Cogn Neurodyn (2016) 10:437–451 DOI 10.1007/s11571-016-9392-2

RESEARCH ARTICLE

Stability and synchronization analysis of inertial memristive neural networks with time delays R. Rakkiyappan1 • S. Premalatha1 • A. Chandrasekar1 • Jinde Cao2,3

Received: 15 December 2015 / Revised: 15 April 2016 / Accepted: 26 May 2016 / Published online: 14 June 2016 Ó Springer Science+Business Media Dordrecht 2016

Abstract This paper is concerned with the problem of stability and pinning synchronization of a class of inertial memristive neural networks with time delay. In contrast to general inertial neural networks, inertial memristive neural networks is applied to exhibit the synchronization and stability behaviors due to the physical properties of memristors and the differential inclusion theory. By choosing an appropriate variable transmission, the original system can be transformed into first order differential equations. Then, several sufficient conditions for the stability of inertial memristive neural networks by using matrix measure and Halanay inequality are derived. These obtained criteria are capable of reducing computational burden in the theoretical part. In addition, the evaluation is done on pinning synchronization for an array of linearly coupled inertial memristive neural networks, to derive the condition using matrix measure strategy. Finally, the two numerical simulations are presented to show the effectiveness of acquired theoretical results. Keywords Inertial memristive neural networks  Matrix measure  Halanay inequality  Synchronization  Pinning control & Jinde Cao [email protected] R. Rakkiyappan [email protected] 1

Department of Mathematics, Bharathiar University, Coimbatore, Tamil Nadu 641 046, India

2

Department of Mathematics, Southeast University, Nanjing 210096, China

3

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Introduction In 1971, Chua discovered the fourth element in circuit theory and named it as memristor (short for memory resistor). Chua mathematically revealed that memristor has the relationship between the magnetic flux and electric charge as in Chua (1971). Almost after 40 years, Stanley Williams and his group formulated the practical memristor in May 2008 in Strukov et al. (2008). The rapid variation of voltage at certain instants leads to irregular change of memristance which is similar to the switching behaviors in a dynamical system. The most fascinating trait of the memristor is that it can memorize the direction of flow of electric charge in the past. Thus, it performs as a forgetting and remembering (memory) process in human brains and hence the memristor is acknowledged well and its potential applications are in next generation computers and powerful brain-like computers. In order to replicate the artificial neural networks of human brain better, the conventional resistor of self feedback connection weights and connection weights of the primitive neural networks are replaced by the memristor. Time delay is a common phenomenon that describes the fact that the future state of a system depends not only on the present state but also on the past state, and often encountered in many fields such as engineering, biological and economical systems. In reality, time delays often occur in many systems due to the finite switching speed of amplifiers in electronic neural networks or due to the finite signal propagation time in biological networks. In neural networks, the inner delay time that frequently occurs in the processing of information storage and telecommunication affects the dynamical behavior of the networks. Hence it is reasonable to consider time delay in modeling dynamical networks (Cao et al. 2016). In neural networks literature,

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research work on second-order states are very few compare to the first-order states. The second order states of the system is due to the inertial term or influence of inductance. There are some biological background for the inclusion of an inductance term in a neural system. For example, the membrane of a hair cell in the semicircular canals of some animals can be described by equivalent circuits that contain an inductance (Jagger and Ashmore 1999; Ospeck et al. 2001). The addition of inductance makes the membrane to have an electrical tuning, filtering behaviors and so on. The inertia can be treated as a helpful tool in the generation of chaos in neural systems. From the literature review of inertial neural networks, the bifurcation in a single inertial neuron model is discussed in He et al. (2012), Li et al. (2004), Liu et al. (2009) and the stability of an inertial two-neuron system in Wheeler and Schieve (1997). Furthermore, the stability analysis of Bidirectional Associative Memory (BAM) inertial neural networks has been discussed in Cao and Wan (2014), Yunkuan and Chunfang (2012), Zhang et al. (2015) and Ke and Miao (2013) deals with the inertial Cohen–Grossberg type neural networks in the literature. The stability analysis of neural networks with time delay has received much more attention. Noticeable results on the stability analysis of neural networks with time delay has been projected by various authors. For instance, in Yunkuan and Chunfang (2012), the stability of an inertial BAM neural network with time delay is discussed by constructing suitable Lyapunov functional. Further, the stability of an inertial BAM neural network with time delay has been investigated by matrix measure theory in Cao and Wan (2014). The authors in Qi et al. (2014), Rakkiyappan et al. (2015), deals with the stability of a class of memristor-based recurrent neural networks with time delays using Lyapunov method and Banach contraction principle. In Chen et al. (2015), the global asymptotic stability of fractional memristor-based delayed neural networks has been discussed by employing the comparison theorem of fractional-order linear systems with time delay. Synchronization of a complex network is a fascinating phenomena which is observed in fields such as physical, biological, chemical, technological, etc., and it has potential applications in biological systems, chemical reactions, secure communication, image processing and so on (Pan et al. 2015; Yang and Cao 2014, 2012). Synchronization of coupled inertial neural networks means that multiple neural networks can achieve a common trajectory, such as a common equilibrium, limit cycle or chaotic trajectory. The authors in Dai et al. (2016) have analyzed, the problem of neutral-type coupled neural networks with Markovian switching parameters by placing the adaptive controllers to part of nodes, and the sufficient conditions for exponential synchronization are drawn with the help of Lyapunov

