fj,uj*)+∑i=1n‍ajai[|bij|LRjp(gj,uj*)+|cij|LRjp(hj,uj*)])×(dj-∑i=1n‍ajai|aij|LRjp(fj,uj*))-1}<0.
By Theorem 8, the solution y(t) of the functional differential equation
(64)dy(t)dt=et[A-1FAC(y(t))+A-1GAC(yt(s))],hhhhhhhhhhhhhhhhhhhhht≥0,t≠tk,Δy(tk)=y(tk+)-y(tk-)=Ik(y(tk)),k∈N,y0=ϕ~∈C([-τ,0],C-1A-1(Rn)),
satisfies
(65)∥y(t)-C-1A-1u*∥≤e-σt·sup⁡-τ≤s≤0∥ϕ~(s)-C-1A-1u*∥1hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀t≥0,
where σ is the unique positive solution of (57). It is obvious that y(t)=C-1A-1v(t) is the solution of (64) if v(t) is a solution of model (12). Consequently, the equilibrium point u* of model (12) is globally exponentially stable; that is,
(66)∥v(t)-u*∥1≤eσt·max⁡1≤i≤naimin⁡1≤i≤nai·max⁡-τ≤s≤0∥ψ(s)-u*∥1
holds for t≥0, where σ is unique positive solution of (57), τ=max⁡1≤j≤n{τj,ςj}, τj=-ln⁡pj>0, ςj=-ln⁡qj>0, and ψi∈C([-τ,0],R). By Theorem 10, we derive that the solution u(t) of model (1) initiated from ϕ∈C([r,1],Rn) satisfies inequality (56) for t≥1. That is to say, the solution of model (1) is globally asymptotically stable.Remark 14.
Theorems12 and 13 provide a sufficient condition (54) to the uniqueness and global asymptotic stability of the equilibrium point of impulsive CNNs (1) with multiproportional delays and partially Lipschitz continuous activation functions. On the one hand, the proportional delay is time varying, unbounded, and monotonic and the model (1) does not require the proportional delays to meet any other condition. Hence, compared with the results in papers [11–23] with the constant, bounded time varying, or unbounded distributed delays, our results are new. Moreover, the stability of CNNs with general unbounded time varying delays in [37, 38] and proportional delays in [7–10] has been investigated. Compared with these results, our results are their generalizations because model (1) has impulsive perturbations. On the other hand, model (1) only requires activation functions to be partially Lipschitz continuous. In fact, partial Lipschitz continuity is less conservative; that is, it does not meet conventional assumptions, such as, boundedness, global Lipschitz continuity, or monotonicity. Compared with these excellent results on neural networks with globally Lipschitz continuous [25–30], bounded and monotonic [31], or bounded [24, 32] activation functions, our results are new. Furthermore, our results are also generalizations of the papers [24, 29, 35] and even improvement of the paper [35] with globally Lipschitz continuous activation functions and there indeed exists a nonlinear globally Lipschitz continuous map T on Ω such that LΩp(T,x) is strictly less than LΩ(T) for any x∈Ω.


