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Impulsive control of a class of multiple unstable neural networks
Journal of Inequalities and Applications volume 2021, Article number: 39 (2021)
Abstract
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 μstable. An example is offered to demonstrate the effectiveness of the control strategy by comprehensive computer simulations.
Introduction
Recently, the multistability 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 [1–35]). For example, Cao et al. proved that the CGNNs with multistability and multiperiodicity could find \(2^{n}\) locally exponentially stable equilibrium points in [1]. The paper [26] revealed the coexistence of unstable and stable equilibrium points of a class of nneuron recurrent neural networks model with timevarying delays. In [31], Nie et al. investigated a class of nneuron 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 timevarying delays could have \(3^{n}\) equilibrium points, of which \(3^{n}2^{n}\) are unstable and the remaining ones are locally μstable. By the abovementioned 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 [36–52]. In [41], the authors studied the delaydependent 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 μstable, 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 nneuron 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 \(f_{j}(x_{j}(t\tau (t)))\) are current activation functions of the jth neuron, and \(\tau (t)\) 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.
Suppose that model (1) has the initial condition
where \(\varphi _{i}(s)\in C((\infty, 0],\mathbb{R}), i=1,2,\ldots,n\). Let \((x_{1}(t),x_{2}(t),\ldots,x_{n}(t))\) and \(x^{\star }=(x_{1}^{\star }, x_{2}^{\star },\ldots,x_{n}^{\star })\) stand for a solution and an equilibrium point of model (1), respectively. Then \(x^{\star }\) is said to be μstable if there exist a positive constant M and a nondecreasing function \(\mu (t)\) with \({\lim_{t \to +\infty } \mu (t)=+\infty }\) such that
Imitating [32], we can divide the \(R^{n}\) into \(3^{n}\) nonintersection subregions. Let Φ be a set of these subregions, and let \((\infty,+\infty )=(\infty,p_{i})\cup [p_{i},q_{i}]\cup (q_{i},+ \infty ),i=1,2,\ldots,n\). One can get
We define the index subsets for each \(\prod_{i=1}^{n} w_{i}\in \Phi \) as \(N_{1}=\{i\mid w_{i}=(\infty, p_{i}), i=1,2,\ldots,n\}\), \(N_{2}=\{i\mid w_{i}=[p_{i}, q_{i}],i=1,2,\ldots,n\}\), \(N_{3}=\{i\mid w_{i}=(q_{i}, +\infty ), i=1,2,\ldots,n\}\), and obviously \(N_{1}\cup N_{2}\cup N_{3}=\{1,2,\ldots,n\}\). Moreover, we also can separate the set Φ into two parts \(\Phi _{1}\) and \(\Phi _{2}\), where \(\Phi _{1}=\{\prod_{i=1}^{n} w_{i}\mid w_{i}=(\infty,p_{i})\text{ or }(q_{i},+\infty ), i=1,2,\ldots,n \}, \Phi _{2}=\Phi \Phi _{1}\). Obviously, there are \(2^{n}\) and \(3^{n}2^{n}\) elements in \(\Phi _{1}\) and \(\Phi _{2}\), respectively.