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stability theory, stochastic analysis and matrix theory. Outer synchronization of partially coupled dynamical networks via pinning impulsive controller has been discussed in Lu et al. (2015). The global exponential synchronization of coupled neural networks with stochastic perturbations and mixed time-varying delays has been discussed in Wang et al. (2015) and synchronization criteria have been derived based on multiple Lyapunov theory. Two types of coupled neural networks with reaction–diffusion terms have been considered in Wang et al. (2016) and the general criterion for ensuring network synchronization has been derived by pinning a small fraction of nodes with adaptive feedback controllers. A sufficient condition for the exponential synchronization of fractional-order complex networks via pinning impulsive control has been derived using Lyapunov function and Mittag-Leffler function in Wang et al. (2015). In Yang et al. (2015), exponential synchronization of neural networks with discontinuous activations with mixed delays has been discussed by combining state feedback control and impulsive control techniques. Moreover, Pinning synchronization of coupled inertial delayed neural networks using matrix measure and Lyapunov– Krasovskii functional has been done in Hu et al. (2015). To the best of our knowledge, upto now no work in the literature have been carried out on the stability and synchronization problem for inertial Memristive neural networks (MNNs) with time-delay by using matrix measure and Halanay inequality. Motivated by the above discussion, in this paper the stability of inertial memristive neural networks and pinning synchronization of coupled inertial memristive neural networks are presented. The main contribution of this paper are as follows: (1) The analysis on inertial MNNs is discussed. (2) The stability of inertial MNNs using matrix measure strategy is introduced. (3) In the literature, the stability of memristive-based first order system and fractional-order system are discussed but no work is done on inertial MNNs. Further, the synchronization of inertial MNNs is considered and pinning feedback control is used to synchronize the coupled inertial MNNs to the objective trajectory. In general, the matrix measure method utilizes the information of matrix elements, especially the diagonal elements of a matrix more sufficiently. The rest of the paper is organized as follows. In ‘‘Problem formulation’’ section, model description and preliminaries are presented. Stability of the equilibrium point is investigated in ‘‘Stability analysis’’ section, and the pinning synchronization analysis of inertial MNNs are stated in ‘‘Synchronization analysis’’ section. In ‘‘Numerical simulation’’ section, some examples are given to show the validity of our results. Finally, in ‘‘Conclusion’’ section, conclusions are drawn.

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Notations Throughout this paper, Rn and Rnn denotes the n-dimensional Euclidean space and the set of all n  n matrices, respectively. aij ¼ maxf^ aij ; aij g, aij ¼ min ^  f^ aij ; aij g, bij ¼ maxfbij ; bij g, bij ¼ minfb^ij ; bij g for i; j ¼ 1; 2; . . .; n. ars ¼ maxf^ ars ; ars g; ars ¼ minf^ ars ; ars g; brs ¼ ^  ^  maxfbrs ; brs g; brs ¼ minfbrs ; brs g for r; s ¼ 1; 2; . . .; n. A ¼ ðars Þnn , A ¼ ðars Þnn , B ¼ ðbrs Þnn , B ¼ ðbrs Þnn . cofu; vg denotes closure of the convex hull generated by real numbers u and v or real matrices u and v. AC1 ð½0; 1; Rn Þ denote the space of differential functions 0 x : ½0; 1 ! R, whose first derivative, x , is absolutely continuous. In and diagfa1 ; a2 ; . . .; an g denotes the identity matrix and the diagonal matrix of order n  n. INn stands for the identity matrix with Nn dimension. Let A and B be the arbitrary matrices, then A  B denotes the Kronecker product of the matrices A and B.

where Ui ðvÞ; Wi ðvÞ 2 Cð1Þ ð½s; 0; Rn Þ, Cð1Þ ð½s; 0; Rn Þ denotes the set of all n-dimensional continuously differentiable functions defined on the interval ½s; 0 with s ¼ sup t  0 fsðtÞg. To proceed, the following assumption, definitions and lemmas are given. Assumption 1 The activation function fi satisfies the Lipschitz condition, i.e., there exist constants li [ 0 such that 8 x, y 2 R, we have jfi ðxÞ  fi ðyÞj  li jx  yj;

i ¼ 1; 2; . . .; n:

Definition 1 (Benchohra et al. 2010) A function si 2 AC1 ðð0; 1Þ; Rn Þ is said to be a solution of (1), (3) if s00i ðtÞ þ di s0i ðtÞ 2 Fðt; si ðtÞÞ almost everywhere on [0, 1], where di [ 0 and the function si satisfies conditions (3). For each si 2 Cð½0; 1; Rn Þ, define the set of selections of F by,

Problem formulation

SF;si ¼ fvi 2 L1 ð½0;1;Rn Þ : vi ðtÞ 2 Fðt;si ðtÞÞ a.e. t 2 ½0;1g:

In this paper, we consider the following inertial memristive delayed neural networks, n X d2 si ðtÞ dsi ðtÞ  c ¼ d s ðtÞ þ aij ðsi ðtÞÞfj ðsj ðtÞÞ i i i dt2 dt j¼1 n X bij ðsi ðtÞÞfj ðsj ðt  sðtÞÞÞ þ Ii ; ð1Þ þ

By applying the theories of set-valued maps and differential inclusion to system (1) as stated in Definition 1, P where di [ 0 and Fðsi ; tÞ ¼ ci si ðtÞ þ nj¼1 aij ðsi ðtÞÞfj P ðsj ðtÞÞ nj¼1 bij ðsi ðtÞÞfj ðsj ðt  sðtÞÞÞ þ Ii . The system (1) can be written as the following differential inclusion: n X d2 si ðtÞ dsi ðtÞ 2 ci si ðtÞ þ þ di co½aij ; aij fj ðsj ðtÞÞ 2 dt dt j¼1 ð4Þ n X co½bij ; bij fj ðsj ðt  sðtÞÞÞ þ Ii ; þ

j¼1

where si ðtÞ is the state vector of the ith neuron, di [ 0, ci [ 0 are constants, ci denotes the rate with which the ith neuron will reset its potential to the resetting state in isolation when disconnected from the networks and external inputs. The second derivative of si ðtÞ is known as inertial term of system (1). aij ðsi ðtÞÞ and bij ðsi ðtÞÞ represent the memristive connective weights which changes based on the feature of memristor and current–voltage characteristic. They are given as,  jsi ðtÞj  Ti ; a^ij ; aij ðsi ðtÞÞ ¼ aij ; jsi ðtÞj [ Ti ; ( ð2Þ b^ij ; jsi ðtÞj  Ti ; bij ðsi ðtÞÞ ¼ bij ; jsi ðtÞj [ Ti ; for i; j ¼ 1; 2; . . .; n in which switching jumps Ti [ 0, a^ij , aij , b^ij , bij are known constants with respect to memristance. fi denotes the nonlinear activation function of the ith neuron at time t. Ii is the external input of the ith neuron. The time delay of the system (1) is defined as sðtÞ  0. The initial conditions of the system (1) is given as, si ðvÞ ¼ Ui ðvÞ and

dsi ðvÞ ¼ Wi ðvÞ; dv

s  v  0;

