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Design disturbance attenuating controller for memristive recurrent neural networks with mixed timevarying delays
Advances in Difference Equations volume 2018, Article number: 189 (2018)
Abstract
This paper investigates the design of disturbance attenuating controller for memristive recurrent neural networks (MRNNs) with mixed timevarying delays. By applying the combination of differential inclusions, setvalued maps and Lyapunov–Razumikhin, a feedback control law is obtained in the simple form of linear matrix inequality (LMI) to ensure disturbance attenuation of memristorbased neural networks. Finally, a numerical example is given to show the effectiveness of the proposed criteria.
1 Introduction
It is well known that the neural networks are so important that they have been widely applied in various areas such as reconstructing moving images, signal processing, pattern recognition, optimization problems and so on (for reference, see [1–48]). During the recent years, more and more researchers have paid attention to a new model named statedependent switching recurrent neural networks whose connection weights vary due to their states. Generally speaking, such switching neural networks have been entitled memristive neural networks or memristorbased neural networks. Therefore, let us recall the brief development of memristive neural networks in the following. In 1971, Dr. Chua (see [10]) firstly advised that a fourth basic circuit element should exist. Different from the other three elements—the resistor, the inductor, and the capacitor—the fourth one was named the memristor. According to Chua’s theory, the memristor must have important and distinctive ability. Precisely, with the rapid development of science, a prototype of the memristor had been built by some scientists from HP Labs until 2008 (see [11]). The memristor, which not only shares many properties of resistors but also shares the same unit of measurement, is a twoterminal element whose characteristic lies in its variable resistance called memristance. Memristance, depending on how much electric charge has been passed through the memristor in a special direction, is its distinctive ability. The ability contributes to its memorizing the passed quantity of electric charge. Therefore, since 2008, its potential applications have become more and more popular in many aspects such as generation computer, powerful brainlike neural computer, and so on. There is no doubt that it has initiated the worldwide concern with the emergence of the memristor (see [11–30]). For the neural networks, the first job is considering whether they are stable or not. Therefore, a lot of scholars have studied the memristive neural networks’ multitudinous stability such as asymptotical stability, global stability, and exponential stability (see [12, 13, 15, 17–20]). Moreover, as far as we know, the passivity theory plays an important role in the analysis of the stability of dynamical systems, nonlinear control, and other areas. Thus, some researchers have investigated passivity or dissipativity criteria on MRNNs (see [14, 16, 23–25, 27–30]).
On the other hand, neural networks with timevarying delays are unavoidable to subject to persistent disturbance. How to solve persistent disturbance for delayed neural networks is still an open problem. Therefore, He et al. [31] studied the problem of disturbance attenuating controller design for delayed cellular neural networks (DCNNs). In this paper, authors designed a feedback control law to guarantee disturbance attenuation for DCNNs by employing Lyapunov–Razumikhin theorem. However, firstly, this paper just discussed disturbance attenuation for delayed cellular neural networks, so the activation function was assumed only to be \(f(x(\cdot ))=0.5(x(\cdot)+1x(\cdot)1)\). As is well known to us, there are still Hopfield neural networks, except cellular neural networks. Both of them belong to recurrent neural networks. Thus, how to design disturbance attenuating controller for general neural networks is our first motivation. Secondly, it is noted that the results in this paper were derived for systems only with discrete delays. Another type of time delay is distributed delay. Systems with distributed delay can be applied in the modeling of feeding systems and combustion chambers in a liquid monopropellant rocket motor with pressure feeding. So, how to solve the persistent disturbance for delayed neural networks with both discrete and distributed timevarying delays remains some room to certain extent.
