1 Introduction

In recent years, memristive neurodynamic systems have become one of the most widely researched topics in the computing architecture enabled by memristors [1–16]. A multiport memristive neurodynamic system is a promising electronic system for its neuromorphic memristor memory [5]. It is worth mention that using memristors as synaptic connections in neuromorphic electronic systems has been suggested with different neural architectures. One of the main characteristics hindering the neural architecture of memristive neuromorphic systems is the sneak-paths phenomenon. Consequently, various interesting nonlinear properties of memristive neuromorphic systems have emerged [1–4, 6–16].

Recently, some works [1, 3, 6–9, 13–15] have demonstrated the interest in studying global stability for some types of memristive neurodynamic systems, such as fractional-order memristive neurodynamic systems [1] and integral-order memristive neurodynamic systems [3, 6–9, 13–15]. Periodic oscillation in neurodynamic systems is an interesting dynamic effect, furthermore, the analysis of persistent oscillation for memristive neurodynamic systems is able to reveal crucial features of the dynamics [2, 16]. In an associative memory model based on memristors, multiple stable states are necessary. The multistability of memristive neurodynamic systems has been investigated in [4, 10]. To control the memristive physiological states, some control strategies of exponential stabilization for memristive neurodynamic systems are proposed in [11]. Consider, for example, the dynamic-state sequence objects where, under the universal need of multi-attractor, the multi-object flow closely relates to a multi-attractor of the network cluster. Thus, Wu and Zeng [12] discussed the Lagrange stability of memristive neurodynamic systems with various feedback functions.

Despite some clear progress on the qualitative analysis of memristive neurodynamic systems, the issue of the memristor multiport effect has not been fully used in the existing studies, except [1, 2, 13], when dealing with the qualitative analysis of such systems. As is well known, the memristor multiport effect explains how intelligent behavior arises from neural models. Ignoring the memristor multiport effect in developing computing platforms based on memristive neuromorphic systems is very unreasonable. Once abandoning the memristor multiport effect, designers of these systems face a major problem: whether such neuromorphic system can accurately simulate memristive intelligent behavior in a mixed analog-digital integrated circuit.

In this paper, we consider the global exponential stability and global asymptotical stability for a class of delayed memristive neural networks. The analytical method differs from those considered in most of the existing literature on qualitative analysis of memristive neurodynamic systems, where the memristor multiport effect is ignored. The obtained algebraical criteria, which can be directly derived from the network parameters, are easily testable. Consequently, a physical realization of such a system is rather simple.

The rest of this paper is organized as follows. Section 2 presents the network model and preliminaries. Section 3 details the analytical framework and theoretical results. Section 4 provides two illustrative examples. Section 5 gives the concluding remarks.

2 Model description and preliminaries

Based on [1, 13], in this paper, consider a class of delayed memristive neural networks described by the following delay differential equations: for \(i=1,2,\ldots,n\),

$$\begin{aligned} \dot{x}_{i}(t)= - d_{i}x_{i}(t) + \sum _{j = 1}^{n} a_{ij}\bigl(x_{i}(t) \bigr)f_{j}\bigl(x_{j}(t-\tau_{ij}) \bigr)+u_{i}, \end{aligned}$$
(1)

where \(x_{i}(t)\) denotes the memristive neuron state, \(d_{i}>0\) is the self-inhibition, \(0\leq\tau_{ij}\leq\tau\) (\(\tau\geq0\) is a constant) is the transmission delay, feedback function \(f_{i}(\cdot)\) is bounded and \(f_{i}(0)=0\), \(u_{i}\) denotes the external input, \(a_{ij}(x_{i}(t))\) is the connection memristive weight, which is defined as

$$ a_{ij}\bigl(x_{i}(t)\bigr)=\left \{ \begin{array}{@{}l@{\quad}l} \hat{a}_{ij}, & {x_{i}(t)>0}, \\ \check{a}_{ij}, & {x_{i}(t)< 0}, \end{array} \right . $$
(2)

for \(i,j=1,2,\ldots,n\), where \(\hat{a}_{ij}\) and \(\check{a}_{ij}\) are constants.

The initial condition of neurodynamic system (1) is assumed to be

$$\begin{aligned} x(t)=\bigl(x_{1}(t),x_{2}(t),\ldots,x_{n}(t) \bigr)^{T} =\phi(t) =\bigl(\phi_{1}(t),\phi_{2}(t), \ldots,\phi_{n}(t)\bigr)^{T}, \quad {t_{0}-\tau} \leq{t}\leq {t_{0}}, \end{aligned}$$
(3)

where \(\phi_{i}(t)\in\mathcal{C}([{t_{0}-\tau},t_{0}],\Re)\), \(i=1,2,\ldots,n\).