ð3Þ

j¼1

or equivalently, for i; j ¼ 1; 2; . . .; n there exist measurable functions a~ij ðsi ðtÞÞ 2 co½aij ; aij ; b~ij ðsi ðtÞÞ 2 co½bij ; bij ;such that n X d2 si ðtÞ dsi ðtÞ ¼ ci si ðtÞ þ þ di a~ij ðsi ðtÞÞfj ðsj ðtÞÞ 2 dt dt j¼1 n X b~ij ðsi ðtÞÞfj ðsj ðt  sðtÞÞÞ þ Ii ; þ j¼1

ð5Þ where i ¼ 1; 2; . . .; n. Consider the following variable transformation to the system (5): p1i ðtÞ ¼ si ðtÞ; p2i ðtÞ ¼

dsi ðtÞ þ si ðtÞ: dt

Then we have the system (5) as,

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8 p_ ðtÞ ¼ p1i ðtÞ þ p2i ðtÞ; > < 1i P p_ 2i ðtÞ ¼ ci p1i ðtÞ  di p2i ðtÞ þ nj¼1 a~ij ðp1i ðtÞÞfj ðp1j ðtÞÞ > P : þ n b~ij ðp1i ðtÞÞfj ðp1j ðt  sðtÞÞÞ þ Ii ; j¼1

ð6Þ where ci ¼ ci þ 1  di and di ¼ di  1 with the initial values p1i ðvÞ ¼ Ui ðvÞ and p2i ðvÞ ¼ Wi ðvÞ þ Ui ðvÞ; s  v  0. Definition 2 For the system pi ðtÞ 2 f ðpi Þ. Since 0 2 f ð0Þ it follows that pi ¼ 0 is the equilibrium point. i.e., 0 ¼ p 1i ðtÞ þ p 2i ðtÞ; ¼ ci p 1i ðtÞ  di p 2i ðtÞ n n X X b~ij ðp 1i ðtÞÞfj ðp 1j ðtÞÞ þ Ii : a~ij ðp 1i ðtÞÞfj ðp 1j ðtÞÞ þ þ j¼1

Definition 3 (He and Cao 2009) The matrix measure of a real square matrix W ¼ ðwij Þnn is as follows: kIn þ hWkp  1 ; lp ðWÞ ¼ limþ h!0 h where k  kp is an induced matrix norm on Rnn , In is an identity matrix and p ¼ 1; 2; 1; x. When the matrix norm, j

n X

jwij j;

i¼1 n X

kWk1 ¼ max i

kWk2 ¼

jwij j;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kmax ðW T WÞ;

kWkx ¼ max

j¼1

j

n X i¼1

xi jwij j: xj

We can obtain matrix measures as, ( ) n X l1 ðWÞ ¼ max wjj þ jwij j ; j



i¼1;i6¼j

 W þW l2 ðWÞ ¼ kmax ; 2 ( ) n X jwij j ; l1 ðWÞ ¼ max wii þ i

(

T

j¼1;i6¼j

) n X xi lx ðWÞ ¼ max wjj þ jwij j ; j x i¼1;i6¼j j where kmax ðÞ represents the maximum eigenvalue of the matrix ðÞ and xi [ 0 for i ¼ 1; 2; . . .; n are any constant numbers. Remark 1 lp ðWÞ is the one-sided directional derivative of the mapping k  kp : Rnn ! Rþ at the point In , in the direction of W. Matrix norm k  kp holds the non-negative property. But lp ðÞ is not restricted to be non-negative. It may take negative values since it emphasizes more information about the diagonal elements of the matrix.

123

jsi ðtÞ  s i j2  Megt kU  s k2 ;

t [ 0;

where i ¼ 1;    ; n, sðtÞ ¼ ðs1 ðtÞ; s2 ðtÞ; . . .; sn ðtÞÞT is a solution of system (1) with the initial value (3). Lemma 1 (Chandrasekar et al. 2014) Under Assumption 1, if fj ð Tj Þ ¼ 0 ðj ¼ 1; 2; . . .; nÞ then jKðaij ðuj ÞÞfj ðuj Þ  Kðaij ðvj ÞÞfj ðvj Þj  a~uij kj juj  vj j; for i; j ¼ 1; 2; . . .; n, i.e., ðuj ÞÞ; gij ðvj Þ 2 Kðaij ðvj ÞÞ;

for

any

gij ðuj Þ 2 Kðaij

jgij ðuj Þfj ðuj Þ  gij ðvj Þfj ðvj Þj  a~uij kj juj  vj j;

j¼1

kWk1 ¼ max

Definition 4 (Ke and Miao 2013) The equilibrium point s of the system (1) is said to be globally exponentially stable, if there exist constants g [ 0 and M [ 0 such that

aij j; j aij jg. for i; j ¼ 1; 2; . . .; n, where a~uij ¼ max fj^ Lemma 2 If the activation function fi is bounded, i.e., jfi ðxÞj  Mi , for each i ¼ 1; 2; . . .; n then for any given input I ¼ diagfI1 ; . . .; In g, there exists an equilibrium for (1). Proof It is clear that s is an equilibrium of (1) if and only if it is a solution of the following equation n n X X aij ðsi ðtÞÞfj ðs j Þ þ bij ðsi ðtÞÞfj ðs j Þ þ Ii ¼ 0: ci s i þ j¼1

j¼1

By applying the theories of set-valued maps and differential inclusion, we have ci s i þ

n X

co½aij ; aij fj ðs j Þ þ

j¼1

n X

co½bij ; bij fj ðs j Þ þ Ii 2 0;

j¼1

or equivalently, for i; j ¼ 1; 2; . . .; n, there exist measurable functions a~ij ðs i ðtÞÞ 2 co½aij ; aij ; b~ij ðs i ðtÞÞ 2 co½bij ; bij  , such that ci s i þ

n X

a~ij ðs i ðtÞÞfj ðs j Þ þ

j¼1

n X

b~ij ðs i ðtÞÞfj ðs j Þ þ Ii ¼ 0:

j¼1

Noting that ci [ 0, jfi ðxÞj  Mi ; 8i; j ¼ 1; 2; . . .; n, one yields that,  " #  1 X n   js i j , jhi ðs 1 ; . . .; s n Þj ¼  ð~ aij þ b~ij Þfj ðs j Þ þ Ii ;  ci j¼1 

n 1X ðj~ aij j þ jb~ij jÞMj þ jIi j; ci j¼1

, Di ;

i ¼ 1; 2; . . .; n:

Thus the function hðsÞ ¼ ðh1 ; h2 ; . . .; hn ÞT maps D ¼ ½D1 ; D1   . . .  ½Dn ; Dn  into itself. According to

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Brower’s fixed point theorem, the existence of an equilibrium is obtained. h Lemma 3 (Vidyasagar 1993) Let k  kp be an induced matrix norm on Rnn and lp ðÞ be the corresponding matrix measure. Then lp ðÞ has the following properties: 1. 2. 3. 4. 5.

