Impulsive control of a class of multiple unstable neural networks

This paper addresses the issue of stability of a class of multiple unstable Cohen–Grossberg neural networks(CGNNs) under impulsive control. Some novel sufficient conditions are given to make the unstable equilibrium points of the model locally μ-stable. An example is offered to demonstrate the effectiveness of the control strategy by comprehensive computer simulations.


Introduction
Recently, the multi-stability of neural network models has attracted extensive attention because of its wide application in the pattern recognition. Many experts and scholars contributed to this topic (see ). For example, Cao et al. proved that the CGNNs with multi-stability and multi-periodicity could find 2 n locally exponentially stable equilibrium points in [1]. The paper [26] revealed the co-existence of unstable and stable equilibrium points of a class of n-neuron recurrent neural networks model with time-varying delays. In [31], Nie et al. investigated a class of n-neuron competitive neural networks and showed that the systems have exactly 5 n equilibrium points, and 5 n -3 n among them are unstable. Based on the partition space method, [32] proved that a class of CGNNs with unbounded time-varying delays could have 3 n equilibrium points, of which 3 n -2 n are unstable and the remaining ones are locally μ-stable. By the above-mentioned references, we can see that most literature focused on the properties of multiple stable equilibrium points of the system. Still, few papers considered the properties of those unstable equilibrium points. Hence it is a challenging problem.
It is common knowledge that impulsive control is a very effective and economical method to address the unstable or chaotic neural networks, and its main idea is to add a pulse into the network topology to control the state of the system. In the past few years, many significant results on impulsive control neural network have been proposed, see [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]. In [41], the authors studied the delay-dependent passivity analysis of impulsive neural networks by using functional and inequality method and compared the system model with impulsive control and without impulsive control, extended the recent results of passivity. [45] introduced new sandwich control systems with impulse time windows and illustrated the stability of the chaotic system by using impulsive. Li et al. in [50] added impulse inputs in unstable neural networks to keep the unstable equilibrium point or the chaotic system stable. Hence it may be a good idea to investigate the stability of unstable equilibrium points of multiple systems by way of impulsive control. Motivated by the above discussions, we investigated the stability of multiple unstable CGNNs in [32] by introducing a pulse into the system and obtained some sufficient conditions to make unstable equilibrium points of the models locally μ-stable, which generalized the results of paper [50]. The arrangement of this article is as follows. In the second section, the Cohen-Grossberg model and some preliminary conclusions are given. The main results are given and proved in the third section. The corollaries and comparisons with the existing literature are given in the fourth section. Section 5 gives a numerical example with simulation to illustrate the effectiveness of the control strategy. At the end of this paper, the conclusion is made.

System description and preliminaries
This article focuses on a class of n-neuron multiple unstable neural networks under some conditions described by the following equations: where x i (t) represents the current state of the ith neuron; a i (x i (t)) denotes the amplification function of the ith neuron; and b i (x i (t)) is the inhibition behavior function of the ith neuron; g j (x j (t)) and f j (x j (tτ (t))) are current activation functions of the jth neuron, and τ (t) is a nonnegative function and denotes the delay of transmission; c ij is the connection weight of the ith neuron and jth neuron, and d ij denotes their delayed feedback connection weight; I i is a constant and denotes the external input of the ith neuron.
In addition, we also denote by C = (c ij ) n×n and D = (d ij ) n×n the connection weight matrices. Other hypotheses and notations of this article are consistent with the literature [32], no more explanation.
By Theorems 1-3 of paper [32], we know that model (1) has 3 n equilibrium points, 3 n -2 n among them are unstable, and others are locally μ-stable. Here, we present only the results in [32] as lemmas directly without proof.
then there exists at least an equilibrium point of (1) in n i=1 w i .

Lemma 3 ([32], Theorem 3)
For any n i=1 w i ∈ 2 , given that (2) holds. If there exist some positive constants ξ 1 , . . . , ξ n such that where λ max To discuss the stability under impulsive control of unstable equilibrium points of model (1), the following two lemmas are useful.

Lemma 4 ([11])
Let Q be a positive definite matrix. Then, for any y, z ∈ R n , 2y T z ≤ y T Q -1 y + z T Qz.
is equivalent to one of the following conditions:

Impulsive control strategy and main results
For the unstable equilibrium points of model (1), we consider designing an impulsive control strategy to make the unstable equilibrium points stable in each subregion of 2 . For any subregion n i=1 w i ∈ 2 , assume that x is one unstable equilibrium point in n i=1 w i of model (1). Then we introduce the following impulsive control on account of x at discrete instances: where (1) and (8) into the matrix equation shown below: where , and lim t→∞ τ (t) = +∞.