Let \(x_{t}\in\mathcal{C}([{t_{0}-\tau},t_{0}],\Re^{n})\) be defined by \(x_{t}(s)=x(t+s)\), \({t_{0}-\tau}\leq{s}\leq{t_{0}}\), and (3) can be rewritten as

$$x_{t_{0}}=\phi\in\mathcal{C}\bigl([{t_{0}- \tau},t_{0}],\Re^{n}\bigr). $$

In addition, we also assume that the feedback function \(f_{i}(\cdot)\) (\(i=1,2,\ldots,n\)) satisfies the Lipschitz condition with the Lipschitz constant \(l_{i}>0\), i.e.,

$$ \bigl\vert f_{i}(\chi)-f_{i}(\widehat{\chi})\bigr\vert \leq{l_{i}}\vert \chi -\widehat{\chi} \vert ,\quad i=1,2,\ldots,n, \forall\chi,\widehat{\chi}\in\Re. $$
(4)

Remark 1

From a systems-theoretic point of view, neurodynamic system (1) is basically a state-dependent switched network cluster. For analyzing and controlling the state-dependent switched network cluster, nonsmooth analysis will be devoted to dealing with neurodynamic system (1). The theory and application of conventional nonlinear systems have been extensively studied over the past few decades; see [17–40]. However, little attention has been paid to the switched network cluster [41].

In this paper, solutions of all the systems considered are in the following interpreted in Filippov’s sense. \(K(\mathcal{P})\) denotes closure of the convex hull of set \(\mathcal{P}\). \(\operatorname{co} \{\widetilde{\Pi},\widehat{\Pi} \}\) denotes closure of the convex hull generated by real numbers \(\widetilde{\Pi}\) and \(\widehat{\Pi}\). Let \(\overline{a}_{ij}=\max \{\hat{a}_{ij},\check{a}_{ij} \}\), \(\underline{a}_{ij}=\min \{\hat{a}_{ij},\check{a}_{ij} \}\), \(\tilde{a}_{ij}=\max \{\vert \hat{a}_{ij}\vert ,\vert \check{a}_{ij}\vert \}\), for \(i,j=1,2,\ldots,n\).

When considering neurodynamic system (1), throughout this paper, let us define the set-valued maps as follows:

$$ K\bigl(a_{ij}\bigl(x_{i}(t)\bigr)\bigr)= \left \{ \begin{array}{@{}l@{\quad}l} \hat{a}_{ij}, & {x_{i}(t)>0}, \\ \operatorname{co} \{\hat{a}_{ij},\check{a}_{ij} \}, & x_{i}(t)=0,\\ \check{a}_{ij}, & {x_{i}(t)< 0}. \end{array} \right . $$
(5)

Obviously, for \(i,j=1, 2,\ldots,n\),

$$\operatorname{co} \{\hat{a}_{ij},\check{a}_{ij} \}=[ \underline {a}_{ij},\overline{a}_{ij}]. $$

By the theory of differential inclusions, from (1), for \(i=1, 2,\ldots,n\),

$$\begin{aligned} \dot{x}_{i}(t)\in -d_{i}x_{i}(t) + \sum_{j=1}^{n}K\bigl(a_{ij} \bigl(x_{i}(t)\bigr)\bigr)f_{j} \bigl(x_{j}(t- \tau_{ij})\bigr)+u_{i}. \end{aligned}$$
(6)

A solution \(x(t)=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))^{T}\) in the sense of Filippov of neurodynamic system (1), with initial condition \(x(s)=\phi(s)\), \(s\in[t_{0}-\tau, t_{0}]\), is absolutely continuous on any compact interval of \([t_{0},+\infty)\), and

$$\begin{aligned} \dot{x}_{i}(t)\in -d_{i}x_{i}(t) +\sum _{j=1}^{n}K\bigl(a_{ij}\bigl(x_{i}(t) \bigr)\bigr)f_{j} \bigl(x_{j}(t-\tau_{ij}) \bigr)+u_{i}. \end{aligned}$$

Remark 2

Filippov solutions are mainly used to analyze nonlinear switched systems. On the concept of Filippov solutions, one may refer to the well-known publication [41].