For each W 2 Rnn , the limit indicated in Definition 3 exists and is well-defined; kWkp  lp ðWÞ  kWkp ; lp ðaWÞ ¼ alp ðWÞ, 8 a  0, 8W 2 Rnn and in general, lp ðWÞ 6¼ lp ðWÞ; maxflp ðAÞ  lp ðBÞ; lp ðBÞ  lp ðAÞg  lp ðA þ BÞ  lp ðAÞ þ lp ðBÞ, 8 A; B 2 Rnn ; lp ðÞ is a convex function, i.e., lp ðaA þ ð1  aÞBÞ  alp ðAÞ þ ð1  aÞlp ðBÞ, 8 a 2 ½0; 1, 8 A; B 2 Rnn .

Lemma 4 (Halanay 1966) Let k1 and k2 be constants with k1 [ k2 [ 0 and y(t) is a non-negative continuous function defined on ½t0  s; þ1 which satisfies the following inequality for t  t0

~ Þ [ lkBk ~ p [ 0; ðlp ðGÞ þ lkAk ð7Þ  p  In In , C ¼ C þ In  D, D ¼ D  In where G ¼ C D and l ¼ max1  i  n fli g. Then the equilibrium ðp 1 ðtÞ; p 2 ðtÞÞT of the system (6) is globally exponentially stable for p ¼ 1; 2; 1; x. Proof Let us define the error between the trajectory of the system (6) and the corresponding equilibrium ðp 1 ðtÞ; p 2 ðtÞÞT as,     p1 ðtÞ  p 1 ðtÞ e1 ðtÞ eðtÞ ¼ ¼ : p2 ðtÞ  p 2 ðtÞ e2 ðtÞ According to the error system, the upper-right Dini derivative of keðtÞkp with respect to ‘t’ is calculated as follows: keðt þ hÞkp  keðtÞkp h!0 h _ þ oðhÞkp  keðtÞkp keðtÞ þ heðtÞ : ¼ limþ h!0 h

Dþ keðtÞkp ¼ limþ

Dþ yðtÞ   k1 yðtÞ þ k2 yðtÞ; where yðtÞ, supts  s  t yðsÞ. Then, yðtÞ  yðt0 Þerðtt0 Þ , where r is a bound on the exponential convergence rate and is the unique positive solution of r ¼ k1  k2 ers , where the upper right Dini derivative Dþ yðtÞ is defined as Dþ yðtÞ ¼ limþ h!0

We have, 1 e1i ðtÞ þ e2i ðtÞ n C B P B ci e1i ðtÞ  di e2i ðtÞ þ ½~ aij ðp1i ðtÞÞfj ðp1j ðtÞÞ  a~ij ðp 1i ðtÞÞfj ðp 1j ðtÞÞ C C B ¼B j¼1 C: C B n A @ P þ ½b~ij ðp1i ðtÞÞfj ðp1j ðt  sðtÞÞÞ  b~ij ðp 1i ðtÞÞfj ðp 1j ðtÞÞ 0



e_1i ðtÞ e_2i ðtÞ



yðt þ hÞ  yðtÞ ; h

j¼1

where h ! 0þ means that h approaches zero from the right-hand side. Lemma 5 (He and Cao 2009) Under Assumption 1, let lp ðÞ be the corresponding matrix measure associated with the induced matrix norm k  kp on Rnn . Then lp ðAFðeðtÞÞÞ  lp ðA LÞ; n fn ðen ðtÞÞ 1 ðtÞÞ where FðeðtÞÞ ¼ diag f1 ðe e1 ðtÞ ; . . .; en ðtÞ g, fl1 ; . . .; ln g, p ¼ 1; 1; x  maxf0; aii g; i ¼ j; A ¼ ðaij Þnn ¼ aij ; i 6¼ j:

Under the Assumption 1, further

Theorem 1

According to Lemma 1, the above equation can be written as follows: 1 e1i ðtÞ þ e2i ðtÞ n C   B B ci e1i ðtÞ  di e2i ðtÞ þ P a~u ðfj ðp1j ðtÞÞ  fj ðp ðtÞÞÞ C e_1i ðtÞ ij 1j C B ¼B j¼1 C; C B e_2i ðtÞ n A @ P þ b~uij ðfj ðp1j ðt  sðtÞÞÞ  fj ðp 1j ðtÞÞÞ 0

j¼1

L ¼ diag and

where a~uij ¼ maxfj^ aij j; j aij jg and b~uij ¼ maxfjb^ij j; jbij jg. The vector form of the above equation is,  _ ¼ eðtÞ

Stability analysis In this section, we will investigate the exponential stability of inertial memristive neural networks using matrix measure strategy. The main result is given in the following theorem.

e_1 ðtÞ e_2 ðtÞ



0

1 e1 ðtÞ þ e2 ðtÞ B ~ ðp1 ðtÞÞ  f ðp ðtÞÞÞ C ¼ @ Ce1 ðtÞ  De2 ðtÞ þ Aðf A; 1 ~ ðp1 ðt  sðtÞÞÞ  f ðp 1 ðtÞÞÞ þBðf

ð8Þ

where C ¼ C þ In  D, D ¼ D  In , A~ ¼ ð~ auij Þnn and B~ ¼ ðb~uij Þnn . We have the upper right Dini-derivative as,