For convenience, let \(\hat{A}=\operatorname{diag}\{\bar{a}_{1},\bar{a}_{2},\ldots,\bar{a}_{n}\}\) and \(\check{A}=\operatorname{diag}\{ \underline{a}_{1}, \underline{a}_{2}, \ldots, \underline{a}_{n}\}\) be two positive diagonal matrices. Denote by \(a(x(t))=(a_{1}(x_{1}(t)),\ldots, a_{n}(x_{n}(t)))\) the amplification function of (1), where, for each i, \(a_{i}(u)\) is nonnegative continuous and satisfies
Denote by \(b(x(t))=(b_{1}(x_{1}(t)),\ldots, b_{n}(x_{n}(t)))\) the inhibition behavior function, where \(b_{i}(u)\) is an odd function that grows monotonically, and there exists a positive matrix \(B=\operatorname{diag}\{{b}_{1},{b}_{2},\ldots,{b}_{n}\}\) such that
Denote by \(g(x(t))=(g_{1}(x_{1}(t)),\ldots,g_{n}(x_{n}(t)))\) and \(f(x(t))=(f_{1}(x_{1}(t)),\ldots, f_{n}(x_{n}(t)))\) the activation functions, where \(g_{j}(\cdot )\) and \(f_{j}(\cdot )\) are continuous linear nondecreasing piecewise function or continuous nonlinear nondecreasing sigmoid function, and one can find some constants \(p_{j}\leq q_{j},m_{j}\leq M_{j},m_{j}^{\prime }\leq M_{j}^{\prime },m_{j}^{ \prime \prime }\leq M_{j}^{\prime \prime }\), so that
Let \(\Sigma ^{g}=\operatorname{diag}\{\bar{\sigma }_{1},\bar{\sigma }_{2},\ldots, \bar{\sigma }_{n}\}\) and \(\Delta ^{f}=\operatorname{diag}\{\bar{\delta }_{1},\bar{\delta }_{2},\ldots, \bar{\delta }_{n}\}\), where \(m_{j}=\min \{ m_{j}^{\prime }, m_{j}^{\prime \prime }\}\), \(M_{j}=\min \{M_{j}^{\prime }, M_{j}^{\prime \prime }\}\), \(\bar{\sigma }_{j}=\max \{\bar{\sigma }_{j}^{l}, \bar{\sigma }_{j}^{m}, \bar{\sigma }_{j}^{r} \}\), \(\bar{\delta }_{j}=\max \{{\bar{\delta }}_{j}^{l},{\bar{\delta }}_{j}^{m},{ \bar{\delta }}_{j}^{r} \}\), \(j=1,2,\ldots,n\). Obviously, both of \(\Sigma ^{g}\) and \(\Delta ^{f}\) are two positive matrices.
In addition, we also denote by \(C=(c_{ij})_{n\times n}\) and \(D=(d_{ij})_{n\times n}\) 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 μstable. Here, we present only the results in [32] as lemmas directly without proof.
Lemma 1
([32], Theorem 1)
For any \(\prod_{i=1}^{n} w_{i}\in \Phi \), if
then there exists at least an equilibrium point of (1) in \(\prod_{i=1}^{n} w_{i}\).
Lemma 2
([32], Theorem 2)
For any \(\prod_{i=1}^{n} w_{i}\in \Phi _{1}\), given that
and the nondecreasing function \(\mu (t)>0\) with
where \(\alpha \geq 0, \beta \geq 0\), and \(T^{\ast }\geq 0\). Then \(x^{\star }\) is μstable in \(\prod_{i=1}^{n} w_{i}\) (locally μstable in \(\Phi _{1}\)) if there exist some positive constants \(\zeta _{1},\zeta _{2},\ldots,\zeta _{n}\) such that
where \(i=1,2,\ldots,n\).
Lemma 3
([32], Theorem 3)
For any \(\prod_{i=1}^{n} w_{i}\in \Phi _{2}\), given that (2) holds. If there exist some positive constants \(\xi _{1},\ldots,\xi _{n}\) such that
where
then \(x^{\star }\) in \(\prod_{i=1}^{n} w_{i}\in \Phi _{2}\) is unstable.
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 \(y, z\in \mathbb{R}^{n}\), \(2y^{T}z\leq y^{T}Q^{1}y+z^{T}Qz\).
Lemma 5
([22])
The LMI $Q=\left(\begin{array}{cc}{Q}_{11}& {Q}_{12}\\ {Q}_{12}^{T}& {Q}_{22}\end{array}\right)<0$ with \(Q_{11}=Q_{11}^{T}\), \(Q_{22}=Q_{22}^{T}\) is equivalent to one of the following conditions:

(i)
\(Q_{22}<0\), \(Q_{11}Q_{12}Q_{22}^{1}Q_{12}^{T}<0\).

(ii)
\(Q_{11}<0\), \(Q_{22}Q_{12}^{T}Q_{11}^{1}Q_{12}<0\).