Definition 1

A constant vector \(x^{*}= (x_{1}^{*}, x_{2}^{*},\ldots, x_{n}^{*})^{T}\) is called an equilibrium point of neurodynamic system (1), if for \(i= 1, 2,\ldots,n\),

$$\begin{aligned} 0\in-d_{i}x_{i}^{*}+\sum_{j=1}^{n}K \bigl(a_{ij}\bigl(x_{i}^{*}\bigr)\bigr)f_{j} \bigl(x_{j}^{*}\bigr)+u_{i}. \end{aligned}$$

Definition 2

The equilibrium point \(x^{*}= (x_{1}^{*}, x_{2}^{*},\ldots, x_{n}^{*})^{T}\) of neurodynamic system (1) is said to be globally exponentially stable, if there exist constants \(\kappa> 0\) and \(\tilde{\kappa}> 0\) such that

$$\begin{aligned} \bigl\vert {{x}}_{i}(t,t_{0},x_{0})-x_{i}^{*} \bigr\vert \leq \kappa\bigl\vert {{x}}_{i}(0)-x_{i}^{*}\bigr\vert \exp \bigl\{ -\tilde{\kappa }(t-t_{0}) \bigr\} ,\quad \forall{t \geq{t_{0}}}, i=1, 2,\ldots,n. \end{aligned}$$

Definition 3

The equilibrium point \(x^{*}= (x_{1}^{*}, x_{2}^{*},\ldots, x_{n}^{*})^{T}\) of neurodynamic system (1) is said to be globally asymptotically stable, if it is locally stable and globally attractive.

The following two lemmas play an important role in discussing the qualitative behavior of neurodynamic system (1).

Lemma 1

For neurodynamic system (1) there exists at least one equilibrium point.

In subsequent sections, we will denote by \(x^{*}= (x_{1}^{*}, x_{2}^{*},\ldots, x_{n}^{*})^{T}\) the equilibrium point of (1).

Lemma 2

For neurodynamic system (1),

$$\begin{aligned} \bigl\vert K\bigl(a_{ij}(x_{i})\bigr)f_{j}(x_{j})-K \bigl(a_{ij}(y_{i})\bigr)f_{j}(y_{j})\bigr\vert \leq \tilde{a}_{ij}{l_{j}}\vert x_{j}-y_{j}\vert ,\quad \forall i,j=1,2,\ldots,n, \forall{x_{i},y_{i}\in\Re}. \end{aligned}$$

Lemmas 1 and 2 can be proved using standard arguments as the Lemmas 1 and 2 in [13], respectively.

Remark 3

Lemma 2 of this paper makes good use of the memristor multiport effect. It establishes a kind of internal ties between system attribute and its neurodynamic approaches.

3 Main results

According to Lemma 1, neurodynamic system (1) has the equilibrium point \(x^{*}= (x_{1}^{*}, x_{2}^{*},\ldots, x_{n}^{*})^{T}\); we shift the equilibrium point \(x^{*}= (x_{1}^{*}, x_{2}^{*},\ldots, x_{n}^{*})^{T}\) to the origin by the translation \(z(t)= x(t)-x^{*}\) in the differential inclusion (6), which results in

$$\begin{aligned} \dot{z}_{i}(t)\in -d_{i}z_{i}(t) +\sum _{j=1}^{n}K\bigl(a_{ij}\bigl(z_{i}(t) \bigr)\bigr)f_{j} \bigl(z_{j}(t-\tau_{ij})\bigr), \quad i=1, 2,\ldots,n, \end{aligned}$$
(7)

where

$$\begin{aligned} K\bigl(a_{ij}\bigl(z_{i}(t)\bigr)\bigr)f_{j} \bigl(z_{j}(t-\tau _{ij})\bigr)=K\bigl(a_{ij} \bigl(z_{i}(t)+x_{i}^{*}\bigr)\bigr)f_{j} \bigl(z_{j}(t-\tau_{ij})+x_{j}^{*}\bigr) - K \bigl(a_{ij}\bigl(x_{i}^{*}\bigr)\bigr)f_{j} \bigl(x_{j}^{*}\bigr). \end{aligned}$$
(8)

According to Lemma 2,

$$ \bigl\vert K\bigl(a_{ij}\bigl(z_{i}(t)\bigr) \bigr)f_{j}\bigl(z_{j}(t-\tau_{ij})\bigr)\bigr\vert \leq \tilde{a}_{ij}{l_{j}}\bigl\vert {z_{j}(t-\tau_{ij})}\bigr\vert . $$
(9)