Motivated by the above mentioned discussion, the problem of disturbance attenuating controller design is extended for memristorbased neural networks. To the best of our knowledge, there has not been any paper to discuss the disturbance attenuating controller design for MRNNs, which motivates our study. Our objective is to give an effective feedback control law to ensure disturbance attenuation and obtain a description of the bounded attractor set for MRNNs with mixed timevarying delays. The main contribution of this paper lies in the following aspects: first of all, this paper is the first one to investigate the disturbance attenuating controller for MRNNs, which is sure to strengthen the systematic research theory for MRNNs and must further enrich the basis of application for MRNNs. Then, comparing to the existing paper [31] about the disturbance attenuating controller design, the studied systems not only contain the more general activation functions but also include both discrete timevarying delay and distribute timevarying delays;a feedback control law is designed in the simple form of linear matrix inequality (LMI) to ensure disturbance attenuation of memristorbased neural networks by employing multiple theories such as differential inclusions, setvalued maps, and Lyapunov–Razumikhin.
2 Problem statement and preliminaries
Throughout this paper, solutions of all the systems considered in the following are intended in Filippov’s sense (see [1, 36]). \([\cdot ,\cdot]\) represents the interval. The superscripts ‘−1’ and ‘T’ stand for the inverse and transpose of a matrix, respectively. \(P>0\) (\(P\geqslant0\), \(P<0\), \(P\leqslant0\)) means that the matrix P is symmetric positive definite (positivesemi definite, negative definite, and negativesemi definite). \(\Vert\cdot\Vert\) refers to the Euclidean vector norm. \(R^{n}\) denotes an ndimensional Euclidean space. \(\mathcal{C}([\rho,0],R^{n})\) represents a Banach space of all continuous functions. \(R^{{m}\times{n}}\) is the set of \(m\times n\) real matrices. ∗ denotes the symmetric block in a symmetric matrix. For matrices \(\mathcal{M}=(m_{ij})_{m\times{n}}\), \(\mathcal {N}=(n_{ij})_{m\times{n}}\), \(\mathcal{M}\gg\mathcal{N}\) (\(\mathcal{M}\ll \mathcal{N}\)) means that \(m_{ij}\gg{n}_{ij}\) (\(m_{ij}\ll{n}_{ij}\)) for \(i=1,2,\ldots,m\), \(j=1,2,\ldots,n\). And by the interval matrix \([\mathcal {M},\mathcal{N}]\), it follows that \(\mathcal{M}\ll\mathcal{N}\). For \(\forall\mathcal{L}=(l_{ij})_{m\times{n}}\in[\mathcal{M},\mathcal{N}]\), it means \(\mathcal{M}\ll\mathcal{L}\ll\mathcal{N}\), i.e., \(m_{ij}\lll _{ij}\ll{n}_{ij}\) for \(i=1,2,\ldots,m\), \(j=1,2,\ldots,n\). \(\operatorname{co}\{\Pi_{1},\Pi _{2}\}\) denotes the closure of the convex hull generated by real numbers \(\Pi_{1}\) and \(\Pi_{2}\). Let \(\bar{a}_{i}=\max\{\hat{a}_{i},\check{a}_{i}\} \), \(\underline{a}_{i}=\min\{\hat{a}_{i},\check{a}_{i}\}\), \(\bar {b}_{ij}=\max\{\hat{b}_{ij},\check{a}_{ij}\}\), \(\underline{b}_{ij}=\min \{\hat{b}_{ij},\check{b}_{ij}\}\), \(\bar{c}_{ij}=\max\{\hat {c}_{ij},\check{c}_{ij}\}\), \(\underline{c}_{ij}=\min\{\hat {c}_{ij},\check{c}_{ij}\}\), \(\bar{d}_{ij}=\max\{\hat{d}_{ij},\check {d}_{ij}\}\), \(\underline{d}_{ij}=\min\{\hat{d}_{ij},\check{d}_{ij}\}\). Matrix dimensions, if not explicitly stated, are assumed to be compatible with algebraic operations.
