Asymptotic Stability and Exponential Stability of Impulsive Delayed Hopfield Neural Networks

and Applied Analysis 3 Definition 4 (see [11]). Assume that y(t) = y(σ, φ)(t) is the solution of (4) through (σ, φ). Then the zero solution of (4) is said to be (1) uniformly stable, if for any σ ≥ t 0 and ε > 0, there exists a δ = δ(ε) > 0 such that φ ∈ PC δ (σ) implies that ‖y(σ, φ)(t)‖ < ε, t ≥ σ; (2) uniformly asymptotically stable, if it is uniformly stable, and there exists a δ > 0 such that for any ε > 0, σ ≥ t 0 , there is a T = T(ε) > 0 such that φ ∈ PC δ (σ) implies that ‖y(σ, φ)(t)‖ < ε, t ≥ σ + T; (3) globally exponentially stable, if for any φ ∈ PC([−τ, 0],R), there exist constants M,μ > 0 such that 󵄩 󵄩 󵄩 󵄩 y (t) 󵄩 󵄩 󵄩 󵄩 < M 󵄩 󵄩 󵄩 󵄩 φ 󵄩 󵄩 󵄩 󵄩τ e −μ(t−σ) , t ≥ σ. (6) In this paper, we always assume that the following assumption holds: (H 0 ) there exist constants M,N > 0 such that Ω T (y)Ω (y) ≤ My T y, Γ T (y τ ) Γ (y τ ) ≤ Ny T τ y τ . (7) In addition, we have the following basic lemmas. Lemma 5 (see [24]). For any vectors a, b ∈ R, the inequality ±2a T b ≤ a T Xa + b T X −1 b (8) holds, in which X is any n × n matrix with X > 0. Lemma 6 (see [25]). Assume that there exist constants P,Q > 0 and m(t) ∈ PC([t 0 − τ,∞),R + ) such that (i) for t = t k , m(t k ) ≤ γ k m(t − k ), γ k > 0 are constants and satisfymax k∈Z+ {1/γ k , 1} < P/Q; (ii) for t ≥ t 0 , t ̸ = t k , D + m(t) ≤ −Pm (t) + Q?̃? (t) , (9) where ?̃?(t) = sup t−τ≤s≤t m(s). Then for t ≥ t 0 , m(t) ≤ ?̃? (t 0 )( ∏ t0<tk≤t γ k )e −λ(t−t0) , (10) where λ satisfies the following inequality: 0 < λ ≤ P − Qmax k∈Z+ { 1 γ k , 1} ⋅ e λτ . (11) 3. Main Results In this section, we will establish some theorems which provide sufficient conditions for uniformly asymptotically stable and global exponential stability of system (1). Theorem 7. The equilibrium point the system (1) is uniformly asymptotically stable, if there exists n × n symmetric, and positive definite matrix P satisfies the following conditions: (H 1 ) η ≐ ∏∞ k=1 max{η k , 1} < ∞, where η k is the largest eigenvalue of P−1D k PD k ; (H 2 ) λ 3 < (−M−N)/λ 1 , where λ 1 is the smallest eigenvalue of P and λ 3 is the largest eigenvalue of P−1(CP + PC + PAA T P + PBB T P). Proof. First, we will prove that the zero solution of system (4) is uniformly stable. For any ε > 0, we may choose a δ > 0 such that δ ≤ √(λ 1 /ηλ 2 )ε, where λ 2 is the largest eigenvalue of P. For any σ ≥ t 0 , φ ∈ PC δ (σ), let y(t) = y(σ, φ)(t) be a solution of (4) through (σ, φ), σ ≥ t 0 (for convenience, that we assume σ = t 0 ); then we can prove that ‖y(t)‖ < ε, t ≥ t 0 . Consider the following Lyapunov function: V(t, y(t)) = y T (t)Py(t); then we have λ 1 󵄩 󵄩 󵄩 󵄩 y (t) 󵄩 󵄩 󵄩 󵄩 2 ≤ V (t, y (t)) ≤ λ 2 󵄩 󵄩 󵄩 󵄩 y (t) 󵄩 󵄩 󵄩 󵄩 2 . (12) By virtue of Lemma 5, we obtain for t ∈ [t k , t k+1 ), k = 1, 2, . . ., D + V (t, y (t)) 󵄨 󵄨 󵄨 󵄨(4) = (y T (t)) 󸀠 Py (t) + y T (t) Py 󸀠 (t) = (Cy (t) + AΩ (y (t)) + BΓ (y