From (7)-(9),

$$\begin{aligned} \bigl\vert \dot{z}_{i}(t)\bigr\vert \leq -d_{i}\bigl\vert z_{i}(t)\bigr\vert +\sum_{j=1}^{n} \tilde{a}_{ij}{l_{j}}\bigl\vert {z_{j}(t- \tau_{ij})}\bigr\vert ,\quad i\in \{1,2,\ldots,n \}. \end{aligned}$$
(10)

Theorem 1

The equilibrium point \(x^{*}= (x_{1}^{*}, x_{2}^{*},\ldots, x_{n}^{*})^{T}\) of neurodynamic system (1) is globally exponentially stable, if any one of the conditions (11) and (12) holds:

$$\begin{aligned}& d_{i}>\sum_{j=1}^{n} \tilde{a}_{ij}{l_{j}},\quad i=1, 2,\ldots,n, \end{aligned}$$
(11)
$$\begin{aligned}& d_{i}>\sum_{j=1}^{n} \tilde{a}_{ji}{l_{i}},\quad i=1, 2,\ldots,n. \end{aligned}$$
(12)

Proof

Consider the comparison systems of (10)

$$\begin{aligned} \bigl\vert \dot{\tilde{z}}_{i}(t)\bigr\vert = -d_{i} \bigl\vert \tilde{z}_{i}(t)\bigr\vert +\sum _{j=1}^{n}\tilde {a}_{ij}{l_{j}} \bigl\vert {\tilde{z}_{j}(t-\tau_{ij})}\bigr\vert ,\quad i \in \{1,2,\ldots,n \}, \end{aligned}$$
(13)

where \(\tilde{z}_{i}(t_{0},z_{0})={z}_{i}(t_{0})\), \(i=1, 2,\ldots,n\).

According to the comparison theorem [22],

$$\bigl\vert {{z}}_{i}(t,t_{0},z_{0})\bigr\vert \leq\bigl\vert {\tilde {z}}_{i}(t,t_{0},z_{0}) \bigr\vert , \quad{\forall{t\geq{t_{0}}}}, i=1, 2,\ldots,n. $$

When (11) holds, choose a sufficiently small positive constant \(\rho> 0\) such that

$$ d_{i}-\rho>\sum_{j=1}^{n} \tilde{a}_{ij}{l_{j}}, \quad i=1, 2,\ldots,n. $$
(14)

Let \(\hat{z}_{i}(t)=\exp \{\rho{t} \}\tilde{z}_{i}(t)\), \({i=1, 2,\ldots,n}\). Then

$$\begin{aligned} \bigl\vert \dot{\hat{z}}_{i}(t)\bigr\vert = (-d_{i}+ \rho)\bigl\vert \hat{z}_{i}(t)\bigr\vert +\sum _{j=1}^{n}\tilde {a}_{ij}{l_{j}} \bigl\vert {\hat{z}_{j}(t-\tau_{ij})}\bigr\vert ,\quad i \in \{1,2,\ldots,n \}, \end{aligned}$$
(15)

where \(\hat{z}_{i}(t_{0})=\exp \{\rho{t_{0}} \}{z}_{i}(t_{0})\), \(i=1, 2,\ldots,n\).

Based on the work in [17], it follows that the zero solution of (15) is globally asymptotically stable if and only if the conditions (16) and (17) hold:

$$\begin{aligned}& \bigl[(-d_{i}+\rho)\delta_{ij}+\tilde{a}_{ij}{l_{j}} \bigr]_{n\times{n}} \mbox{ is a Hurwitz matrix}, \end{aligned}$$
(16)
$$\begin{aligned}& \operatorname{det} \bigl[\bigl(i\omega\delta_{ij}+(d_{i}- \rho)\bigr)\delta _{ij}-\tilde{a}_{ij}{l_{j}} \exp \{-i\omega\tau_{ij} \} \bigr]_{n\times{n}} \neq0, \quad\forall \omega\in\Re, \end{aligned}$$
(17)

where

$$\begin{aligned} \delta_{ij}=\left \{ \begin{array}{@{}l@{\quad}l} 1, & i= j, \\ 0, & i\neq j, i,j=1, 2,\ldots,n. \end{array} \right . \end{aligned}$$

Applying the Groshgorin circular disc theorem [22], by (14), it implies that (16) is true.