Definition 1
Let h(t) be a solution to model (9). Then model (9) is said to be locally μstable, if one can find a constant M > 0 satisfying that 2 , and μ(t) is a continuously differentiable and nondecreasing function on [0, +∞).
Remark 1 The definition of local μ-stability here includes some famous stabilities such as local asymptotic stability, local Lipschitz stability, and so on. Besides, we design an impulsive control strategy {t i , ϒ i } i∈Z + (8) to stabilize the unstable equilibrium points of system (1).

Theorem 2 Let x be one unstable equilibrium point in
then model (9) is locally μ-stable. Furthermore, x under impulsive control strategy {t i , ϒ i } i∈Z + (8) is locally μ-stable, and so model (1) can increase 3 n -2 n locally μ-stable equilibrium points.
Proof We use a similar method as that in [50] to prove the theorem. Let φ = 0. Then we just have to prove the following inequality: where Firstly, if s P (t) ≤ γ M is not true when k = 1, then there is ν 2 ∈ (0, t 1 ) so that h(ν 2 ) ∈ n i=1 w i , and then By (18) and (19), it can be seen that there must exist ν 1 ∈ [0, ν 2 ) and h(ν 1 ) ∈ n i=1 w i in 2 so that For any t ∈ [ν 1 , ν 2 ], by (19) and (20), it follows that Meanwhile, by Theorem 1, we can get which is a contradiction. Hence s P (t) ≤ γ M, and k = 1 holds.
Remark 2 Theorem 2 shows that the impulse control can make the unstable regions stable and also increases the stable equilibrium points of model (1).

Corollaries and comparisons
On the basis of lemmas and theorems above, the following conclusions are drawn and compared with those in the existing literature.
Remark 3 The net self-inhibition function b i (x i (t)) in model (30) is monotone increasing and odd, which contains the case of Ref. [50]. Hence model (30) is more general.
Remark 4 Ref. [50] studied the stability of unstable systems with unbounded time-varying delays at some certain discrete time for HNN model (30) and derived some control results to stabilize neural networks with an unstable equilibrium point by the impulsive control. However, we studied in the present paper the stability of multiple unstable equilibrium points.

Numerical example
Example Consider the following two-dimensional CGNNs model: Let μ * (t) = 1 + 0.2t if t ≥ 0 and μ * (t) = 1 if t < 0. Then we know that the hypothesis of Eq. (36) and μ * (t) satisfy condition (6) by calculation. Therefore, by Lemmas 1-3, there are nine equilibrium points in model (36), four of which are μ-stable, and others are unstable. Running program [x, fval] = fsolve( myfun7 , x0) with Matlab software for model (36) in each subregion, one can obtain the nine equilibrium points of (36) as follows: Trace the solutions of model (36) with 150 initial conditions, the dynamics of x 1 (t) and x 2 (t) are depicted in the above three graphs of Fig. 1, which show that there are four locally μ-stable equilibrium points, which is in accord with our results.
It is easy to get that the stable equilibrium points of model (36) are more than before adding impulse, and just right one equilibrium point exists in each region of model (36). Specifically, Figs. 2-6 show that the other five equilibrium points are unstable, while they are locally μ-stable after adding impulse, which verifies the effectiveness of the control strategy and the correctness of the obtained results.  Remark 5 The activation functions in the example of Ref. [50] without time delay and with time delay are the same, but they are different in the present paper. Therefore, the simulation of this paper is closer to the results of the theory.

Conclusion
Impulsive control of multiple unstable CGNNs with unbounded time-varying delays is studied in this article. Ref. [32] proved that there exist multiple equilibrium points, and some of them are unstable in model (1). For those unstable equilibrium points, we introduce an impulsive control strategy into the unstable region to ensure that system (1) is μ-stable in each local region of 2 . In Sect. 4, we conclude some results of other models and point out the advantages of model (28). Meanwhile, we summarize that model (1) is μ-stable in each local region of R n under impulsive control, including the asymptotically stable, log-stable, and log-log-stable. In addition, we also show the effectiveness of impulsive control strategy by one example and its comprehensive numerical simulations. From the results of this article, we see that it is an effective method to study the stability of multiple unstable CGNNs by introducing impulse inputs. Therefore, we can investigate the stability of other multiple unstable system by employing the impulsive control strategy further.