## 5. Illustrative Example

In this section, we present an illustrative example to verify effectiveness of our method.Example 1.
Consider impulsive CNNs with proportional delays and partially Lipschitz continuous activation functions(67)u˙(t)=-Du(t)+Af(u(t))+Bg(u(pt))+Ch(u(qt))+I,t≥1,t≠tk,Δu(tk)=Ik(u(tk)),t=tk,k∈N,u(s)=ϕ(s),r≤s≤1,
where
(68)D=(5000800010),A=(-0.80.20.5-0.7-1.30.61.5-0.42.1),B=(-0.30.5-0.91.4-0.81.20.2-0.1-0.5),C=(-10.61.30.4-1.6-0.7-1.7-0.30.6),I=(000),Ik=(-γ1k-γ2k-γ3k),p=(0.70.50.3),q=(0.20.90.5),fj(x)=gj(x)=hj(x)=f(x)=0.5x(sin2x+cos⁡x) for x∈R and j=1,2,3, Δu1(tk)=-γ1ku1(tk), Δu2(tk)=-γ2ku2(tk), and Δu3(tk)=-γ3ku3(tk), t1<t2<⋯, is a strictly increasing sequence such that lim⁡k→+∞tk=+∞ and  γ1k=1+0.9sin2(2k2+k),γ2k=1.8+0.2sin(1+k2), and γ3k=1.4+0.5cos⁡2(2+3k2), r=min⁡1≤j≤3{pj,qj}=0.2. ϕ1(s)=5+s2,ϕ2(s)=-2+s3 and ϕ3(s)=-4+s for r≤s≤1.
From the definition off(x) we can conclude
(69)|f(y)-f(x)|=|y(sin2y+cos⁡y)-x(sin2x+cos⁡x)|≤0.5(2+3|x|)|y-x|,
holds for x,y∈R, which means that f is partially Lipschitz continuous on R. It is easily verified that u*=(0,0,0)T is the equilibrium point of the model (67) and LRp(f,0)=0.625; that is, LRjp(fj,0)=LRjp(gj,0)=LRjp(hj,0)=0.625 for j=1,2,3. Consequently, the criteria of the papers [24, 29] are not applied to this the model (67) because they require activation functions to be globally Lipschitz continuous. Moreover, none of the stability criteria in [11–23] is applied to model (67) since the delay factors (1-pj)t and (1-qj)t are unbounded and monotonic and not required to meet any other additional conditions. Although the papers [7–10] have investigated the stability of CNNs with proportional delays, these methods are not able to apply to model (67) because model (67) enjoys the impulsive perturbations.
However, takinga1=3, a2=6 and a3=5, we obtain
(70)max⁡1≤j≤31dj∑i=13‍{LRjp(fj,0)ajai|aij|+LRjp(gj,0)ajai|bij|+LRjp(hj,0)ajai|cij|}=max⁡{0.6738,0.5672,0.6115}=0.6738<1,
that is, the condition (54) holds for a1=3, a2=6, and a3=5. Moreover, 0≤γik≤2 for i=1,2,3. According to Theorems 10 and 12, we conclude that the equilibrium point u*=(0,0,0)T of the model (67) is globally asymptotically stable and the solution satisfies the following inequality (Figure 1):
(71)∥u(t)∥1=|u1(t)|+|u2(t)|+|u3(t)|≤2t-σsup⁡0.2≤s≤1⁡∥ϕ(s)∥1,t≥1,
where u(t)=(u1(t),u2(t),u3(t))T is the solution of (67) initiated from ϕ∈C([0.2,1],R3) and σ is the unique positive solution of the equation 0.1273σ-1+0.5613eσ=0.Figure 1
The simulation for the solution to the neural networks (67) at the impulsive moments 2,4,6,….Remark 2.
It needs to be pointed out that the impulsive instants are only selected as2,4,6,… in the simulation of this example to simplify the simulation, which is obviously not enough to illustrate the impulsive effect. In order to accurately characterize wider of impulses, the papers [39, 40] proposed the concepts of average dwell time and average impulsive interval. Moreover, the papers [41, 42] presented single impulsive controller and pinning impulsive stabilization criterion, respectively. Consequently, their methods are recommended to simulations with more general impulses.


## 6. Conclusions and Further Work

By means of relative nonlinear measure and transformation, this paper has discussed global asymptotic stability of impulsive cellular neural networks with proportional delays and partially Lipschitz activation functions. We have obtained the novel criterion of uniqueness and global asymptotic stability of the equilibrium point of this CNNs model. Our method does not require conventional assumptions on global Lipschitz continuity, boundedness, and monotonicity of activation functions and proportional delays to meet other requirements, which demonstrates that our criteria derived are less restrictive than some existing ones and that they are generalizations and improvements of some existing ones. Finally, the example with three cells has illustrated that our method is effective and that our results are correction.Our method only requires the activation functions of the cellular networks to be partially Lipschitz continuous. The relative weak assumption makes our results applicable to more general engineering problems. In the future, we attempt to design a cellular neural networks model to solve optimization problems with some constraints, where unbounded (or nonmonotonic, nonglobally Lipspchitz continuous) activations are required such that these constraints are satisfied.



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*Source: 832892-2014-07-23.xml*