On the other hand,

$$\begin{aligned}[b] \vert i\omega+d_{i}-\rho \vert -\sum _{j=1}^{n}\tilde {a}_{ij}{l_{j}} \bigl\vert \exp \{-i\omega\tau_{ij} \}\bigr\vert &\geq\bigl\vert \operatorname{Re}({i\omega+d_{i}-\rho})\bigr\vert -\sum_{j=1}^{n}\tilde{a}_{ij}{l_{j}} \bigl\vert \exp \{-i\omega\tau _{ij} \}\bigr\vert \\ &=\vert d_{i}-\rho \vert -\sum_{j=1}^{n} \tilde{a}_{ij}{l_{j}}>0,\quad i=1, 2,\ldots,n. \end{aligned} $$

Combining with (14) and the Groshgorin circular disc theorem [22], (17) holds, then the zero solution of (15) is globally asymptotically stable. Meanwhile,

$$\begin{aligned} &\bigl\vert {{z}}_{i}(t,t_{0},z_{0})\bigr\vert \leq\bigl\vert {\tilde {z}}_{i}(t,t_{0},z_{0}) \bigr\vert = \exp \bigl\{ -\rho(t-t_{0}) \bigr\} \bigl\vert { \hat{z}}_{i}\bigl(t,t_{0},\exp \{\rho{t_{0}} \}z_{0}\bigr)\bigr\vert ,\\ &\quad {\forall{t\geq{t_{0}}}}, i=1, 2,\ldots,n. \end{aligned}$$

Therefore, the zero solution of (7) is globally exponentially stable, i.e., the equilibrium point \(x^{*}= (x_{1}^{*}, x_{2}^{*},\ldots, x_{n}^{*})^{T}\) of (1) is globally exponentially stable. □

When (12) holds, the proof can be presented using standard arguments similar to the proof above.

Theorem 2

The equilibrium point \(x^{*}= (x_{1}^{*}, x_{2}^{*},\ldots, x_{n}^{*})^{T}\) of neurodynamic system (1) is globally asymptotically stable, if any one of the conditions (18) and (19) holds:

$$\begin{aligned}& d_{i}\geq\sum_{j=1}^{n} \tilde{a}_{ji}{l_{i}}, \quad i=1, 2,\ldots,n, \end{aligned}$$
(18)
$$\begin{aligned}& d_{i}\geq\sum_{j=1}^{n} \tilde{a}_{ij}{l_{j}},\quad i=1, 2,\ldots,n. \end{aligned}$$
(19)

Proof

When (18) holds, consider the following functional:

$$V(t)=\sum_{i=1}^{n}\bigl\vert z_{i}(t)\bigr\vert +\sum_{i=1}^{n} \sum_{j=1}^{n}\int_{t-\tau_{ij}}^{t} \tilde{a}_{ij}{l_{j}}\bigl\vert z_{j}( \vartheta)\bigr\vert \,\mathrm{d}\vartheta. $$

Calculating the upper right Dini derivative of \(V(t)\) along the solution \(z(t,t_{0}, z_{0})\) of (7),

$$\begin{aligned} D^{+}V(t) \leq&-\sum_{i=1}^{n}d_{i} \bigl\vert z_{i}(t)\bigr\vert + \sum_{i=1}^{n} \sum_{j=1}^{n}\tilde{a}_{ij}{l_{j}} \bigl\vert {z_{j}(t-\tau _{ij})}\bigr\vert +\sum _{i=1}^{n}\sum_{j=1}^{n} \tilde{a}_{ij}{l_{j}}\bigl\vert z_{j}(t)\bigr\vert \\ &{}- \sum_{i=1}^{n}\sum _{j=1}^{n}\tilde{a}_{ij}{l_{j}} \bigl\vert {z_{j}(t-\tau _{ij})}\bigr\vert \\ =& -\sum_{i=1}^{n}d_{i}\bigl\vert z_{i}(t)\bigr\vert +\sum_{i=1}^{n} \sum_{j=1}^{n}\tilde{a}_{ij}{l_{j}} \bigl\vert z_{j}(t)\bigr\vert \\ =&\sum_{i=1}^{n} \Biggl[-d_{i}+ \sum_{j=1}^{n}\tilde{a}_{ji}{l_{i}} \Biggr]\bigl\vert z_{i}(t)\bigr\vert \\ \leq&0. \end{aligned}$$

This implies that the zero solution of (7) is globally asymptotically stable, i.e., the equilibrium point \(x^{*}= (x_{1}^{*}, x_{2}^{*},\ldots, x_{n}^{*})^{T}\) of (1) is globally asymptotically stable.