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 \(\Phi _{2}\). For any subregion \(\prod_{i=1}^{n} w_{i}\in \Phi _{2}\), assume that \(x^{\star }\) is one unstable equilibrium point in \(\prod_{i=1}^{n} w_{i}\) of model (1). Then we introduce the following impulsive control on account of \(x^{\star }\) at discrete instances:
where \(\Upsilon _{i}\in \mathbb{R}^{n\times n}\) is a control matrix based on the ith pulse. Let \(h(t)=x(t)x^{\star }\). We can transform (1) and (8) into the matrix equation shown below:
where \(\phi (t)=\varphi (t)x^{\star }\), \(A(h(t))=a(h(t)+x^{\star })\), \(B(h(t))=b(h(t)+x^{ \star })b(x^{\star })\), \(G(h(t))=g(h(t)+x^{\star })g(x^{\star })\), \(F(h(t))=f(h(t)+x^{ \star })f(x^{\star })\), and \(\lim_{t\rightarrow \infty }\tau (t)=+\infty \).
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
where \(\\phi \^{2}=\sup_{s\leq 0}\\phi (s)\^{2}\), and \(\mu (t)\) is a continuously differentiable and nondecreasing function on \([0,+\infty )\).
Remark 1
The definition of local μstability here includes some famous stabilities such as local asymptotic stability, local Lipschitz stability, and so on. Besides, we design an impulsive control strategy \(\{t_{i}, \Upsilon _{i}\}_{i\in \mathbb{Z}_{+}}\) (8) to stabilize the unstable equilibrium points of system (1).
Theorem 1
Suppose that there are two constants \(\mu _{1}\geq 1\), \(\mu _{2}>0\) such that
where \(t_{0}=0\), and \(\mu ^{\ast }(t)=\mu (t)\) if \(t\geq 0 \) and \(\mu ^{\ast }(t)=1\) if \(t<0\). Besides, one can find a matrix \(P>0\), two diagonal matrices \(Q_{1}>0\), \(Q_{2}>0\), and three constants \(\lambda _{1}\geq 0, \lambda _{2}\geq 0, \gamma >1\) such that
and
where \(\Pi =P\check{A}\beta \beta \check{A}P+\Sigma ^{g}Q_{1}\Sigma ^{g} \lambda _{1}P, \theta =\sup_{i\in \mathbb{Z}_{+}} \{ t_{i}t_{i1} \} \).
Consider the following positive definite Lyapunov function:
where \(h(t)\) is an arbitrary solution in \(\prod_{i=1}^{n} w_{i}\in \Phi _{2}\) of model (9) with the initial condition \(h(s)=\phi (s), s\leq 0 \). If there exist \(i\in \mathbb{Z}_{+}\) and some \(\nu _{1}<\nu _{2}\in [t_{i1},t_{i})\) such that \(\{h(t)h(t)=(h_{1}(t),h_{2}(t),\ldots,h_{n}(t)), t\in [\nu _{1}, \nu _{2}]\}\subseteq \prod_{i=1}^{n} w_{i}\in \Phi _{2}\), \(\{h(t\tau (t))h(t)=(h_{1}(t),h_{2}(t),\ldots,h_{n}(t)), t\in [ \nu _{1},\nu _{2}]\}\subseteq \prod_{i=1}^{n} w_{i}\in \Phi _{2}\), and
then
Proof
For \(\forall t\in [\nu _{1},\nu _{2}]\), by Lemma 3 and Lemma 4, the right upper Dini derivative of function \(\mathfrak{s}_{P}(t)\) can be inferred from (11) and (13):
Let us take the integral of (14) for t in the interval \([\nu _{1},\nu _{2}]\). By (10) and (12), it can be obtained that
Therefore, one can get \(\mathfrak{s}_{P}(\nu _{2})\leq \gamma \mathfrak{s}_{P}(\nu _{1})\) from (15). □
Theorem 2
Let \(x^{\star }\) be one unstable equilibrium point in \(\prod_{i=1}^{n} w_{i}\in \Phi _{2}\) of model (1). If (10), (11), and (12) hold and
then model (9) is locally μstable. Furthermore, \(x^{\star }\) under impulsive control strategy \(\{t_{i}, \Upsilon _{i}\}_{i\in \mathbb{Z}_{+}}\) (8) is locally μstable, and so model (1) can increase \(3^{n}2^{n}\) locally μstable equilibrium points.