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      eðtÞþh e_1 ðtÞ þoðhÞ keðtÞk p   e_2 ðtÞ p Dþ keðtÞkp ¼ limþ h!0 h       ðtÞþe e 1 2 ðtÞ eðtÞþh þoðhÞ   keðtÞkp n p ¼ limþ h!0 h       eðtÞþh In In eðtÞ keðtÞk p   C D p  limþ h!0 h ~ þkAðf ðp1 ðtÞÞf ðp1 ðtÞÞÞkp ~ ðp1 ðt sðtÞÞÞf ðp 1 ðtÞÞÞkp þkBðf       eðtÞþh In In eðtÞ keðtÞk p   C D p  limþ h!0 h ~ kp1 ðtÞp ðtÞk þ lkAk 1 p p ~ p kp1 ðt sðtÞÞp 1 ðtÞkp þ lkBk kIn þhðGÞkp 1 ~ ke1 ðtÞk keðtÞkp þlkAk p p h ~ p ke1 ðt sðtÞÞkp þlkBk ~ keðtÞk þlkBk ~ keðt sðtÞÞk ¼ lp ðGÞkeðtÞk þlkAk

 limþ h!0

p

p

p

p

p

~ ÞkeðtÞk þlkBk ~ p sup keðvÞkp ðlp ðGÞþlkAk p p tsvt

~ ÞkeðtÞk þlkBk ~ p sup keðvÞkp ; ¼ ðlp ðGÞlkAk p p tsvt

~ ðp1 ðtÞÞf ðp ðtÞÞÞþ B~ where n ¼ Ce1 ðtÞDe2 ðtÞþ Aðf 1 ðf ðp1 ðt sðtÞÞÞf ðp1 ðtÞÞÞ. Using Lemma 4, by considering ~ Þ, k2 ¼ lkBk ~ and the assumption of k1 ¼ ðlp ðGÞþlkAk p

p

~ Þ[lkBk ~ p [0. We have the Theorem 1, ðlp ðGÞþlkAk p result as, keðtÞkp 

sup ts  v  t

keðvÞkp erðtt0 Þ ;

where r is a bound on the exponential convergence rate with

Rosier (2005). Secondly, we have to evaluate its first-order time derivative along the trajectory of the system. If the derivative of a Lyapunov function is decreasing along the system trajectory as time increases, then the system energy is dissipating and the system will finally settle down. However, it should be noted that much of the previous work in the stability analysis of neural networks is based on Lyapunov direct methods, where constructing a proper Lyapunov function is important for the stability analysis. Moreover, to construct a proper Lyapunov function for a given system is very difficult and there are no general rules to follow. Compared with Lyapunov direct method, matrix measure strategy is an efficient tool to address the stability problem of nonlinear systems. Generally, the established results by using matrix measure are more superior than common algebraic criterion due to the fact that matrix measure can not only be taken to be positive value but also it can be a negative value. Inspired by the above method, we introduce the so called matrix measure and Halanay inequality to study the stability of the system (1), and some simple but generic criteria have been derived. Remark 3 Compared to the matrix norm k  kp , the matrix measure, lp ðÞ are sign sensitive which ensure that the obtained results are more precise and less computational burden those obtained by using matrix norm. Corollary 1 Under the Assumption 1 and if further there exists matrix K ¼ diagfn1 ; n2 ; . . .; nn g such that ~ Þ [ lkBk ~ p [ 0; ðlp ðGÞ þ lkAk ð9Þ p   K In , C ¼ C þ In  D, D ¼ holds where G ¼ C D D  In and l ¼ max1  i  n fli g. Then the equilibrium ðp 1 ðtÞ; p 2 ðtÞÞT of the system (6) is globally exponentially stable for p ¼ 1; 2; 1; x. Proof To the system (5), we consider the following transformation: p1i ðtÞ ¼ si ðtÞ;

~ Þ  lkBk ~ p ers : r ¼ k1  k2 e ¼ ðlp ðGÞ þ lkAk p rs

Thus we conclude that, e(t) converges exponentially to zero with a convergence rate r. That is, every trajectory of the system (6) converges exponentially towards the equilibrium ðp 1 ðtÞ; p 2 ðtÞÞT with a convergence rate r. This completes the proof. h Remark 2 The fundamental concept of the Lyapunov direct method is that if the total energy of a system is continuously dissipating, then the system will eventually reach an equilibrium point and remain at that point. Hence, the Lyapunov direct method consists of two steps. Firstly, a suitable scalar function is chosen and this function is referred as Lyapunov function Hahn (1967), Bacciotti and

123

p2i ðtÞ ¼

dsi ðtÞ þ ni si ðtÞ: dt

and proceed as stated before and the proof is similar to the Theorem 1. h Remark 4 The condition of Theorem 1 certainly does not holds for p ¼ 1; 1. On that case, it is better to introduce ni [ 0, i ¼ 1; 2; . . .; n in the variable transformation as in Corollary 1. So that the condition holds for p ¼ 1; 1 and leads to the better performance. Theorem 2

Under the Assumption 1, further

~ p [ 0; ðlp ðGÞ þ lp ðAL ÞÞ [ lkBk

ð10Þ

Cogn Neurodyn (2016) 10:437–451

443

holds where L ¼ diagfl1 ; l2 ; . . .; ln ; 1; 1; . . .; 1g and   0 0 A¼ ~ . Then the equilibrium ðp 1 ðtÞ; p 2 ðtÞÞT of A 0 the system (6) is globally exponentially stable for p ¼ 1; 1; x. Proof Similar to the proof of Theorem 1, first let us calculate the upper-right Dini-derivative as, keðt þ hÞkp  keðtÞkp h!0 h _ þ oðhÞkp  keðtÞkp keðtÞ þ heðtÞ : ¼ limþ h!0 h

Dþ keðtÞkp ¼ limþ

The error system is defined as,     p1 ðtÞ  p 1 ðtÞ e1 ðtÞ eðtÞ ¼ ¼ : p2 ðtÞ  p 2 ðtÞ e2 ðtÞ

  f11 ðp11 ðtÞÞ  f11 ðp 11 ðtÞÞ f11 ðp11 ðtÞÞ  f11 ðp 11 ðtÞÞ ; ; . . .; FðpðtÞÞ  Fðp ðtÞÞ ¼ diag p11 ðtÞ  p 11 ðtÞ p11 ðtÞ  p 11 ðtÞ   0 F1 ðp1 ðtÞÞ  F1 ðp1 ðtÞÞ : ¼ 0 F2 ðp2 ðtÞÞ  F2 ðp 2 ðtÞÞ