When (19) holds,

$$ (d_{i}-\tilde{a}_{ii}{l_{i}})-\sum _{j=1,j\neq{i}}^{n}\tilde {a}_{ij}{l_{j}} \geq0,\quad i=1, 2,\ldots,n. $$
(20)

Let

$$\gamma_{i}=(d_{i}-\tilde{a}_{ii}{l_{i}})- \sum_{j=1,j\neq {i}}^{n}\tilde{a}_{ij}{l_{j}} \geq0,\quad i\in \{1, 2,\ldots,n \}, $$

thus

$$(d_{i}-\tilde{a}_{ii}{l_{i}})- \gamma_{i}=\sum_{j=1,j\neq {i}}^{n} \tilde{a}_{ij}{l_{j}},\quad i\in \{1, 2,\ldots,n \}, $$

then there must exist constants \(\sigma_{i}>0\) (\(i=1, 2,\ldots,n\)) such that

$$ \bigl[(d_{j}-\tilde{a}_{jj}{l_{j}})- \gamma_{j}\bigr]\sigma_{j}-\sum_{i=1,i\neq {j}}^{n} \tilde{a}_{ij}{l_{j}}\sigma_{i}=0,\quad j=1, 2, \ldots,n. $$
(21)

The coefficient matrix of (21) is singular, and combining with (20), the existence of \(\sigma_{i}>0\) (\(i=1, 2,\ldots,n\)) can be understood. In fact, some of the references concerning this claim can be found in Liao et al. [21].

From (21),

$$ (d_{j}-\tilde{a}_{jj}{l_{j}}) \sigma_{j}-\sum_{i=1,i\neq {j}}^{n} \tilde{a}_{ij}{l_{j}}\sigma_{i}= \gamma_{j}\sigma_{j}\geq0,\quad j=1, 2,\ldots,n, $$
(22)

that is,

$$ d_{j}-\sum_{i=1}^{n} \tilde{a}_{ij}{l_{j}}\sigma_{i} \frac{1}{\sigma _{j}}=\gamma_{j}\geq0,\quad j=1, 2,\ldots,n. $$
(23)

Make a full rank linear transform:

$$w(t)=\operatorname{diag}(\sigma_{1},\sigma_{2},\ldots, \sigma_{n}){z(t)}. $$

From (10), it follows that

$$\begin{aligned} \bigl\vert \dot{w}_{i}(t)\bigr\vert \leq -d_{i}\bigl\vert w_{i}(t)\bigr\vert +\sum_{j=1}^{n}{ \sigma_{i}}\tilde {a}_{ij}{l_{j}} \frac{1}{\sigma_{j}}\bigl\vert {w_{j}(t-\tau_{ij})}\bigr\vert ,\quad i\in \{1,2,\ldots,n \}. \end{aligned}$$
(24)

Consider the following functional:

$$\tilde{V}(t)=\sum_{i=1}^{n}\bigl\vert w_{i}(t)\bigr\vert +\sum_{i=1}^{n} \sum_{j=1}^{n}\int_{t-\tau_{ij}}^{t} \sigma_{i}\tilde{a}_{ij}{l_{j}} \frac{1}{\sigma_{j}}\bigl\vert w_{j}(\vartheta)\bigr\vert \,\mathrm{d} \vartheta. $$

Calculating the upper right Dini derivative of \(\widetilde{V}(t)\) along the trajectory of (24),

$$\begin{aligned} D^{+}\widetilde{V}(t)\leq{}&{-}\sum_{i=1}^{n}d_{i} \bigl\vert w_{i}(t)\bigr\vert + \sum_{i=1}^{n} \sum_{j=1}^{n}\sigma_{i} \tilde{a}_{ij}{l_{j}}\frac {1}{\sigma_{j}}\bigl\vert {w_{j}(t-\tau_{ij})}\bigr\vert \\ &{}+\sum_{i=1}^{n}\sum _{j=1}^{n}\sigma_{i}\tilde{a}_{ij}{l_{j}} \frac {1}{\sigma_{j}}\bigl\vert w_{j}(t)\bigr\vert - \sum _{i=1}^{n}\sum_{j=1}^{n} \sigma_{i}\tilde{a}_{ij}{l_{j}}\frac {1}{\sigma_{j}} \bigl\vert {w_{j}(t-\tau_{ij})}\bigr\vert \\ ={}&-\sum_{j=1}^{n}d_{j}\bigl\vert w_{j}(t)\bigr\vert +\sum_{i=1}^{n} \sum_{j=1}^{n}\sigma_{i} \tilde{a}_{ij}{l_{j}}\frac{1}{\sigma_{j}}\bigl\vert w_{j}(t)\bigr\vert \\ ={}&\sum_{j=1}^{n} \Biggl[-d_{j}+ \sum_{i=1}^{n}\sigma_{i}\tilde {a}_{ij}{l_{j}}\frac{1}{\sigma_{j}} \Biggr]\bigl\vert w_{j}(t)\bigr\vert \\ \leq{}&0. \end{aligned}$$

