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Impulsive control of a class of multiple unstable neural networks

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 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 [135]). 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 [3652]. 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:

$$ \textstyle\begin{cases} \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t} =-a_{i}(x_{i}(t)) [b_{i}(x_{i}(t))- \sum_{j=1}^{n}c_{ij}g_{j}(x_{j}(t))-\sum_{j=1}^{n}d_{ij}f_{j}(x_{j}(t- \tau (t)))+I_{i} ], \\ \quad i=1,2,\ldots,n, t \ge 0, \end{cases} $$
(1)

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

$$ x_{i}(s)=\varphi _{i}(s),\quad s\in (-\infty,0], $$

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 μ-stable if there exist a positive constant M and a nondecreasing function \(\mu (t)\) with \({\lim_{t \to +\infty } \mu (t)=+\infty }\) such that

$$ \bigl\vert x_{i}(t)-x^{\star }_{i} \bigr\vert \leq \frac{M}{\mu (t)},\quad i=1,2,\ldots, n. $$

Imitating [32], we can divide the \(R^{n}\) into \(3^{n}\) non-intersection 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

$$ \Phi =\Biggl\{ \prod_{i=1}^{n} w_{i}\mid w_{i}=(-\infty,p_{i}), [p_{i},q_{i}]\text{ or }(q_{i},+\infty )\Biggr\} . $$

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

$$ \underline{a}_{i}\leq a_{i}(u)\leq \bar{a}_{i},\quad u\in (-\infty,+ \infty ), i=1,2,\ldots,n. $$

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

$$ \frac{b_{i}(y)-b_{i}(z)}{y-z}\geq b_{i}, y,\quad z\in (-\infty,+ \infty ), y\neq z, i=1,2,\ldots,n. $$

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

$$\begin{aligned} &m_{j}^{\prime }= {\lim_{x \rightarrow -\infty }} {g_{j}(x)},\qquad M_{j}^{ \prime }= {\lim_{x \rightarrow +\infty }} {g_{j}(x)}, \qquad m_{j}^{ \prime \prime }= {\lim_{x \rightarrow -\infty }} {f_{j}(x)}, \qquad M_{j}^{ \prime \prime }= {\lim_{x \rightarrow +\infty }} {f_{j}(x)}. \\ &0\leq \underline{\sigma }_{j}^{l} \leq \frac{g_{j}(u)-g_{j}(v)}{u-v} \leq \bar{\sigma }_{j}^{l},\qquad 0\leq \underline{\delta }_{j}^{l} \leq \frac{f_{j}(u)-f_{j}(v)}{u-v}\leq \bar{\delta }_{j}^{l},\quad \forall u,v \in (-\infty, p_{j}), \\ &0\leq \underline{\sigma }_{j}^{m} \leq \frac{g_{j}(u)-g_{j}(v)}{u-v} \leq \bar{\sigma }_{j}^{m},\qquad 0\leq \underline{\delta }_{j}^{m} \leq \frac{f_{j}(u)-f_{j}(v)}{u-v}\leq \bar{\delta }_{j}^{m},\quad \forall u,v \in [p_{j},q_{j}], \\ &0\leq \underline{\sigma }_{j}^{r} \leq \frac{g_{j}(u)-g_{j}(v)}{u-v} \leq \bar{\sigma }_{j}^{r},\qquad 0\leq \underline{\delta }_{j}^{r} \leq \frac{f_{j}(u)-f_{j}(v)}{u-v}\leq \bar{\delta }_{j}^{r},\quad \forall u,v \in ( p_{j},+\infty ). \end{aligned}$$

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 μ-stable. 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

$$ \textstyle\begin{cases} -b_{i}(p_{i})+c_{ii}g_{i}(p_{i})+d_{ii}f_{i}(p_{i})+\sum^{n}_{ \stackrel{j=1}{j\neq i}}\max \{(c_{ij}+d_{ij})m_{j},(c_{ij}+d_{ij})M_{j} \}-I_{i} < 0,\\ \quad i \in N_{1}\cup N_{2}, \\ -b_{i}(q_{i})+c_{ii}g_{i}(q_{i})+d_{ii}f_{i}(q_{i})+\sum^{n}_{ \stackrel{j=1}{j\neq i}}\min \{(c_{ij}+d_{ij})m_{j},(c_{ij}+d_{ij})M_{j} \}-I_{i} > 0,\\ \quad i \in N_{2}\cup N_{3}, \end{cases} $$
(2)