We have the error system as,   In In _ ¼ eðtÞ eðtÞ þ AðFðpðtÞÞ  Fðp ðtÞÞÞðpðtÞ C D   0  p ðtÞÞ þ : ~ ðp1 ðt  sðtÞÞÞ  f ðp 1 ðtÞÞÞ Bðf Then, we have the upper-right Dini derivative as, Dþ keðtÞkp    In eðtÞ þ h  C  limþ

  eðtÞ þ AðFðpðtÞÞ  Fðp ðtÞÞÞðpðtÞ  p ðtÞÞ   keðtÞkp D p h!0 h ~ þ kBðf ðp1 ðt  sðtÞÞÞ  f ðp1 ðtÞÞÞkp

Also note that, e1 ðtÞ ¼ ðe11 ðtÞ; e12 ðtÞ; . . .; e1n ðtÞÞ, fi ðp1i ðtÞÞ  fi ðp 1i ðtÞÞ ¼ fi ðp 1i ðtÞþe1i ðtÞÞ  fi ðp 1i ðtÞÞ; f ðp1 ðtÞÞ  f ðp 1 ðtÞÞ ¼ ðf11 ðp11 ðtÞÞ  f11 ðp 11 ðtÞÞ; . . .; f1n ðp1n ðtÞÞ  f1n ðp 1n ðtÞÞÞ: Define the function as,

Now by letting,

In



keðtÞ þ hðG þ AðFðpðtÞÞ  Fðp ðtÞÞÞeðtÞkp  keðtÞkp h ~ ðp1 ðt  sðtÞÞÞ  f ðp 1 ðtÞÞÞkp þ kBðf ~ p keðt  sðtÞÞkp : ¼ lp ðG þ AðFðpðtÞÞ  Fðp ðtÞÞÞÞkeðtÞkp þ lkBk

 limþ h!0

It follows from Lemmas 3 and 5 that, 

lp ðG þ AðFðpðtÞÞ  Fðp ðtÞÞÞÞ  lp ðGÞ þ lp ðAðFðpðtÞÞ

f11 ðp11 ðtÞÞ  f11 ðp 11 ðtÞÞ ; . . .; p11 ðtÞ  p 11 ðtÞ f1n ðp1n ðtÞÞ  f1n ðp 1n ðtÞÞ g; F2 ðp2 ðtÞÞ p1n ðtÞ  p 1n ðtÞ

where L ¼ diagfl1 ; l2 ; . . .; ln ; 1; 1; . . .; 1g. Hence, we have

F2 ðp 2 ðtÞÞ  In :

~ p Dþ keðtÞkp  ðlp ðGÞ þ lp ðAL ÞÞkeðtÞkp þ lkBk

F1 ðp1 ðtÞÞ  F1 ðp 1 ðtÞÞ ¼ diag

 Fðp ðtÞÞÞÞ;  lp ðGÞ þ lp ðAL Þ;

sup ts  v  t

Then, f ðp1 ðtÞÞ  f ðp 1 ðtÞÞ and f ðp2 ðtÞÞ  f ðp 2 ðtÞÞ can be written as,

~ p Dþ keðtÞkp   ðlp ðGÞ  lp ðAL ÞÞkeðtÞkp þ lkBk

f ðp1 ðtÞÞ  f ðp 1 ðtÞÞ ¼ ðF1 ðp1 ðtÞÞ  F1 ðp 1 ðtÞÞÞðp1 ðtÞ  p 1 ðtÞÞ;

Using

f ðp2 ðtÞÞ  f ðp 2 ðtÞÞ ¼ p2 ðtÞ  p 2 ðtÞ ¼ ðF2 ðp2 ðtÞÞ  F2 ðp 2 ðtÞÞÞðp2 ðtÞ  p 2 ðtÞÞ:

 _ ¼ eðtÞ  ¼

e_1 ðtÞ e_2 ðtÞ In C 

þ



1

e1 ðtÞ þ e2 ðtÞ B ~ ðp1 ðtÞÞ  f ðp ðtÞÞÞ C ¼ @ Ce1 ðtÞ  De2 ðtÞ þ Aðf A 1 ~ þBðf ðp1 ðt  sðtÞÞÞ  f ðp 1 ðtÞÞÞ     0 In e1 ðtÞ þ ~ ðp1 ðtÞÞ  f ðp ðtÞÞÞ Aðf e2 ðtÞ D 1  0

~ ðp1 ðt  sðtÞÞÞ  f ðp 1 ðtÞÞÞ Bðf      f ðp1 ðtÞÞ  f ðp 1 ðtÞÞ 0 0 e1 ðtÞ In In þ ¼ f ðp2 ðtÞÞ  f ðp 2 ðtÞÞ A~ 0 C D e2 ðtÞ   0 þ ~ ðp1 ðt  sðtÞÞÞ  f ðp 1 ðtÞÞÞ Bðf      F1 ðp1 ðtÞÞ  F1 ðp 1 ðtÞÞðp1 ðtÞ  p 1 ðtÞÞ In In e1 ðtÞ ¼ þA F2 ðp2 ðtÞÞ  F2 ðp 2 ðtÞÞðp2 ðtÞ  p 2 ðtÞÞ e2 ðtÞ C D   0 : þ ~ Bðf ðp1 ðt  sðtÞÞÞ  f ðp1 ðtÞÞÞ 

sup ts  v  t

keðvÞkp :

4, by considering k1 ¼ ðlp ðGÞþ ~ p and the assumption of our Theolp ðAL ÞÞ, k2 ¼ lkBk ~ p [ 0. We have the rem 2, ðlp ðGÞ þ lp ðAL ÞÞ [ lkBk Lemma



result as,

It follows from (8) that, 0

keðvÞkp ;

keðtÞkp 

sup ts  v  t

keðvÞkp erðtt0 Þ ;

where r is a bound on the exponential convergence rate with ~ p ers : r ¼ k1  k2 ers ¼ ðlp ðGÞ þ lp ðAL ÞÞ  lkBk Thus we conclude that, e(t) converges exponentially to zero with a convergence rate r. That is, every trajectory of the system (6) converges exponentially towards the equilibrium ðp 1 ðtÞ; p 2 ðtÞÞT with a convergence rate r. This completes the proof. h Remark 5 The stability condition of Theorem 1 utilize only the maximum Lipschitz constant and not the information of each li . But the stability condition of Theorem 2