τ )) T Py (t) + y T (t) P (Cy (t) + AΩ (y (t)) + BΓ (y τ )) = y T (t) CPy (t) + Ω T (y (t)) A T Py (t) + Γ T (y τ ) B T Py (t) + y T (t) PCy (t) + y T (t) PAΩ (y (t)) + y T (t) PBΓ (y τ ) = y T (t) (CP + PC) y (t) + 2Ω T (y (t)) A T Py (t) + 2Γ T (y τ ) B T Py (t) ≤ y T (t) (CP + PC) y (t) + Ω T (y (t))Ω (y (t)) + y T (t) PAA T Py (t) + Γ T (y τ ) Γ (y τ ) + y T (t) PBB T Py (t) ≤ y T (t) (CP + PC + PAA T P + PBB T P) y (t) + Ω T (y (t))Ω (y (t)) + Γ T (y τ ) Γ (y τ ) ≤ λ 3 y T (t) Py (t) + My T (t) y (t) + Ny T τ y τ ≤ λ 3 y T (t) Py (t) + Mλ −1 1 y T (t) Py (t) + Nλ −1 1 y T τ Py τ ≤ [λ 3 + Mλ −1 1 ] y T (t) Py (t) + Nλ −1 1 y T τ Py τ . (13) First, it is obvious that for t 0 − τ ≤ t ≤ t 0 , λ 1 󵄩 󵄩 󵄩 󵄩 y(t) 󵄩 󵄩 󵄩 󵄩 2 ≤ V (t, y (t)) ≤ λ 2 δ 2 ≤ η −1 λ 1 ε 2 . (14) Then we can prove that for t ∈ [t 0 , t 1 ), V (t, y (t)) ≤ η −1 λ 1 ε 2 . (15) 4 Abstract and Applied Analysis Suppose that this is not true; then there exists ?̂? ∈ [t 0 , t 1 ) such that V( ?̂?, y( ?̂? ) ) > η −1λ 1 ε . Set ̌ t = sup {t | s ∈ [t 0 , t) , V (s, y (s)) ≤ η −1 λ 1 ε 2 } . (16) It is obvious that ̌ t < ?̂?. Then it follows that (I a ) V(t, y(t)) ≤ η −1λ 1 ε 2 , t ∈ [t 0 , ̌ t); (II a ) V( ̌ t, y( ̌ t)) = η −1λ 1 ε ; (III a ) for any δ > 0, there exists t δ ∈ ( ̌ t, ̌ t + δ) such that V(t δ , y(t δ )) > η −1 λ 1 ε . So V( ̌ t, y ( ̌ t)) = η −1 λ 1 ε 2 ≥ V (t, y τ ) , ̌ t − τ ≤ t ≤ ̌ t. (17) In view of condition (H 2 ), from (13), we obtain D + V( ̌ t, y ( ̌ t)) ≤ [λ 3 + Mλ −1 1 ] y T ( ̌ t) Py ( ̌ t) + Nλ −1 1 y T τ Py τ ≤ [λ 3 + Mλ −1 1 + Nλ −1 1 ]V ( ̌ t, y ( ̌ t)) < 0, (18) which is a contradiction with (III a ). Hence, (15) holds. Considering V (t 1 , y (t 1 )) = y T (t 1 ) Py (t 1 ) = y T (t − 1 )D 1 PD 1 y (t − 1 ) ≤ η 1 y T (t − 1 ) Py (t − 1 ) = η 1 V (t − 1 , y (t − 1 ))


Introduction
In the last several years, Hopfield neural networks (HNN) have received especially considerable attention due to their extensive applications in solving optimization problem, traveling salesman problem, and many other subjects in recent years [1][2][3][4][5][6][7][8][9].In hardware implementation of neural networks, time delays are inevitably present due to the finite switching speeds of the amplifiers.Hence, it is vital to investigate the stability of delayed HNN.Recently, various results for the stability of delayed HNN are obtained via different approaches.In [3], Rakkiyappan and Balasubramaniam studied the exponential stability for fuzzy impulsive neural networks by utilizing the Lyapunov-Krasovskii functional and the linear matrix inequality approach.In [8], Li studied the global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type based on the similar methods.In [9], Xia et al. derived some sufficient conditions for the synchronization problem of coupled identical Yang-Yang type fuzzy cellular neural networks with timevarying delays based on using the invariance principle of functional differential equations.