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

$$ \textstyle\begin{cases} -b_{i}(p_{i})+c_{ii}g_{i}(p_{i})+\sum^{n}_{ \stackrel{j=1}{j\neq i}}\max \{c_{ij}m_{j},c_{ij}M_{j}\}+\sum^{n}_{j=1}\max \{d_{ij}m_{j},d_{ij}M_{j}\}-I_{i} < 0,\\ \quad i \in N_{1}, \\ -b_{i}(q_{i})+c_{ii}g_{i}(q_{i})+\sum^{n}_{ \stackrel{j=1}{j\neq i}}\min \{c_{ij}m_{j},c_{ij}M_{j}\}+\sum^{n}_{j=1}\min \{d_{ij}m_{j},d_{ij}M_{j}\}-I_{i} > 0,\\ \quad i \in N_{3}, \end{cases} $$
(3)

and the nondecreasing function \(\mu (t)>0\) with

$$ \lim_{t\rightarrow +\infty }\mu (t)=+\infty,\qquad 0\leq \sup_{t\geq T^{\ast }} \frac{\dot{\mu }(t)}{\mu (t)}\leq \alpha,\qquad \sup_{t\geq T^{\ast }}\frac{\mu (t)}{\mu (t-\tau (t))}\leq 1+ \beta, $$
(4)

where \(\alpha \geq 0, \beta \geq 0\), and \(T^{\ast }\geq 0\). Then \(x^{\star }\) is μ-stable in \(\prod_{i=1}^{n} w_{i}\) (locally μ-stable in \(\Phi _{1}\)) if there exist some positive constants \(\zeta _{1},\zeta _{2},\ldots,\zeta _{n}\) such that

$$\begin{aligned} &(-\underline{a}_{i}\beta _{i}+\alpha )\zeta _{i}+\sum_{j\in N_{1}} \zeta _{j} \bar{a}_{i}\bar{\sigma }_{j}^{l} \vert c_{{i}{j}} \vert + \sum_{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{\sigma }_{j}^{r} \vert c_{{i}{j}} \vert \\ &\quad{}+(1+\beta ) \biggl(\sum _{j\in N_{1}}\zeta _{j} \bar{a}_{i}\bar{\delta }_{j}^{l} \vert d_{{i}{j}} \vert + \sum _{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{\delta }_{j}^{r} \vert d_{{i}{j}} \vert \biggr)< 0, \end{aligned}$$
(5)

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

$$\begin{aligned} &\min_{i\in N_{2}}\biggl\{ \bigl(-\beta _{i}+c_{{i}{i}} \sigma _{i}^{m}\bigr) \xi _{i} -\sum _{j\in N_{1}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{l} -\sum_{j\in N_{2}} \xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{m} -\sum_{j\in N_{3}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{r} \\ &\quad{}-\sum_{j\in N_{1}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{l} -\sum _{j\in N_{2}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{m} -\sum _{j\in N_{3}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{r}\biggr\} >\max \{\lambda,0\}, \end{aligned}$$
(6)

where

$$\begin{aligned} \lambda \triangleq {}&\max_{i\in N_{1}\cup N_{3}}\biggl\{ (- \beta _{i}\xi _{i} +\sum_{j\in N_{1}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{l} +\sum_{j\in N_{2}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{m} +\sum _{j\in N_{3}} \xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{r} \\ &{}+\sum_{j\in N_{1}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{l} +\sum _{j\in N_{2}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{m} +\sum _{j\in N_{3}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{r}\biggr\} , \end{aligned}$$
(7)

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= ( Q 11 Q 12 Q 12 T Q 22 ) <0 with \(Q_{11}=Q_{11}^{T}\), \(Q_{22}=Q_{22}^{T}\) is equivalent to one of the following conditions:

  1. (i)

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

  2. (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:

$$\begin{aligned} \Delta x(t_{i})=\Upsilon _{i}\bigl(x\bigl(t_{i}^{-} \bigr)-x^{\star }\bigr),\quad i\in \mathbb{Z}_{+}, \end{aligned}$$
(8)

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:

$$ \textstyle\begin{cases} \frac{\mathrm{d}h(t)}{\mathrm{d}t}=A(h(t))[-B(h(t))+CG(h(t))+DF(h(t- \tau (t)))],&t>0, t\neq t_{i}, \\ \Delta h(t_{i})=\Upsilon _{i}h(t_{i}^{-}),&t= t_{i}, i \in \mathbb{Z}_{+}, \\ h(s)=\phi (s),&s\leq 0, \end{cases} $$
(9)

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 μ-stable, if one can find a constant \(M>0\) satisfying that

$$ \Vert h \Vert ^{2}\leq \frac{M}{\mu (t)} \Vert \phi \Vert ^{2},\quad t\geq 0, $$

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 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 \(\{t_{i}, \Upsilon _{i}\}_{i\in \mathbb{Z}_{+}}\) (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

$$ \frac{\mu (t_{i})}{\mu (t_{i-1})}\leq \mu _{1},\qquad \int _{t_{i-1}}^{t_{i}} \frac{\mu (t)}{\mu ^{\ast }(t-\tau (t))}\,\mathrm{d}t\leq \mu _{2},\quad i\in \mathbb{Z}_{+}, $$
(10)

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

$$ \begin{pmatrix} \Pi & P\hat{A}C& P\hat{A}D \\ *& -Q_{1}& 0 \\ *& *& -Q_{2} \end{pmatrix}\leq 0,\qquad \Delta ^{f}Q_{2} \Delta ^{f}\leq \lambda _{2}P, $$
(11)

and

$$ \ln \mu _{1}+\lambda _{1}\theta +\mu _{2} \lambda _{2}\gamma < \ln \gamma, $$
(12)

where \(\Pi =-P\check{A}\beta -\beta \check{A}P+\Sigma ^{g}Q_{1}\Sigma ^{g}- \lambda _{1}P, \theta =\sup_{i\in \mathbb{Z}_{+}} \{ t_{i}-t_{i-1} \} \).

Consider the following positive definite Lyapunov function:

$$ \mathfrak{s}_{P}(t)=\mu ^{\ast }h^{T}Ph, $$

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 \(\nu _{1}<\nu _{2}\in [t_{i-1},t_{i})\) 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

$$ \mathfrak{s}_{P}(s)\leq \gamma \mathfrak{s}_{P}(t),\quad \forall s\in (- \infty,t], $$
(13)

then

$$ \mathfrak{s}_{P}(\nu _{2})\leq \gamma \mathfrak{s}_{P}(\nu _{1}). $$

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):