Proof
We use a similar method as that in [50] to prove the theorem. Let \(\phi \neq 0\). Then we just have to prove the following inequality:
where \(M=\mu (0)\lambda _{\max }(P)\\phi \^{2}\), \(\{h(t)h(t)\in R^{n}, t\in [t_{i1},t_{i})\}\subset \prod_{i=1}^{n} w_{i}\) in \(\Phi _{2}\). Note that
Firstly, if \(\mathfrak{s}_{P}(t)\leq \gamma M\) is not true when \(k=1\), then there is \(\nu _{2}\in (0,t_{1})\) so that \(h(\nu _{2})\in \prod_{i=1}^{n} w_{i}\), and then
By (18) and (19), it can be seen that there must exist \(\nu _{1}\in [0,\nu _{2})\) and \(h(\nu _{1})\in \prod_{i=1}^{n} w_{i}\) in \(\Phi _{2}\) so that
For any \(t\in [\nu _{1},\nu _{2}]\), by (19) and (20), it follows that
Meanwhile, by Theorem 1, we can get
which is a contradiction. Hence \(\mathfrak{s}_{P}(t)\leq \gamma M\), and \(k=1\) holds.
Secondly, suppose that \(\mathfrak{s}_{P}(t)\leq \gamma M\) holds for any \(k\leq N,\forall N \in \mathbb{Z}_{+}\). However, if \(\mathfrak{s}_{P}(t)\leq \gamma M\) is not true when \(n=N+1\), then there exists \(\nu _{2}^{\ast }\in (t_{N},t_{N+1})\) such that \(h(\nu _{2}^{\ast })\in \prod_{i=1}^{n} w_{i}\), and then
With respect to (16), we can obtain that
By (22) and (23), there must exist \(\nu ^{\ast }_{1}\in [t_{N},\nu ^{\ast }_{2})\) so that \(h(\nu _{1}^{\ast })\in \prod_{i=1}^{n} w_{i}\), and then
Furthermore, for any \(t\in [\nu ^{\ast }_{1},\nu ^{\ast }_{2}]\), by (23) and (24), we have
However, we can obtain by Theorem 1
which leads to a contradiction. Thus \(\mathfrak{s}_{P}(t)\leq \gamma M\), and \(k=N+1\) holds.
Finally, by mathematical induction, we get
which implies that (26) satisfies Definition 1. Hence model (9) is μstable in \(\prod_{i=1}^{n} w_{i}\) of \(\Phi _{2}\). Consequently, \(x^{\star }\) under impulsive control (8) is locally μstable, and so model (1) can add \(3^{n}2^{n}\) locally μstable points. □
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.
When \(a_{i}(x_{i}(t))=1\), model (1) converts into the model HNN:
According to conditions (2)–(6), there are at least \(3^{n}\) equilibrium points in model (27), \(2^{n}\) of them in \(\Phi _{1}\) are locally μstable, and the remaining in \(\Phi _{2}\) are unstable.
Let \(h(t)=x(t)x^{\star }, t\geq 0\), where \(x^{\star }\) is an unstable equilibrium point in \(\prod_{i=1}^{n} w_{i}\in \Phi _{2}\) and \(x(t)\) is a solution of (27) with the initial condition \(x(s)=\varphi (s)\in \Phi _{2}, s\in (\infty,0]\). Then model (27) and the impulsive control (8) with respect to \(x^{\star }\) can be turned into
Corollary 1
Under conditions (10), (12), (16), and
where \(\Pi =P\beta \beta P+\Sigma ^{g}Q_{1}\Sigma ^{g}\lambda _{1}P\), model (28) is μstable in \(\prod_{i=1}^{n} w_{i}\). Furthermore, \(x^{\star }\) under impulsive control strategy \(\{t_{i}, \Upsilon _{i}\}_{i\in \mathbb{Z}_{+}}\) (8) is locally μstable, and so model (27) can increase \(3^{n}2^{n}\) locally μstable points.