On the other hand, impulsive differential equations have attracted a great deal of attention due to its potential applications in biological systems, chemical reactions, and various results are obtained; for instance, see [10][11][12][13][14]. Impulses can make unstable systems stable, and stable systems can become unstable after impulse effects.Hence, the stability properties of impulsive HNN with time delays have become an important topic of theoretical studies and have been investigated by many researchers; see [5,6,[15][16][17][18][19][20][21][22].In [5], Zhang and Sun obtained a result for the uniform stability of the equilibrium point of the impulsive HNN systems with time delays by using Lyapunov functions and analysis technique.In [6], global exponential stability of impulsive delay HNN is investigated by applying the piecewise continuous vector Lyapunov function.
The purpose of this paper is to present some sufficient conditions for uniform asymptotic stability and global exponential stability of impulsive HNN with time delays by means of constructing the extended impulsive Halanay inequality which is different from that given in [23], Lyapunov functional methods, and linear matrix inequality approach.
where Λ = {1, 2, . . ., }. ≥ 2 corresponds to the number of units in a neural network; the impulse times   satisfy 0 ≤  0 <  1 < ⋅ ⋅ ⋅ <   < ⋅ ⋅ ⋅ , lim  → +∞   = +∞;   corresponds to the membrane potential of the unit  at time ;   is positive constant;   ,   denote, respectively, the measures of response or activation to its incoming potentials of the unit  at time  and  −   (); constant   denotes the synaptic connection weight of the unit  on the unit  at time ; constant   denotes the synaptic connection weight of the unit  on the unit  at time  −   ();   is the input of the unit ;   () is the transmission delay of the th neuron such that 0 <   () ≤ ,  ≥  0 , and  ∈ Λ.
Assume that system (1) is supplemented with initial conditions of the form where In this paper, we assume that some conditions are satisfied, so that the equilibrium point of (1) without impulse does exist denoted by  * = ( * 1 ,  * 2 , . . .,  *  )  ; see [2,5].Impulsive operator is viewed as perturbation of the equilibrium point of system (1) without impulsive effects.We assume that ∈ R, and  ∈ Λ,  = 1, 2, . ... Since  * is an equilibrium point of (1), one can derive from (1) that the transformation   =   −  *  ,  ∈ Λ, transforms system (1) into the following system: where Clearly,  * is uniformly asymptotically stable for system (1) if and only if the trivial solution of system (3) is uniformly asymptotically stable.Hence, we only need to prove the stability of the trivial solution of system (3).
Then for  ≥  0 , where  satisfies the following inequality:

Main Results
In this section, we will establish some theorems which provide sufficient conditions for uniformly asymptotically stable and global exponential stability of system (1).