$$\begin{aligned} D^{+}\mathfrak{s}_{P}(t)={}&\mu ^{\ast }{'}(t)h^{T}Ph+2 \mu ^{\ast }(t)h^{T}Ph{'} \\ \leq {}&\mu {'}(t)h^{T}Ph+\mu (t)\bigl[h^{T}(-P \check{A}\beta -\beta \check{A}P)h+h^{T}P\hat{A}C Q_{1}C^{T} \hat{A}Ph+h^{T}\Sigma ^{g}Q_{1} \Sigma ^{g}h \\ &{}+h^{T}P\hat{A}D Q_{2}D^{T} \hat{A}Ph+h^{T}\bigl(t-\tau (t)\bigr)\Delta ^{f}Q_{2} \Delta ^{f}h\bigl(t-\tau (t)\bigr)\bigr] \\ ={}&\frac{\mu {'}(t)}{\mu (t)}\mu (t)h^{T}Ph+\mu (t)\bigl[h^{T} \bigl(-P\check{A} \beta -\beta \check{A}P+P\hat{A}C Q_{1}C^{T} \hat{A}P+\Sigma ^{g}Q_{1} \Sigma ^{g} \\ &{}+P\hat{A}D Q_{2}D^{T}\hat{A}P\bigr)h+h^{T} \bigl(t-\tau (t)\bigr)\Delta ^{f}Q_{2} \Delta ^{f}h\bigl(t-\tau (t)\bigr)\bigr] \\ \leq{} &\frac{\mu {'}(t)}{\mu (t)}\mu (t)h^{T}Ph+\mu (t)\bigl[h^{T} \lambda _{1}Ph+h^{T}\bigl(t- \tau (t)\bigr)\lambda _{2}Ph\bigl(t-\tau (t)\bigr)\bigr] \\ ={}&\biggl(\frac{\mu {'}(t)}{\mu (t)}+\lambda _{1}\biggr)\mu (t)h^{T}Ph+\lambda _{2} \frac{\mu (t)}{\mu ^{\ast }(t-\tau (t))}\mu ^{\ast }\bigl(t-\tau (t)\bigr)h^{T}\bigl(t- \tau (t)\bigr)Ph \bigl(t-\tau (t)\bigr) \\ ={}&\biggl(\frac{\mu {'}(t)}{\mu (t)}+\lambda _{1}\biggr)\mathfrak{s}_{P}(t)+ \lambda _{2}\frac{\mu (t)}{\mu ^{\ast }(t-\tau (t))}\mathfrak{s}_{P}\bigl(t- \tau (t)\bigr) \\ \leq {}&\biggl(\frac{\mu {'}(t)}{\mu (t)}+\lambda _{1}+\lambda _{2} \gamma \frac{\mu (t)}{\mu ^{\ast }(t-\tau (t))}\biggr)\mathfrak{s}_{P}(t). \end{aligned}$$
(14)

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

$$\begin{aligned} \ln \frac{\mathfrak{s}_{P}(\nu _{2})}{\mathfrak{s}_{P}(\nu _{1})} &\leq \ln \frac{\mu (t_{i})}{\mu (t_{i-1})}+\lambda _{1} \theta + \lambda _{2}\gamma \int _{t_{k-1}}^{t_{k}} \frac{\mu (t)}{\mu ^{\ast }(t-\tau (t))}\,\mathrm{d}t \\ &\leq \ln \mu _{1}+ \lambda _{1}\theta +\lambda _{2} \gamma \mu _{2} \leq \ln \gamma. \end{aligned}$$
(15)

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

$$ \gamma (I+\Upsilon _{i})^{T}P(I+\Upsilon _{i}) \leq P,\quad i\in \mathbb{Z}_{+}, $$
(16)

then model (9) is locally μ-stable. Furthermore, \(x^{\star }\) under impulsive control strategy \(\{t_{i}, \Upsilon _{i}\}_{i\in \mathbb{Z}_{+}}\) (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 \(\phi \neq 0\). Then we just have to prove the following inequality:

$$ \mathfrak{s}_{P}(t)\leq \gamma M,\quad t\in [t_{i-1},t_{i}), i\in \mathbb{Z}_{+}, $$
(17)

where \(M=\mu (0)\lambda _{\max }(P)\|\phi \|^{2}\), \(\{h(t)|h(t)\in R^{n}, t\in [t_{i-1},t_{i})\}\subset \prod_{i=1}^{n} w_{i}\) in \(\Phi _{2}\). Note that

$$ \mathfrak{s}_{P}(t)=\mu ^{\ast }h^{T}Ph=h^{T}Ph \leq \mu (0)\lambda _{ \max }(P) \Vert \phi \Vert ^{2}\leq \gamma M,\quad t\leq 0. $$
(18)

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

$$ \mathfrak{s}_{P}(\nu _{2})= \gamma M,\qquad \mathfrak{s}_{P}(t)\leq \gamma M,\quad t\leq \nu _{2}. $$
(19)

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

$$ \mathfrak{s}_{P}(\nu _{1})=M,\qquad M\leq \mathfrak{s}_{P}(t) \leq \gamma M,\quad t\in [\nu _{1},\nu _{2}]. $$
(20)