This implies that the zero solution of (7) is globally asymptotically stable, i.e., the equilibrium point \(x^{*}= (x_{1}^{*}, x_{2}^{*},\ldots, x_{n}^{*})^{T}\) of (1) is globally asymptotically stable. □

Remark 4

The question remains, in fact, what the significance is of the physical parameters of a memristive neurodynamic system to stability. Theorems 1 and 2 make use of the physical parameters of neurodynamic system (1) when proving the stability. These algebraical criteria, which only depend on the parameters of neurodynamic system (1), are easily verified. Theoretically, such stability criteria are very useful because they provide some more efficient ways in analysis and design of a memristive neurodynamic system.

Remark 5

By Theorem 1, we can see that neurodynamic system (1) is globally exponentially stable, if

$$d_{i}>\sum_{j=1}^{n} \tilde{a}_{ij}{l_{j}},\quad i=1, 2,\ldots,n, \quad \mbox{or}\quad d_{i}>\sum_{j=1}^{n} \tilde{a}_{ji}{l_{i}}, \quad i=1, 2,\ldots,n. $$

From Theorem 2, we know that neurodynamic system (1) is globally asymptotically stable, if

$$d_{i}\geq\sum_{j=1}^{n} \tilde{a}_{ji}{l_{i}},\quad {i=1, 2,\ldots,n}, \quad \mbox{or}\quad d_{i}\geq\sum_{j=1}^{n} \tilde{a}_{ij}{l_{j}}, \quad {i=1, 2,\ldots,n}. $$

Clearly, the critical conditions

$$d_{i}=\sum_{j=1}^{n} \tilde{a}_{ji}{l_{i}},\quad {i=1, 2,\ldots,n}, \quad \mbox{or}\quad d_{i}=\sum_{j=1}^{n} \tilde{a}_{ij}{l_{j}},\quad {i=1, 2,\ldots,n}, $$

reflect the critical dynamics of memristive neurodynamic system (1).

Remark 6

Recently, some delay-independent stability criteria [10, 12, 13] and delay-dependent stability criteria [14] for memristive neurodynamic systems were reported. Generally, delay-dependent stability criteria are less conservative than delay-independent stability criteria. In order to easily check the stability criteria, stability conditions could best be directly derived from the parameters of the system. Theorems 1 and 2 of this paper, which can be directly derived from the network parameters (without any external parameters), are easily testable. These new results can bring about convenience in applications.

4 Illustrative examples

In this section, two numerical examples are given to illustrate the effectiveness of the obtained algebraical criteria.

Example 1

Consider a memristive neural network model,

$$ \left \{ \begin{array}{@{}l} \dot{x}_{1}(t)= - x_{1}(t) + a_{11}(x_{1}(t))f_{1}(x_{1}(t-1)) + a_{12}(x_{1}(t))f_{2}(x_{2}(t-2))+0.3, \\ \dot{x}_{2}(t)= - x_{2}(t) + a_{21}(x_{2}(t))f_{1}(x_{1}(t-1)) + a_{22}(x_{2}(t))f_{2}(x_{2}(t-2))+ 0.5, \end{array} \right . $$
(25)

where \(f_{i}(\upsilon)=\frac{\vert \upsilon+1\vert -\vert \upsilon-1\vert }{2}\),