In this section, by Krichoff’s current law, a general class of memristorbased recurrent neural networks containing both persistent disturbances and mixed timevarying delays is introduced as follows:
where \(x_{i}(t)\) represents the voltage of the capacitor \(C_{i}\), \(f_{i}(x_{i}(t))\in {R}^{n}\) is the nonlinear activation function, \(u_{i}(t)\) is the input, \(h_{l}(t)\) (\(l=1,2,\ldots,m\)) is the bounded disturbance. \(\tau_{i}(t)\) is the discrete timevarying delay, and \(\rho_{i}(t)\) is the distributed delay. They satisfy the following conditions: \(0\leq\tau_{i}(t)\leq\tau\), \(0\leq \rho_{i}(t)\leq\rho\) (τ and ρ are constants). \(\phi_{i}(t)\) is the initial condition and is bounded and continuously differential on \([\delta,0]\) (\(\delta=\max\{\tau,\rho\}\)). \(g_{il}\) describes the weighting coefficients of the disturbance. \(\breve{a}_{i}\) describes the rate with which each neuron will reset its potential to the resting state in isolation when disconnected from the networks and external inputs. \(\breve{b}_{ij}\), \(\breve{c}_{ij}\), and \(\breve{d}_{ij}\) represent the element of the connection weight matrix, the discretely delayed connection weight matrices, and the distributed delays, respectively. They satisfy the following conditions:
in which switching jump \(T_{i}>0\), \(\hat{d}_{ij}\), \(\check{d}_{ij}\), \(\hat {b}_{ij}\), \(\check{b}_{ij}\), \(\hat{c}_{ij}\), \(\check{c}_{ij}\), \(\hat {a}_{i}\), \(\check{a}_{i}\), \(i,j=1,2,\ldots,n\), are all constant numbers.
Remark 2.1
The clear exposition about the relation between memristances and coefficients of switching system (1) has been given in the works [12, 18]. Thus, researchers can consult [12, 18] to get more information.
From the above description, the studied networks are statedependent switching recurrent neural networks whose connection weights vary according to their states. To translate these statedependent neural networks into the general ones, the next definitions are necessary.
Definition 2.1
Let \(E\subseteq{R}^{n}\), \(x\mapsto {F}(x)\) is called a setvalued map from \(E\hookrightarrow{R}^{n}\) if, for each point x of a set \(E\subseteq{R}^{n}\), there corresponds a nonempty set \(F(x)\subseteq{R}^{n}\).
Definition 2.2
A setvalued map F with nonempty values is said to be upper semicontinuous at \(x_{0}\in{E}\subseteq{R}^{n}\) if, for any open set N containing \(F(x_{0})\), there exists a neighborhood M of \(x_{0}\) such that \(F(M)\subseteq{N}\). \(F(x)\) is said to have a closed (convex, compact) image if, for each \(x\in{E}\), \(F(x)\) is closed (convex, compact).
Definition 2.3
For the differential system \(\frac {dx}{dt}=f(t,x)\), where \(f(t,x)\) is discontinuous in x, the setvalued map of \(f(t,x)\) is defined as follows:
where \(B(x,\epsilon)=\{y: \Vert yx \Vert\leq\epsilon\}\) is the ball of center x and radius ϵ. Intersection is taken over all sets N of measure zero and over all \(\epsilon>0\); and \(\mu(N)\) is the Lebesgue measure of set N.
A Filippov solution of system (1) with initial condition \(x(0)=x_{0}\) is absolutely continuous on any subinterval \(t\in[t_{1},t_{2}]\) of \([0,T]\), which satisfies \(x(0)=x_{0}\), and the differential inclusion:
Firstly, by employing the theories of differential inclusions and setvalued maps, from (1), it follows that
or equivalently, for \(i,j=1,2,\ldots,n\), there exist \(a_{i}\in\operatorname{co}\{\hat {a}_{i},\check{a}_{i}\}\), \(b_{ij}\in\operatorname{co}\{\hat{b}_{ij},\check{b}_{ij}\} \), \(c_{ij}\in\operatorname{co}\{\hat{c}_{ij},\check{c}_{ij}\}\), and \(d_{ij}\in\operatorname{co}\{ \hat{d}_{ij},\check{d}_{ij}\}\) such that
Clearly, \(\operatorname{co}\{\hat{a_{i}},\check{a_{i}}\}=[\bar{a},\underline{a}]\), \(\operatorname{co}\{ \hat{b}_{ij},\check{b}_{ij}\}=[\bar{b}_{ij},\underline{b}_{ij}]\), \(\operatorname{co}\{ \hat{c}_{ij},\check{c}_{ij}\}=[\bar{c}_{ij},\underline{c}_{ij}]\), \(\operatorname{co}\{ \hat{d}_{ij},\check{d}_{ij}\}=[\bar{d}_{ij},\underline{d}_{ij}]\) for \(i,j=1,2,\ldots,n\).