For any \(t\in [\nu _{1},\nu _{2}]\), by (19) and (20), it follows that

$$ \mathfrak{s}_{P}(s)\leq \gamma M\leq \gamma \mathfrak{s}_{P}(t),\quad \forall s\in (-\infty,t]. $$
(21)

Meanwhile, by Theorem 1, we can get

$$ \gamma M=\mathfrak{s}_{P}(\nu _{2})< \gamma \mathfrak{s}_{P}(\nu _{1})= \gamma M, $$

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

$$ \mathfrak{s}_{P}\bigl(\nu _{2}^{\ast }\bigr)= \gamma M, \qquad\mathfrak{s}_{P}(t) \leq \gamma M,\quad t\leq \nu _{2}^{\ast }. $$
(22)

With respect to (16), we can obtain that

$$\begin{aligned} \mathfrak{s}_{P}(t_{N})&=\mu ^{\ast }(t_{N})h^{T}(t_{N})Ph(t_{N})= \mu ^{ \ast }h^{T}\bigl(t_{N}^{-}\bigr) (I+\Upsilon _{N})^{T}P(I+\Upsilon _{N})h \bigl(t_{N}^{-}\bigr) \\ & \leq \frac{1}{\gamma } \mathfrak{s}_{P}\bigl(t_{N}^{-}\bigr)=M. \end{aligned}$$
(23)

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

$$ \mathfrak{s}_{P}\bigl(\nu ^{\ast }_{1}\bigr)=M,\qquad M \leq \mathfrak{s}_{P}(t)\leq \gamma M,\quad t\in \bigl[\nu ^{\ast }_{1}, \nu ^{\ast }_{2}\bigr]. $$
(24)

Furthermore, for any \(t\in [\nu ^{\ast }_{1},\nu ^{\ast }_{2}]\), by (23) and (24), we have

$$ \mathfrak{s}_{P}(s)\leq \gamma M\leq \gamma \mathfrak{s}_{P}(t),\quad \forall s\in (-\infty,t]. $$
(25)

However, we can obtain by Theorem 1

$$ \gamma M=\mathfrak{s}_{P}\bigl(\nu ^{\ast }_{2} \bigr)< \gamma \mathfrak{s}_{P}\bigl( \nu ^{\ast }_{1} \bigr)=\gamma M, $$

which leads to a contradiction. Thus \(\mathfrak{s}_{P}(t)\leq \gamma M\), and \(k=N+1\) holds.

Finally, by mathematical induction, we get

$$\begin{aligned} &\mu (t)\lambda _{\min }(P)h^{T}h\leq \mu ^{\ast }(t)h^{T}Ph= \mathfrak{s}_{P}(t)\leq \gamma \mu (0)\lambda _{\max }(P) \Vert \varphi \Vert ^{2}, \\ &\quad \forall t\in [t_{i-1},t_{i}), i\in \mathbb{Z}_{+}, \end{aligned}$$
(26)

which implies that (26) satisfies Definition 1. Hence model (9) is μ-stable in \(\prod_{i=1}^{n} w_{i}\) of \(\Phi _{2}\). Consequently, \(x^{\star }\) under impulsive control (8) is locally μ-stable, and so model (1) can add \(3^{n}-2^{n}\) locally μ-stable 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:

$$ \textstyle\begin{cases} \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t}=-b_{i}(x_{i}(t))+\sum_{j=1}^{n}c_{ij}g_{j}(x_{j}(t))+ \sum_{j=1}^{n}d_{ij}f_{j}(x_{j}(t-\tau (t)))-I_{i}, \\ \quad i=1,2,\ldots,n, t \ge 0. \end{cases} $$
(27)

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 μ-stable, 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

$$ \textstyle\begin{cases} \frac{\mathrm{d}h(t)}{\mathrm{d}t}=-B(h(t))+CG(h(t))+DF(h(t-\tau (t))),&t>0,t \neq t_{i}, \\ h(t)=\phi (t),&t\leq 0, \\ \Delta h(t_{i})=\Upsilon _{i}h(t_{i}^{-}),&t=t_{i},i\in \mathbb{Z}_{+}. \end{cases} $$
(28)