When \(a_{i}(x_{i}(t))=1\) and \(b_{i}(x_{i}(t))=b_{i} \cdot x_{i}(t)\), model (1) changes into
If (30) meets (4) and the following conditions (31)–(33):
where \(\zeta _{1},\zeta _{2},\ldots,\zeta _{n}\), \(\xi _{1},\xi _{2},\ldots,\xi _{n}\) are positive constants, and
then one can obtain that there exist at least \(3^{n}\) equilibrium points in model (30), \(3^{n}2^{n}\) of them in \(\Phi _{2}\) are unstable, and the remaining \(2^{n}\) points in \(\Phi _{1}\) are locally μstable.
Let \(x^{\star }\) be an unstable equilibrium point in \(\prod_{i=1}^{n} w_{i}\in \Phi _{2}\) and \(x(t)\) be a solution of (30) with the initial condition \(x(s)=\varphi (s)\in \Phi _{2}, s\in (\infty,0]\), and let \(h(t)=x(t)x^{\star }, t\geq 0\). Then model (30) and the impulsive control (8) with respect to \(x^{\star }\) can transform into the following matrix form:
Corollary 2
Under conditions (10),(12),(16), and (29), where \(\Pi =PBBP+\Sigma ^{g}Q_{1}\Sigma ^{g}\lambda _{1}P\), model (35) is μstable in \(\prod_{i=1}^{n} w_{i}\). Furthermore, \(x^{\star }\) under impulsive control strategy \(\{t_{i}, \Upsilon _{i}\}_{i\in \mathbb{Z}_{+}}\) (8) is locally μstable, and so model (30) can increase \(3^{n}2^{n}\) locally μstable points.
Remark 3
The net selfinhibition 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 timevarying 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.
Corollary 3
When \(\mu (t)=1+\varsigma t, \varsigma >0\), and \(\tau (t)= kt, k \in (0,1) \), assume that (2), (3), (6) hold in \(\Phi _{1}\), and satisfy
Let \(\ln (1+\varsigma \theta )+\lambda _{1}\theta +\frac{\theta }{1\tau } \lambda _{2}\gamma <\ln \gamma \). If (11) and (16) hold for \(\Phi _{2}\), then model (1) under the impulsive control (8) is asymptotically stable in each local region of \(\Phi _{2}\).
Corollary 4
When \(\mu (t)=\ln (f+t), f > e\), and \(\tau (t)=t\ln {t}/t\), assume that (2), (3), (6) hold in \(\Phi _{1}\) and satisfy
Let \(\ln {\frac{\ln (f+\theta )}{\ln {f}}}+\lambda _{1}\theta +\ln (1+ \theta /f)\lambda _{2}\gamma <\ln \gamma \). If (11) and (16) hold for \(\Phi _{2}\), then model (1) under impulsive control (8) is logstable in each local region of \(\Phi _{2}\).
Corollary 5
When \(\mu (t)=\ln (f+t)\), \(\tau (t)=(f+t)(f+t)^{\epsilon }\), where \(f > e, \epsilon \in (0,1)\), assume that (2), (3), (6) hold in \(\Phi _{1}\) and satisfy
Let \(\ln {\frac{\ln (f+\theta )}{\ln {f}}}+\lambda _{1}\theta + \frac{\theta }{\epsilon }\lambda _{2}\gamma <\ln \gamma \). If (11) and (16) hold for \(\Phi _{2}\), then model (1) under impulsive control (8) is loglogstable in each region of \(\Phi _{2}\).
Numerical example
Example
Consider the following twodimensional CGNNs model:
where \(a(x)=1+0.2\sin (x)\), \(b_{1}(x_{1}(t))=x_{1}(t)\), \(b_{2}(x_{2}(t))=1.2x_{2}(t)\), \(f(x)= \frac{\vert x+1\vert \vert x1\vert }{2}\),
Let \(\mu ^{\ast }(t)=1+0.2t\) if \(t\geq 0 \) and \(\mu ^{\ast }(t)=1\) if \(t<0\). Then we know that the hypothesis of Eq. (36) and \(\mu ^{\ast }(t)\) satisfy condition (6) by calculation. Therefore, by Lemmas 1–3, there are nine equilibrium points in model (36), four of which are μstable, 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 μstable equilibrium points, which is in accord with our results.