A solution \(x(t)=[x_{1}(t),x_{2}(t),\ldots,x_{n}(t)]^{T}\in{R}^{n}\)(in the sense of Filippov) of system (1) is absolutely continuous on any compact interval of \([0,+\infty]\), and for \(i=1,2,\ldots,n\),
For convenience, transform (1) into the compact form as follows:
or equivalently, there exist \(A^{\ast}\in\operatorname{co}\{\hat{A},\check{A}\}\), \(B^{\ast}\in\operatorname{co}\{\hat{B},\check{B}\}\), \(C^{\ast}\in\operatorname{co}\{\hat{C},\check {C}\}\), and \(D^{\ast}\in\operatorname{co}\{\hat{D},\check{D}\}\) such that
where \(C^{\ast}=C(x)\), \(A^{\ast}=A(x)\), \(B^{\ast}=B(x)\), \(D^{\ast}=D(x)\), \(\hat{A}=(\hat{a}_{i})_{n\times{n}}\), \(\hat{B}=(\hat{b}_{ij})_{n\times {n}}\), \(\hat{C}=(\hat{c}_{ij})_{n\times{n}}\), \(\hat{D}=(\hat {d}_{ij})_{n\times{n}}\), \(\check{A}=(\check{a}_{i})_{n\times{n}}\), \(\check{B}=(\check {b}_{ij})_{n\times{n}}\), \(\check{C}=(\check{c}_{ij})_{n\times{n}}\), \(\check{D}=(\check{d}_{ij})_{n\times{n}}\), \(G=(g_{il})_{n\times{m}}\), \(x(t)=[x_{1}(t),x_{2}(t),\ldots,x_{n}(t)]^{T}\in{R}^{n}\), \(f(x(t))=[f_{1}(x_{1}(t)),f_{2}(x_{2}(t)), \ldots,f_{n}(x_{n}(t))]^{T}\in{R}^{n}\), \(f(x(t\tau(t)))=[f_{1}(x_{1}(t\tau_{1}(t))),f_{2}(x_{2}(t\tau_{2}(t))), \ldots ,f_{n}(x_{n}(t\tau_{n}(t)))]^{T}\in{R}^{n}\), \(y(t)=[y_{1}(t),y_{2}(t),\ldots ,y_{n}(t)]^{T}\in{R}^{n}\), \(u(t)=[u_{1}(t),u_{2}(t),\ldots,u_{n}(t)]^{T}\in{R}^{n}\). The bounded disturbance \(h(t)=[h_{1}(t),h_{2}(t),\ldots, h_{n}(t)]^{T}\in{R}^{n}\) is assumed to belong to the set \(\mathcal{H}=\{hh^{T}h\leq1\}\).
Clearly, \(\operatorname{co}\{\hat{D},\check{D}\}=[\bar{D},\underline{D}]\), \(\operatorname{co}\{\hat {A},\check{A}\}=[\bar{A},\underline{A}]\), \(\operatorname{co}\{\hat{B},\check{B}\}=[\bar {B},\underline{B}]\), \(\operatorname{co}\{\hat{C},\check{C}\}=[\bar{C},\underline {C}]\), where \(\bar{A}=(\bar{a_{i}})_{n\times{n}}\), \(\underline{A}=(\underline {a}_{i})_{n\times{n}}\), \(\bar{B}=(\bar{b}_{ij})_{n\times{n}}\), \(\underline {B}=(\underline{b}_{ij})_{n\times{n}}\), \(\bar{C}=(\bar{c}_{ij})_{n\times {n}}\), \(\underline{C}=(\underline{c}_{ij})_{n\times{n}}\), \(\bar{D}=(\bar {d}_{ij})_{n\times{n}}\), \(\underline{D}=(\underline{d}_{ij})_{n\times{n}}\).