Corollary 1

Under conditions (10), (12), (16), and

$$ \begin{pmatrix} \Pi & PC& PD \\ *& -Q_{1}& 0 \\ *& *& -Q_{2} \end{pmatrix}\leq 0, \qquad \Delta ^{f}Q_{2}\Delta ^{f}\leq \lambda _{2}P, $$
(29)

where \(\Pi =-P\beta -\beta P+\Sigma ^{g}Q_{1}\Sigma ^{g}-\lambda _{1}P\), model (28) is μ-stable 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 μ-stable, and so model (27) can increase \(3^{n}-2^{n}\) locally μ-stable points.

When \(a_{i}(x_{i}(t))=1\) and \(b_{i}(x_{i}(t))=b_{i} \cdot x_{i}(t)\), model (1) changes into

$$ \textstyle\begin{cases} \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t}=-b_{i} \cdot x_{i}(t)+\sum_{j=1}^{n}c_{ij}g_{j}(x_{j}(t))+ \sum_{j=1}^{n}d_{ij}f_{j}(x_{j}(t-\tau (t)))-I_{i}, \\ \quad i=1,2,\ldots,n, t \ge 0. \end{cases} $$
(30)

If (30) meets (4) and the following conditions (31)(33):

$$\begin{aligned} &\textstyle\begin{cases} -b_{i}p_{i}+c_{ii}g_{i}(p_{i})+\sum^{n}_{ \stackrel{j=1}{j\neq i}}\max \{c_{ij}m_{j},c_{ij}M_{j}\}+\sum^{n}_{j=1}\max \{d_{ij}m_{j},d_{ij}M_{j}\}-I_{i} < 0, \\ -b_{i}q_{i}+c_{ii}g_{i}(q_{i})+\sum^{n}_{ \stackrel{j=1}{j\neq i}}\min \{c_{ij}m_{j},c_{ij}M_{j}\}+\sum^{n}_{j=1}\min \{d_{ij}m_{j},d_{ij}M_{j}\}-I_{i} > 0, \\ \quad i=1,2,\ldots,n. \end{cases}\displaystyle \end{aligned}$$
(31)
$$\begin{aligned} &(-\underline{a}_{i}\beta _{i}+\alpha )\zeta _{i}+\sum_{j\in N_{1}} \zeta _{j} \bar{a}_{i}\bar{\sigma }_{j}^{l} \vert c_{{i}{j}} \vert + \sum_{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{\sigma }_{j}^{r} \vert c_{{i}{j}} \vert \\ &\quad{}+(1+\beta ) \biggl(\sum _{j\in N_{1}}\zeta _{j} \bar{a}_{i}\bar{\delta }_{j}^{l} \vert d_{{i}{j}} \vert + \sum _{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{\delta }_{j}^{r} \vert d_{{i}{j}} \vert \biggr)< 0, \end{aligned}$$
(32)
$$\begin{aligned} &\min_{i\in N_{2}}\biggl\{ \bigl(-b_{i}+c_{{i}{i}} \sigma _{i}^{m^{\ast }}\bigr) \xi _{i} -\sum _{j\in N_{1}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{l} -\sum_{j\in N_{2}} \xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{m} -\sum_{j\in N_{3}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{r} -\sum_{j\in N_{1}} \xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{l} \\ &\quad{}-\sum_{j\in N_{2}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{m} -\sum _{j\in N_{3}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{r}\biggr\} >\max \{\lambda,0\}, \end{aligned}$$
(33)

where \(\zeta _{1},\zeta _{2},\ldots,\zeta _{n}\), \(\xi _{1},\xi _{2},\ldots,\xi _{n}\) are positive constants, and

$$\begin{aligned} \lambda \triangleq {}&\max_{i\in N_{1}\cup N_{3}}\biggl\{ (-b_{i} \xi _{i} +\sum_{j\in N_{1}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{l} +\sum _{j\in N_{2}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{m} +\sum _{j\in N_{3}}\xi _{j} \vert c_{{i}{j}} \vert \bar{\sigma }_{j}^{r} \\ &{}+\sum_{j\in N_{1}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{l} +\sum _{j\in N_{2}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{m} +\sum _{j\in N_{3}}\xi _{j} \vert d_{{i}{j}} \vert \bar{\delta }_{j}^{r}\biggr\} , \end{aligned}$$
(34)

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 μ-stable.