With the functions and parameters given above, we can find that \(\check{A}=\operatorname{diag}\{0.8,0.8\}\), \(\hat{A}=\operatorname{diag}\{1.2,1.2\}\), \(\Sigma ^{g}=\operatorname{diag} \{1.564,1.564\}\), \(\Delta ^{f}=\operatorname{diag}\{1,1\}\), \(\lambda _{1}=18\). And we can obtain the following results by resorting to Matlab LMI control toolbox:
Let \(\lambda _{2}=11, \gamma =3.5, \theta =0.0186\), and \(t_{i}=0.018i\). Then we can deduce the following impulsive control matrix by (16):
Under the impulsive control matrix (37), the state trajectory curve of model (36) can be obtained with the same 150 initial solutions, which is the below three graphs of Fig. 1. 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 μstable 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 timevarying 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 μstable in each local region of \(\Phi _{2}\). 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 μstable in each local region of \(R^{n}\) under impulsive control, including the asymptotically stable, logstable, and loglogstable. 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.
Availability of data and materials
Not applicable.
References
 1.
Cao, J., Feng, G., Wang, Y.: Multistability and multiperiodicity of delayed Cohen–Grossberg neural networks with a general class of activation functions. Physica D 237(13), 1734–1749 (2008)
 2.
Cheng, C., Shi, C.: Complete stability in multistable delayed neural networks. Neural Comput. 21(3), 719–740 (2009)
 3.
Zhang, L., Yi, Z., Zhang, S., Heng, P.: Activity invariant sets and exponentially stable attractors of linear threshold discretetime recurrent neural networks. IEEE Trans. Autom. Control 54(6), 1341–1347 (2009)
 4.
Nie, X., Cao, J.: Multistability of competitive neural networks with timevarying and distributed delays. Nonlinear Anal., Real World Appl. 10, 928–942 (2009)
 5.
Huang, Z., Song, Q., Feng, C.: Multistability in networks with selfexcitation and highorder synaptic connectivity. IEEE Trans. Circuits Syst. I, Regul. Pap. 57(8), 2144–2155 (2010)
 6.
Zeng, Z., Huang, T., Zheng, W.: Multistability of recurrent neural networks with timevarying delays and the piecewise linear activation function. IEEE Trans. Neural Netw. 21(8), 1371–1377 (2010)
 7.
Huang, G., Cao, J.: Delaydependent multistability in recurrent neural networks. Neural Netw. 23(2), 201–209 (2010)
 8.
Wang, L., Lu, W., Chen, T.: Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions. Neural Netw. 23, 189–200 (2010)
 9.
Kaslik, E., Sivasundaram, S.: Impulsive hybrid discretetime Hopfield neural networks with delays and multistability analysis. Neural Netw. 24(4), 370–377 (2011)
 10.
Nie, X., Cao, J.: Multistability of secondorder competitive neural networks with nondecreasing saturated activation functions. IEEE Trans. Neural Netw. 22(11), 1694–1708 (2011)
 11.
Wang, L., Chen, T.: Complete stability of cellular neural networks with unbounded timevarying delays. Neural Netw. 36, 11–17 (2012)
 12.
Zeng, Z., Zheng, W.: Multistability of neural networks with timevarying delays and concaveconvex characteristics. IEEE Trans. Neural Netw. Learn. Syst. 23(2), 293–305 (2012)
 13.
Marco, M., Forti, M., Grazzini, M., Pancioni, L.: Limit set dichotomy and multistability for a class of cooperative neural networks with delays. IEEE Trans. Neural Netw. Learn. Syst. 23(9), 1473–1485 (2012)
 14.
Wang, L., Chen, T.: Multistability of neural networks with Mexicanhattype activation functions. IEEE Trans. Neural Netw. Learn. Syst. 23(11), 1816–1826 (2012)
 15.
Du, Y., Li, Y., Xu, R.: Multistability and multiperiodicity for a general class of delayed Cohen–Grossberg neural networks with discontinuous activation functions. Discrete Dyn. Nat. Soc. 917835, 1–11 (2013)
 16.