$$\begin{aligned}& a_{11}\bigl(x_{1}(t)\bigr) =\left \{ \begin{array}{@{}l@{\quad}l} {0.4}, & {x_{1}(t)>0}, \\ {0.3}, & {x_{1}(t)< 0}, \end{array} \right .\qquad a_{12}\bigl(x_{1}(t)\bigr)=\left \{ \begin{array}{@{}l@{\quad}l} {0.4}, & {x_{1}(t)>0}, \\ {0.3}, & {x_{1}(t)<0}, \end{array} \right . \\& a_{21}\bigl(x_{2}(t)\bigr)=\left \{ \begin{array}{@{}l@{\quad}l} {0.3}, & {x_{2}(t)>0}, \\ {0.2}, & {x_{2}(t)<0}, \end{array} \right .\qquad a_{22}\bigl(x_{2}(t)\bigr)=\left \{ \begin{array}{@{}l@{\quad}l} {0.3}, & {x_{2}(t)>0}, \\ {0.2}, & {x_{2}(t)<0}. \end{array} \right . \end{aligned}$$

Clearly,

$$\begin{aligned}& d_{1}-\sum_{j=1}^{2} \tilde{a}_{1j}=0.2>0, \end{aligned}$$
(26)
$$\begin{aligned}& d_{2}-\sum_{j=1}^{2} \tilde{a}_{2j}=0.4>0, \end{aligned}$$
(27)

the criterion condition (11) in Theorem 1 is satisfied. Figure 1 shows the time-domain behavior of the state variables \(x_{1}\) and \(x_{2}\) of (25).

Figure 1
figure 1

Transient behaviors of trajectories of system ( 25 ).

Full size image

Example 2

Consider a memristive neural network model,

$$ \left \{ \begin{array}{@{}l} \dot{x}_{1}(t)= - x_{1}(t) + a_{11}(x_{1}(t))f_{1}(x_{1}(t-1)) + a_{12}(x_{1}(t))f_{2}(x_{2}(t-2))+3, \\ \dot{x}_{2}(t)= - x_{2}(t) + a_{21}(x_{2}(t))f_{1}(x_{1}(t-1)) + a_{22}(x_{2}(t))f_{2}(x_{2}(t-2))+ 5, \end{array} \right . $$
(28)

where \(f_{i}(\upsilon)=\frac{\exp \{\upsilon \}-\exp \{ -\upsilon \}}{\exp \{\upsilon \}+\exp \{ -\upsilon \}}\),

$$\begin{aligned}& a_{11}\bigl(x_{1}(t)\bigr)=\left \{ \begin{array}{@{}l@{\quad}l} {0.7}, & {x_{1}(t)>0}, \\ {0.3}, & {x_{1}(t)< 0}, \end{array} \right .\qquad a_{12}\bigl(x_{1}(t)\bigr)=\left \{ \begin{array}{@{}l@{\quad}l} {0.3}, & {x_{1}(t)>0}, \\ {0.1}, & {x_{1}(t)<0}, \end{array} \right . \\& a_{21}\bigl(x_{2}(t)\bigr)=\left \{ \begin{array}{@{}l@{\quad}l} {0.6}, & {x_{2}(t)>0}, \\ {0.4}, & {x_{2}(t)<0}, \end{array} \right .\qquad a_{22}\bigl(x_{2}(t)\bigr)=\left \{ \begin{array}{@{}l@{\quad}l} {0.4}, & {x_{2}(t)>0}, \\ {0.2}, & {x_{2}(t)<0}. \end{array} \right . \end{aligned}$$

Clearly,

$$\begin{aligned}& d_{1}-\sum_{j=1}^{2} \tilde{a}_{1j}=0, \end{aligned}$$
(29)
$$\begin{aligned}& d_{2}-\sum_{j=1}^{2} \tilde{a}_{2j}=0, \end{aligned}$$
(30)

the criterion condition (19) in Theorem 2 is satisfied. Figure 2 depicts the time-domain behavior of the state variables \(x_{1}\) and \(x_{2}\) of (28).

Figure 2
figure 2

Transient behaviors of trajectories of system ( 28 ).

Full size image

5 Concluding remarks

The memristive neurodynamic system exhibits well-characterized analog switching effects in electrical characteristics for devices formed by multiport architectures. Rigorously analyzing the nonlinear dynamics of memristive neurodynamic system is very essential for its neuromorphic computing applications. This paper is concerned with the global exponential stability and global asymptotical stability for a class of delayed memristive neural networks. The newly proposed approach is based on the comparison method and stability theory within a nonsmooth analysis framework. Quite differently, these stability criteria are formulated via some simple algebraical inequalities. The characteristic analysis of memristive neurodynamic systems might also reveal the electrical behavior of analog memristor devices.