123

444

Cogn Neurodyn (2016) 10:437–451

utilizes the information of each Lipschitz constant li . Thus, ~ as lp ðAL Þ by letting some terms, we make the term kAk p in Theorem 2 and hence the result is more precise than Theorem 1. But the condition of Theorem 2 holds only for p ¼ 1; 1; x.

positive constant which represents network coupling strength. C is the inner coupling matrix. The matrix W ¼ ðWkl ÞNN is the constant coupling configuration matrix which is considered to be diffusive. i.e., Wkl  0 ðk 6¼ lÞ P and Wkk ¼  Nl¼1;l6¼k Wkl . Also, W is not required to be symmetric or irreducible. The initial condition of the system (13) is given as,

Synchronization analysis Consider the following memristive delayed neural networks, n X d2 yr ðtÞ dyr ðtÞ  c ¼ d y ðtÞ þ ars ðyðtÞÞfs ðys ðtÞÞ r r r dt2 dt s¼1 n X brs ðyðtÞÞfs ðys ðt  sðtÞÞÞ þ Ir ; þ s¼1

r ¼ 1; . . .; n;

xk ðvÞ ¼ Uk ðvÞ and

dxk ðvÞ ¼ Wk ðvÞ; dv

s  v  0; ð14Þ

where Uk ðvÞ, Wk ðvÞ 2 Cð1Þ ð½s; 0; Rn Þ. The isolated node of network (13) is given as, d2 sðtÞ dsðtÞ  CsðtÞ þ AðsðtÞÞf ðsðtÞÞ þ BðsðtÞÞf ðsðt ¼ D 2 dt dt  sðtÞÞÞ þ I:

ð11Þ

ð15Þ

where yðtÞ ¼ ðy1 ðtÞ; . . .; yn ðtÞÞ 2 Rn . dr [ 0, cr [ 0 are constants, cr denotes the rate with which the rth neuron will reset its potential to the resetting state in isolation when disconnected from the networks and external inputs. Ir ¼ ðI1 ; . . .; In ÞT and f ðyðtÞÞ ¼ ðf1 ðy1 ðtÞÞ; . . .; fn ðyn ðtÞÞÞT denotes the external input and the non-linear activation function respectively. AðyðtÞÞ ¼ ðars ðyðtÞÞÞnn and BðyðtÞÞ ¼ ðbrs ðyðtÞÞÞnn are memristive connection weights which changes based on the feature of memristor and current–voltage characteristics are defined as,  jyðtÞj  Tr ; a^rs ; ars ðyðtÞÞ ¼ ars ; jyðtÞj [ Tr ; ( ð12Þ jyðtÞj  Tr ; b^rs ; brs ðyðtÞÞ ¼ brs ; jyðtÞj [ Tr ;

The connection weight matrices of A(s(t)) and B(s(t)) are defined as in (12). The initial condition of the system (15) is given as,

T

for r; s ¼ 1; 2; . . .; n in which switching jumps Tr [ 0, a^rs , ars , b^rs , brs are known constants with respect to memristance. When N inertial MNNs are coupled by a network, we can obtain the following array of linearly coupled inertial MNNs with the dynamics of the kth node as, d2 xk ðtÞ dxk ðtÞ  Cxk ðtÞ þ Aðxk ðtÞÞf ðxk ðtÞÞ ¼ D 2 dt dt þ Bðxk ðtÞÞf ðxk ðt  sðtÞÞÞ þ I   N X dxl ðtÞ þ xl ðtÞ ; þa Wkl C dt l¼1

dsðvÞ ¼ WðvÞ; dv

s  v  0;

ð16Þ

where UðvÞ, WðvÞ 2 Cð1Þ ð½s; 0; Rn Þ. By applying the theories of set-valued maps and differential inclusion to (15) as stated in Definition 1 we have, d2 sðtÞ dsðtÞ 2 CsðtÞ þ co½A; Af ðsðtÞÞ þD dt2 dt þ co½B; Bf ðsðt  sðtÞÞÞ þ I;

ð17Þ

or equivalently, there exist measurable functions e e AðsðtÞÞ 2 co½A; A; BðsðtÞÞ 2 co½B; B, such that d2 sðtÞ dsðtÞ e ¼ CsðtÞ þ AðsðtÞÞf ðsðtÞÞ þD dt2 dt e þ BðsðtÞÞf ðsðt  sðtÞÞÞ þ I:

ð18Þ

Consider the following transformation to the system (18), pðtÞ ¼ sðtÞ; qðtÞ ¼

ð13Þ

where k ¼ 1; . . .; N. xk ðtÞ ¼ ðxk1 ðtÞ; . . .; xkn ðtÞÞT 2 Rn is the state of the kth neural network. D ¼ diagfd1 ; d2 ; . . .; dn g and C ¼ diagfc1 ; c2 ; . . .; cn g denote the positive definite matrices. Aðxk ðtÞÞ and Bðxk ðtÞÞ are memristive connection weights defined in (12). a is the

123

sðvÞ ¼ UðvÞ and

dsðtÞ þ sðtÞ: dt

Then we have the system (18) as, 1 0 pðtÞ þ qðtÞ   _ pðtÞ C B e ¼ @ CpðtÞ  DqðtÞ þ AðpðtÞÞf ðpðtÞÞ A; _ qðtÞ e þ BðpðtÞÞf ðpðt  sðtÞÞÞ þ I

ð19Þ

where C ¼ C þ In  D and D ¼ D  In . The pinning controlled networks of the system (13) is given as,