Let \(C=\frac{\underline{C}+\bar{C}}{2}\), \(A=\frac{\underline{A}+\bar {A}}{2}\), \(B=\frac{\underline{B}+\bar{B}}{2}\), \(D=\frac{\underline {D}+\bar{D}}{2}\), ∀ \(A^{\ast}\in\operatorname{co}\{\hat{A},\check{A}\}\), \(B^{\ast}\in\operatorname{co}\{ \hat{B},\check{B}\}\), \(C^{\ast}\in\operatorname{co}\{\hat{C},\check{C}\}\), \(D^{\ast}\in \operatorname{co}\{\hat{D},\check{D}\}\), \(A^{\ast}=A+\Delta{A}(t)\), \(B^{\ast}=B+\Delta {B}(t)\), \(C^{\ast}=C+\Delta{C}(t)\), \(D^{\ast}=D+\Delta{D}(t)\), (4) can be described as follows:
Moreover, if \(\hat{a}_{i}=\check{a}_{i}\), \(\hat{c}_{ij}=\check {c}_{ij}\), \(\hat{b}_{ij}=\check{b}_{ij}\), \(\hat{d}_{ij}=\check{d}_{ij}\) (\(i,j=1,2,\ldots,n\)), (5) can be expressed as follows:
Suppose the state feedback to be \(u=Fx\), then system (6) is changed into
Moreover, throughout this paper, the neuron activation functions are assumed to satisfy the following assumption.
Assumption 2.1
The neuron activation function \(f(x(t))\) satisfies
where \(l_{j}>0\) is a known real constant.
To get the main results in this paper, the definition of disturbance attenuation is introduced as follows.
Definition 2.4
Given system (6), the controller \(u=Fx\) is called disturbance attenuating if systems (7) satisfy the following conditions:

(1)
When \(h(t)=0\), systems (7) are globally asymptotically stable;

(2)
When \(h(t)\neq0\), there exists a bounded attractor for systems (7).
Remark 2.2
The attractor of systems (7) is the invariant set Ω, which not only lies in the fact that all the trajectories beginning from it will retain in it for any \(h\in {\mathcal{H}}\), but also subjects to the condition that any trajectories beginning from outside the set will ultimately go into the set for any \(h\in\mathcal{H}\).
To establish the feedback controller for systems (7), the following lemmas will be used in this paper.
Lemma 2.1
(Lyapunov–Razumikhin theorem [35])
Consider the following functional differential equation:
Assume that \(\phi\in{C}_{n,\tau}\) and the map \(f(\phi):{C}_{n,\tau }\mapsto{R}^{n}\) is continuous and Lipschitzian in ϕ and \(f(0)=0\). Suppose that \(u(s)\), \(\nu(s)\), \(w(s)\), and \(p(s)\in{R}^{+}\mapsto {R}^{+}\) are scalar, continuous, and nondecreasing functions, \(u(s)\), \(\nu(s)\), \(w(s)\) positive for \(s>0\), \(u(0)=\nu(0)=0\) and \(p(s)>s\) for \(s>0\). If there are a continuous function \(V:R^{n}\mapsto{R}\) and a positive number ρ such that, for all \(x_{t}\in{M}_{V(\rho)}:=\{\phi \in{C}_{n,\tau}:V(\phi(\theta))\leq\rho,\forall\theta\in[\tau,0]\}\), the following conditions hold:

(1)
\(u( \Vert x \Vert )\leq{V}(x)\leq\nu( \Vert x \Vert )\);

(2)
\(\dot{V}(x(t))\leqw( \Vert x \Vert )\), if \(V(x(t+ \theta ))< p (V (x(t) ) )\).