Wang, L., Chen, T.: Multiple μstability of neural networks with unbounded timevarying delays. Neural Netw. 53, 109–118 (2014)
 17.
Cheng, C., Huang, Z.: Nontypical multistability in neural networks with distributed delays. Neurocomputing 121, 207–217 (2013)
 18.
Zhou, B., Song, Q.: Boundedness and complete stability of complexvalued neural networks with time delay. IEEE Trans. Neural Netw. Learn. Syst. 24(8), 1227–1238 (2013)
 19.
Huang, Z., Raffoul, Y., Cheng, C.: Scalelimited activating sets and multiperiodicity for threshold networks on time scales. IEEE Trans. Cybern. 44(4), 488–499 (2014)
 20.
Nie, X., Zheng, W.: Multistability and instability of neural networks with discontinuous nonmonotonic piecewise linear activation functions. IEEE Trans. Neural Netw. Learn. Syst. 26(11), 2901–2913 (2015)
 21.
Nie, X., Zheng, W.: Multistability of neural networks with discontinuous nonmonotonic piecewise linear activation functions and timevarying delays. Neural Netw. 65, 65–79 (2015)
 22.
Nie, X., Zheng, W.: Complete stability of neural networks with nonmonotonic piecewise linear activation functions. IEEE Trans. Circuits Syst. II, Express Briefs 62(10), 1002–1006 (2015)
 23.
Nie, X., Zheng, W., Cao, J.: Coexistence and local μstability of multiple equilibrium points for memristive neural networks with nonmonotonic piecewise linear activation functions and unbounded timevarying delays. Neural Netw. 84, 172–180 (2016)
 24.
Yang, W., Wang, Y., Zeng, Z., Zheng, D.: Multistability of discretetime delayed Cohen–Grossberg neural networks with secondorder synaptic connectivity. Neurocomputing 164, 252–261 (2015)
 25.
Liang, J., Gong, W., Huang, T.: Multistability of complexvalued neural networks with discontinuous activation functions. Neural Netw. 84, 125–142 (2016)
 26.
Liu, P., Zeng, Z., Wang, J.: Multistability analysis of a general class of recurrent neural networks with nonmonotonic activation functions and timevarying delays. Neural Netw. 79, 117–127 (2016)
 27.
Liu, P., Zeng, Z., Wang, J.: Multistability of recurrent neural networks with nonmonotonic activation functions and mixed time delays. IEEE Trans. Syst. Man Cybern. 46(4), 512–523 (2016)
 28.
Chen, X., Zhao, Z., Song, Q., Hu, J.: Multistability of complexvalued neural networks with timevarying delays. Appl. Math. Comput. 294, 18–35 (2017)
 29.
Tan, M., Xu, D.: Multiple μstability analysis for memristorbased complexvalued neural networks with nonmonotonic piecewise nonlinear activation functions and unbounded timevarying delays. Neurocomputing 275, 2681–2701 (2018)
 30.
Nie, X., Liang, W., Cao, J.: Multistability analysis of competitive neural networks with Gaussianwavelettype activation functions and unbounded timevarying delays. Appl. Math. Comput. 356, 449–468 (2019)
 31.
Nie, X., Liang, W., Cao, J.: Multistability and instability of competitive neural networks with nonmonotonic piecewise linear activation functions. Nonlinear Anal., Real World Appl. 45, 799–821 (2019)
 32.
Chen, Y., Jia, S.: Multiple stability and instability of Cohen–Grossberg neural network with unbounded timevarying delays. J. Inequal. Appl. 178, 1–14 (2019)
 33.
Chen, J., Chen, B., Zeng, Z., Jiang, P.: Eventbased synchronization for multiple neural networks with time delay and switching disconnected topology. IEEE Trans. Cybern. 99, 1–11 (2020)
 34.
Huang, Y., Yuan, X., Yang, X., Long, H.: Multiple Lagrange stability and Lyapunov asymptotical stability of delayed fractionalorder Cohen–Grossberg neural networks. Chin. Phys. B 29, 229–238 (2020)
 35.