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:

$$ \textstyle\begin{cases} \frac{\mathrm{d}h(t)}{\mathrm{d}t}=-Bh(t)+CG(h(t))+DF(h(t-\tau (t))),&t>0,t \neq t_{i}, \\ h(t)=\phi (t),&t\leq 0, \\ \Delta h(t_{i})=\Upsilon _{i}h(t_{i}^{-}),&t=t_{i},i\in \mathbb{Z}_{+}. \end{cases} $$
(35)

Corollary 2

Under conditions (10),(12),(16), and (29), where \(\Pi =-PB-BP+\Sigma ^{g}Q_{1}\Sigma ^{g}-\lambda _{1}P\), model (35) is μ-stable 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 μ-stable, and so model (30) can increase \(3^{n}-2^{n}\) locally μ-stable points.

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.

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

$$\begin{aligned} &(-\underline{a}_{i}\beta _{i}+\varsigma )\zeta _{i}+\sum_{j \in N_{1}}\zeta _{j} \bar{a}_{i}\bar{\sigma }_{j}^{l} \vert c_{{i}{j}} \vert +\sum_{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{\sigma }_{j}^{r} \vert c_{{i}{j}} \vert \\ &\quad{}+(1+\varsigma \tau ) \biggl(\sum _{j\in N_{1}} \zeta _{j}\bar{a}_{i}\bar{\delta }_{j}^{l} \vert d_{{i}{j}} \vert + \sum _{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{\delta }_{j}^{r} \vert d_{{i}{j}} \vert \biggr)< 0. \end{aligned}$$

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

$$ -\underline{a}_{i}\beta _{i}\zeta _{i}+\sum _{j\in N_{1}} \zeta _{j}\bar{a}_{i} \bar{\sigma }_{j}^{l} \vert c_{{i}{j}} \vert + \sum _{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{ \sigma }_{j}^{r} \vert c_{{i}{j}} \vert +\biggl( \sum_{j\in N_{1}}\zeta _{j} \bar{a}_{i} \bar{\delta }_{j}^{l} \vert d_{{i}{j}} \vert + \sum _{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{ \delta }_{j}^{r} \vert d_{{i}{j}} \vert \biggr)< 0. $$

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 log-stable 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

$$ -\underline{a}_{i}\beta _{i}\zeta _{i}+\sum _{j\in N_{1}} \zeta _{j}\bar{a}_{i} \bar{\sigma }_{j}^{l} \vert c_{{i}{j}} \vert + \sum _{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{ \sigma }_{j}^{r} \vert c_{{i}{j}} \vert + \frac{1}{\epsilon }\biggl(\sum_{j\in N_{1}} \zeta _{j}\bar{a}_{i}\bar{\delta }_{j}^{l} \vert d_{{i}{j}} \vert + \sum_{j\in N_{3}}\zeta _{j}\bar{a}_{i}\bar{\delta }_{j}^{r} \vert d_{{i}{j}} \vert \biggr)< 0. $$

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 log-log-stable in each region of \(\Phi _{2}\).