Then the solution \(x(t)\equiv0\) of (9) is asymptotically stable. Moveover, the set \({M}_{V(\rho)}\) is an invariant set inside the domain of attraction. Further, if \(u(s)\rightarrow\infty\) as \(s\rightarrow\infty\), then the solution \(x(t)\equiv0\) of (9) is globally stable.
Lemma 2.2
([27])
Let H, E, and \(G(t)\) be real matrices of appropriate dimensions with \(G(t)\) satisfying \(G(t)^{T}G(t)\leq{I}\). Then, for any scalar \(\varepsilon>0\),
3 Main results
In this paper, the disturbance attenuation is investigated for memristive recurrent neural networks with mixed timevarying delays. According to Definition 2.4, the condition is constructed for the global asymptotic stability of systems (7) when \(h(t)=0\). Secondly, it is proved that there exists a bounded attractor for systems (7) when \(h(t)\neq0\). For convenience, denote \(L=\operatorname{diag}\{ l_{1},l_{2},\ldots,l_{n}\}\).
Theorem 3.1
Under Assumption 2.1, the memristive neural network (7) with \(h(t)=0\) under a disturbance attenuating controller \(u(t)=Fx(t)\) is asymptotically stable if there exist matrices \(Q>0\), F, positive constants \(\varepsilon_{i}\) (\(i=0,1,2\)), and any given positive constant ϵ such that the following inequality holds:
Proof
Consider the following Lyapunov–Razumikhin function candidate:
Taking the timederivative of \(V(t)\) along the solution of (7) when \(h(t)=0\), the timederivative of \(V(t)\) is
If there exist positive constants \(\varepsilon_{i}\) (\(i=0,1,2\)), by employing Lemma 2.2, it is easy to obtain
Moreover, according to Assumption 2.1, it is not difficult to get
Suppose \(p(s)=r\cdot{s}\) with \(r>1\) in Lemma 2.1, it is easy to get that \(p(s)>s\) for \(s>0\). Due to the condition \(V(x(\theta))\leq {p}(V(x(t)))\), \(\theta\in[t\tau(t),t]\), that is, \(x^{T}(\theta)Qx(\theta )\leq{p}x^{T}(t)Qx(t)\). Thus, it is obvious that
In addition, choosing the controller to be \(u(t)=Fx(t)\), it is easy to obtain
Because (11) holds, \(r>1\) is chosen to guarantee
Thus, combining (13) with (14), it is not difficult to obtain
It is obvious that \(\dot{V}(t)\) is negative definite. According to Lyapunov stability theory, systems (7) when \(h(t)=0\) are asymptotically stable. This completes our first step. Next, when \(h(t)\neq0\), it is proved that there really exists a bounded attractor for systems (7). □
Theorem 3.2
Under Assumption 2.1, the memristive neural network (7) with \(h(t)\neq0\) has a disturbance attenuating controller \(u(t)=Fx(t)\) with an attractor as \(\Phi=\{ xx^{T}Qx\leq1\}\) if there exist matrices \(Q>0\), M, F, positive constants \(\varepsilon_{i}\) (\(i=0,1,2\)), and any given positive constant ϵ such that the following inequality holds:
where
Proof
Consider the same Lyapunov–Razumikhin function candidate:
Taking the timederivative of \(V(t)\) along the solution of (7) when \(h(t)\neq0\), the timederivative of \(V(t)\) is
After the same discussion as that in Theorem 3.1, let \(M=QF\), it is easy to obtain
Applying Lemma 2.2 to the term \(2x^{T}(t)QGh(t)\), for the given positive ϵ, it is easy to get
Because the bounded disturbances are assumed to belong to the set \(\mathcal{H}=\{hh^{T}h\leq1\}\), it is easy to obtain
Thus, it follows that
Applying the Schur complement to (16), it is equivalent to
By choosing \(r>1\), it implies that
Thus, it follows that
Obviously, \(\dot{V}(t)\) is negative outside the set Φ, the trajectories beginning from outside the set Φ will ultimately access the set Φ for any \(h\in\mathcal{H}\). Therefore, Φ is the invariant set of systems (7). So far, condition (2) of disturbance attenuation has been constructed. Meanwhile, the disturbance attenuating controller \(u(t)=Fx(t)\) with an attractor as \(\Phi=\{xx^{T}Qx\leq1\}\) has been designed. This completes our proof. □
Remark 3.1
In comparison to the published paper [31], our paper’s contribution lies in three aspects: Firstly, the studied memristive neural networks are more popular at present; secondly, the activation function is not needed to be strict to be \(f(x(\cdot))=0.5(x(\cdot)+1x(\cdot)1)\), but rather it is relaxed to just satisfy Lipschitz conditions; thirdly, the discussed model not only contains discrete timevarying delay but also includes distributed timevarying delay. Therefore, our results are more general to be well applied.