Wan, P., Sun, D., Zhao, M., Wan, L., Jin, S.: Multistability and attraction basins of discretetime neural networks with nonmonotonic piecewise linear activation functions. Neural Netw. 122, 231–238 (2020)
 36.
Agranovich, G., Litsyn, E., Slavova, A.: Impulsive control of a hysteresis cellular neural network model. Nonlinear Anal. Hybrid Syst. 3, 65–73 (2009)
 37.
Lu, J., Kurths, J., Cao, J., et al.: Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy. IEEE Trans. Neural Netw. 23, 285–292 (2012)
 38.
Yang, X., Cao, J., Qiu, J.: pmoment exponential stochastic synchronization of coupled memristorbased neural networks with mixed delays via delayed impulsive control. Neural Netw. 65, 80–91 (2015)
 39.
Liu, X., Wang, Q.: Impulsive stabilization of highorder Hopfieldtype neural networks with timevarying delays. IEEE Trans. Neural Netw. 19, 71–79 (2008)
 40.
Li, X., Regan, D., Akca, H.: Global exponential stabilization of impulsive neural networks with unbounded continuously distributed delays. IMA J. Appl. Math. 80, 85–99 (2015)
 41.
Li, L., Jian, J.: Delaydependent passivity analysis of impulsive neural networks with timevarying delays. Neurocomputing 168, 276–282 (2015)
 42.
Guan, Z., Liu, Z., Feng, G., et al.: Synchronization of complex dynamical networks with timevarying delays via impulsive distributed control. IEEE Trans. Circuits Syst. I, Regul. Pap. 57, 2182–2195 (2010)
 43.
Guan, Z., Hill, D., Shen, X.: Hybrid impulsive and switching systems and application to control and synchronization. IEEE Trans. Autom. Control 50, 1058–1062 (2005)
 44.
Chen, W., Lu, X., Zheng, W.: Impulsive stabilization and impulsive synchronization of discretetime delayed neural networks. IEEE Trans. Neural Netw. 26, 734–748 (2015)
 45.
Feng, Y., Li, C., Huang, T.: Sandwich control systems with impulse time windows. Int. J. Mach. Learn. Cybern. 8, 2009–2015 (2017)
 46.
Li, X., Zhang, X., Song, S.: Effect of delayed impulses on inputtostate stability of nonlinear systems. Automatica 76, 378–382 (2017)
 47.
Stamova, I., Stamov, T., Li, X.: Global exponential stability of a class of impulsive cellular neural networks with supremums. Int. J. Adapt. Control Signal Process. 28, 1227–1239 (2014)
 48.
Yao, F., Deng, F.: Stability of impulsive stochastic functional differential systems in terms of two measures via comparison approach. Sci. China Inf. Sci. 55, 1313–1322 (2012)
 49.
Li, C., Wu, S., Feng, G., et al.: Stabilizing effects of impulses in discretetime delayed neural networks. IEEE Trans. Neural Netw. 22, 323–329 (2011)
 50.
Li, X., Song, S., Wu, J.: Impulsive control of unstable neural networks with unbounded timevarying delays. Sci. China Inf. Sci. 61, 012203 (2018)
 51.
Khan, H., Khan, A., Jarad, F., Shah, A.: Existence and data dependence theorems for solutions of an ABCfractional order impulsive system. Chaos Solitons Fractals 131, 109477 (2020)
 52.
Khan, H., Khan, A., Abdeljawad, T., Alkhazzan, A.: Existence results in Banach space for a nonlinear impulsive system. Adv. Differ. Equ. 2019, 18 (2019)
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestion.
Funding
This work was supported in part by the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202001217), the Foundation of Chongqing Three Gorges University (Grant No. 18ZDPY07), Chongqing Municipal Key Laboratory of Institutions of Higher Education (Grant No.[2017]3), Program of Chongqing Development and Reform Commission (Grant No. 2017[1007]).
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Jia, S., Chen, Y. Impulsive control of a class of multiple unstable neural networks. J Inequal Appl 2021, 39 (2021). https://doi.org/10.1186/s13660021025671
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Keywords
 Cohen–Grossberg neural network
 Multiple unstable equilibrium points
 Impulsive control
 μstability