Theorem 7. The equilibrium point the system (1) is uniformly asymptotically stable, if there exists  ×  symmetric, and positive definite matrix  satisfies the following conditions: Proof.First, we will prove that the zero solution of system ( 4) is uniformly stable.For any  > 0, we may choose a  > 0 such that  ≤ √( First, it is obvious that for  0 −  ≤  ≤  0 , Then we can prove that for  ∈ [ 0 ,  1 ), Suppose that this is not true; then there exists t ∈ [ 0 ,  1 ) such that ( t, ( t It is obvious that ť < t.Then it follows that (  ) for any  > 0, there exists In view of condition ( 2 ), from (13), we obtain which is a contradiction with (  ).Hence, (15) holds.Considering we will prove that for Suppose that this is false; then we can define Similarly, we can obtain (  ) for any  > 0, there exists In fact, if ⃗ − ≥  1 , then it is obvious that inequality (22) holds.
By induction hypothesis, we may prove, in general, that for  ∈ [  ,  +1 ), that is, Finally, we arrive at Therefore, we obtain ‖()‖ < ,  ≥  0 .In view of the choice of , the zero solution of ( 4) is uniformly stable; that is, the equilibrium point of ( 1) is uniformly stable.
We also have two cases.
( 1 ) If τ1 is not impulsive point, that is to say τ1 >  + , then considering the definition of τ1 , we have By the same argument as the above mentioned, we obtain that (48) still holds.Hence, from (13), we get which implies that which is a contradiction with (29).So τ1 is some impulsive point.
( 1 ) If τ1 is some impulsive point, that is to see τ1 =  + , then from the definition of τ1 , it is clear that which implies that On the other hand, note that inequality (48) still holds; from ( 13) and (29), we have ≤  (τ 1 ,  (τ 1 )) which implies that That means which contradicts (29).Hence, the first situation  2 >  + is impossible.
( 2 ) If  2 =  + , then by the same arguments as in the proof in ( 1 ) and (43), we have Then let The rest of the arguments are omitted.Finally we can find our desirable contradiction.Hence, (36) holds.With above mentioned, the same arguments as before, if we replace  1 with  ⋆ , then there exists a  2 =  1 +  1 =  ⋆ + 2  Remark 9.For using the less conservative conditions in Theorem 7, our results obviously improve some results established in the earlier references.In [5], condition (  ) = ( −  ) holds for all  ∈ Z + ; here note in our Theorem 7 that we only require that the solutions satisfy the hypothesis ( 1 ) at impulsive points.In addition, our conditions are without requirement of the range of the largest eigenvalues of  −1    on (0, 1), which are milder than the restrictions in [5].
By utilizing Lemma 6, we will give some sufficient conditions for globally exponential stability of the equilibrium point of system (1).Theorem 10.Assume that there exists  ×  symmetric and positive definite matrix  such that  1 > 0 is the smallest eigenvalue of ,  3 is the largest eigenvalue of  −1 ( +  +    +   ),   is the largest eigenvalue of  −1     , and  1 ,  3 and   satisfy the following conditions: ( 4 ) there exist constants U(> 0), (≥ 0) such that  <  and the following inequality then we get (77) By a straightforward calculation, we obtain that the largest characteristic root  3 ≈ −3.885 < −2.By Theorem 7, the equilibrium point of system (73) is uniformly asymptotically stable with impulses (74) for any  > 0, which is shown in Figure 1(a).However, the criteria in [5] are invalid here.In fact, condition (  ) = ( −  ) is not satisfied here.Moreover, because of the impulsive effect, the criteria in [2] are also invalid here.Therefore, our results are less conservative than those given result in [2,5].

Conclusion
The uniform asymptotic stability and global exponential stability of impulsive HNN with time delays are considered in this paper.Some new stability conditions are obtained by means of constructing the extended impulsive Halanay inequality, Lyapunov functional methods, and linear matrix inequality approach.Moreover, our results can be applied to the case not covered in some other existing criteria.Hence, the results extend and improve the earlier publications.An example is given to illustrate the feasibility of the results and the effects of impulses.