Numerical example

Example

Consider the following two-dimensional CGNNs model:

$$ \textstyle\begin{cases} \frac{\mathrm{d}x_{i}(t)}{\mathrm{d}t}=-a(x_{i}(t)) [b_{i}(x_{i}(t))- \sum_{j=1}^{n}c_{ij}g(x_{j}(t))-\sum_{j=1}^{n}d_{ij}f(x_{j}(t-\tau (t)))+I_{i}], \\ \quad i=1,2, t \ge 0, \end{cases} $$
(36)

where \(a(x)=1+0.2\sin (x)\), \(b_{1}(x_{1}(t))=x_{1}(t)\), \(b_{2}(x_{2}(t))=-1.2x_{2}(t)\), \(f(x)= \frac{\vert x+1\vert -\vert x-1\vert }{2}\),

$$\begin{aligned} &g(x)=\textstyle\begin{cases} \tanh (0.2x)-\tanh (1)+\tanh (0.2),&x< -1, \\ \tanh (x),&-1\leq x\leq 1, \\ \tanh (0.2x)+\tanh (1)-\tanh (0.2),&x>1, \end{cases}\displaystyle \quad\tau (t)=0.2t,\\ &C=(c_{ij})= \begin{pmatrix} 3.5& 0.2 \\ 0.4& 4.8 \end{pmatrix},\qquad D=(d_{ij})= \begin{pmatrix} 0.4& 0.2 \\ 0.2& 0.5 \end{pmatrix},\qquad I_{1}=I_{2}= \begin{pmatrix} -0.3 \\ -0.6 \end{pmatrix}. \end{aligned}$$

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 13, 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:

$$\begin{aligned} &x(1)=(-3.8676,5.9164)^{T},\qquad x(2)=(-0.2792,6.4788)^{T},\\ & x(3)=(6.1086,7.4167)^{T},\qquad x(4)=(-4.6061,0.0284)^{T},\\ & x(5)=(-0.0852,-0.1349)^{T},\qquad x(6)=(5.2870,-0.3416)^{T}, \\ &x(7)=(-5.3083,-6.1639)^{T},\qquad x(8)=(0.0586,-5.2811)^{T},\\ & x(9)=(4.8374,-4.3921)^{T}. \end{aligned}$$

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.

Figure 1
figure1

The state trajectories and phase diagrams without impulsive input (three graphs above) and with impulsive input (three graphs below)of model (36)

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:

$$\begin{aligned} &P= \begin{pmatrix} 0.4605& 0.0069 \\ 0.0069& 0.4558 \end{pmatrix},\qquad Q_{1}= \begin{pmatrix} 1.7450 & 0 \\ 0& 1.7450 \end{pmatrix},\\ &Q_{2}= \begin{pmatrix} 4.6120& 0 \\ 0 & 4.6120 \end{pmatrix}. \end{aligned}$$

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):

$$ \Upsilon _{i}= \begin{pmatrix} -0.7 & 0 \\ 0& -0.7 \end{pmatrix}. $$
(37)

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. 26 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.

Figure 2
figure2

Transient behavior and state trajectories of \(x_{1}\) and \(x_{2}\) nearby the equilibrium point \(x(4)=(-4.6061,0.0284)^{T}\) without impulsive input in the left graphs, and with impulsive input in the right graphs

Figure 3
figure3

Transient behavior and state trajectories of \(x_{1}\) and \(x_{2}\) near the equilibrium point \(x(2)=(-0.2792,6.4788)^{T}\) without impulsive input in the left graphs, and with impulsive input in the right graphs

Figure 4
figure4

Transient behavior and state trajectories of \(x_{1}\) and \(x_{2}\) near the equilibrium point \(x(5)=(-0.0852,-0.1349)^{T}\) without impulsive input in the left graphs, and with impulsive input in the right graphs

Figure 5
figure5

Transient behavior and state trajectories of \(x_{1}\) and \(x_{2}\) near the equilibrium point \(x(8)=(0.0586,-5.2811)^{T}\) without impulsive input in the left graphs, and with impulsive input in the right graphs

Figure 6
figure6

Transient behavior and state trajectories of \(x_{1}\) and \(x_{2}\) near the equilibrium point \(x(6)=(5.2870,-0.3416)^{T}\) without impulsive input in the left graphs, and with impulsive input in the right graphs

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 \(\Phi _{2}\). 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 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.

Availability of data and materials

Not applicable.

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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/s13660-021-02567-1

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Keywords

  • Cohen–Grossberg neural network
  • Multiple unstable equilibrium points
  • Impulsive control
  • μ-stability
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