Remark 3.2
Recently, many scholars have studied different kinds of control theories about MRNNS such as exponential synchronization control [12], finitetime synchronization control [13], exponential lag adaptive synchronization control [18], lag synchronization control [19], and so on. However, to the best of our knowledge, there has not been any paper to discuss the disturbance attenuating controller design for MRNNs. This paper is the first one to investigate the disturbance attenuating controller for MRNNs, which is sure to strengthen the systematic research theory for MRNNs and must further enrich the basis of application for MRNNs.
4 Numerical examples
In this section, one example is presented to demonstrate the effectiveness of our results.
Example 4.1
Consider a twoneuron memristive neural network containing both persistent disturbances and mixed timevarying delays:
where
Meanwhile, the discrete timevarying delay is assumed to be \(\tau (t)=1+0.4\operatorname{sin}(5t)\), and the distributed timevarying delay is supposed to be \(\rho(t)=0.81\operatorname{cos}(t)\). In addition, the activation functions are assumed to be \(f_{i}(x_{i})=0.5(x_{i}+1x_{i}1)\) (\(i=1,2\)). Moreover, the disturbance \(h(t)=[0.03\operatorname{cos}t;0.02\operatorname{sin}t]\). Particularly, if we choose that \(\varepsilon_{0}=\varepsilon_{1}=\varepsilon_{1}=\epsilon=1\), by solving LMI (16), we obtain
Figure 1 demonstrates the state trajectories of neural network (7) with \(u(t)=Fx(t)\) when \(h(t)=0\). From Fig. 1, it shows that the neural networks are globally asymptotically stable under the feedback controller \(u(t)\). Figure 2 describes the disturbance attenuating controller \(u(t)=Fx(t)\) with an attractor as \(\Phi=\{xx^{T}Qx\leq1\}\) for neural network (7).
Remark 4.1
Comparatively speaking, although the feedback controller law is established in the form of bilinear matrix inequality (BMI), it can be easily solved by alternatively fixing some parameters and optimizing the rest. However, the LMIs in [31] are at least four, which is obviously difficult to be solved.
5 Conclusions
In this paper, the famous differential inclusions, setvalued maps, and Lyapunov–Razumikhin are employed to design a feedback controller law for MRNNs. A feedback controller law is obtained with less computation burden. In the future, other approach, such as the delaypartitioning technique, can be employed to further reduce the conservativeness of the obtained result.
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Acknowledgements
This work was supported in part by the National Science Foundation of China under Grant 61202045, Grant 11501475,Grant 61703060, in part by the Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and its Applications (No. 18TD0013),and in part by the Program of Science and Technology of Sichuan Province of China under Grant No. 2016JY0067.
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Xiao, J., Zhong, S. & Xu, F. Design disturbance attenuating controller for memristive recurrent neural networks with mixed timevarying delays. Adv Differ Equ 2018, 189 (2018). https://doi.org/10.1186/s1366201816418
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DOI: https://doi.org/10.1186/s1366201816418