Cogn Neurodyn (2016) 10:437–451

d2 xk ðtÞ dxk ðtÞ  Cxk ðtÞ þ Aðxk ðtÞÞf ðxk ðtÞÞ ¼ D dt2 dt þ Bðxk ðtÞÞf ðxk ðt  sðtÞÞÞ þ I   N X dxl ðtÞ þ xl ðtÞ þa Wkl C dt l¼1   dðxk ðtÞ  sðtÞÞ þ ðxk ðtÞ  sðtÞÞ ;  ark C dt

445 1 yk ðtÞ þ zk ðtÞ B e k ðtÞÞf ðyk ðtÞÞ þ Bðy e k ðtÞÞ C   B Cyk ðtÞ  Dzk ðtÞ þ Aðy C y_k ðtÞ C B N ¼B P C: B  f ðyk ðt  sðtÞÞÞ þ I  a lkl Cðzl ðtÞÞ  ark C z_k ðtÞ A @ l¼1 0

ð20Þ

where k ¼ 1; 2; . . .; N and rk ¼ 1 if the node is pinned, otherwise rk ¼ 0. Aðxk ðtÞÞ, Bðxk ðtÞÞ and the initial condition of the system are given in (12) and (14) respectively. By applying the theories of set-valued maps and differential inclusion to the form as stated in Definition 1 to the system (20) we have, d2 xk ðtÞ dxk ðtÞ 2 Cxk ðtÞ þ co½A; Af ðxk ðtÞÞ þD 2 dt dt þ co½B; Bf ðxk ðt  sðtÞÞÞ þ I   N X dxl ðtÞ þ xl ðtÞ þa Wkl C dt l¼1   dðxk ðtÞ  sðtÞÞ þ ðxk ðtÞ  sðtÞÞ ;  ark C dt or equivalently, there exist measurable functions e k ðtÞÞ 2 co½A; A; Bðx e k ðtÞÞ 2 co½B; B, such that Aðx d2 xk ðtÞ dxk ðtÞ e k ðtÞÞf ðxk ðtÞÞ ¼ Cxk ðtÞ þ Aðx þD 2 dt dt e k ðtÞÞf ðxk ðt  sðtÞÞÞ þ I þ Bðx   N X dxl ðtÞ þ xl ðtÞ þa Wkl C dt k¼1   dðxk ðtÞ  sðtÞÞ  ark C þ ðxk ðtÞ  sðtÞÞ : dt Consider the following transformation to the above system, yk ðtÞ ¼ xk ðtÞ; dxk ðtÞ þ xk ðtÞ: zk ðtÞ ¼ dt Then we have, 1 yk ðtÞ þ zk ðtÞ B Cy ðtÞ  Dz ðtÞ þ Aðy e k ðtÞÞf ðyk ðtÞÞ þ Bðy e k ðtÞÞf ðyk ðt  sðtÞÞÞ C C B k k ¼B C; N A @ P þI þ a Wkl Cðzl ðtÞÞ  ark Cðzk ðtÞ  qðtÞÞ 0



y_k ðtÞ z_k ðtÞ



l¼1

where C ¼ C þ In  D and D ¼ D  In . Now, let us introduce a Laplacian matrix L ¼ ðlkl ÞNN of the coupling network and define L ¼ W. Then we have,

ð21Þ

 Cðzk ðtÞ  qðtÞÞ

Let us define the error dynamical network as,     e1k ðtÞ yk ðtÞ  pðtÞ ek ðtÞ ¼ ¼ : e2k ðtÞ zk ðtÞ  qðtÞ Subtracting (19) from (21), we obtain the following, 1 yk ðtÞþzk ðtÞþpðtÞqðtÞ C B e e   B Cyk ðtÞDzk ðtÞþ Aðyk ðtÞÞf ðyk ðtÞÞþ Bðyk ðtÞÞf ðyk ðt sðtÞÞÞ C C B e_1k ðtÞ N C; P e_k ðtÞ ¼ ¼B C B a lkl Cðzl ðtÞÞark Cðzk ðtÞqðtÞÞþCpðtÞþDqðtÞ e_2k ðtÞ C B A @ l¼1 e e  AðpðtÞÞf ðpðtÞÞ BðpðtÞÞf ðpðt sðtÞÞÞ 0

According to Lemma 1, the above system can be written as follows: 0

1 e1k ðtÞ þ e2k ðtÞ B ~~ ðy ðtÞÞ  f ðpðtÞÞÞ C C   B Ce1k ðtÞ  De2k ðtÞ þ Aðf k B C e_1k ðtÞ C; ~ ¼B ~ ðyk ðt  sðtÞÞÞ  f ðpðt  sðtÞÞÞÞ Bðf þ B C e_2k ðtÞ B C N @ A P a lkl Cðe2l ðtÞÞ  ark Cðe2k ðtÞÞ l¼1

where A~~ ¼ max fjars j; jars jg and B~~ ¼ max fjbrs j; jbrs jg: Using the Kronecker product, the above system can be written in the following compact form, 0



e_1 ðtÞ e_2 ðtÞ



1 e1 ðtÞ þ e2 ðtÞ B ~ ~ ðyk ðtÞÞ  f ðpðtÞÞÞ C B ðIN  CÞe1 ðtÞ  ðIN  DÞe2 ðtÞ þ ðIN  AÞðf C C: ¼B B C ~ ~ ðyk ðt  sðtÞÞÞ  f ðpðt  sðtÞÞÞÞ @ A þðIN  BÞðf aððL þ RÞ  CÞe2 ðtÞ

ð22Þ Lemma 6 (Cao et al. 2006) For matrices A, B, C and D with appropriate dimensions, one has 1. 2. 3.

ðaAÞ  B ¼ A  ðaBÞ. ðA þ BÞ  C ¼ A  C þ B  C. ðA  BÞðC  DÞ ¼ ðACÞ  ðBDÞ.

Definition 5 The linearly coupled inertial MNNs (13) is said to be exponentially synchronized if the error system (22) is exponentially stable, i.e., there exist two constants a [ 1 and b [ 0 such that keðtÞkp  a

sup t0 s  h  t0

keðhÞkp ebðtt0 Þ ;

t  t0 :

Now, we present the synchronization result in the following theorem based on the above transformations.

123

446

Cogn Neurodyn (2016) 10:437–451

Theorem 3

Under the Assumption 1, further

~~ Þ [ lkI  Bk ~~ [ 0; ð23Þ ðlp ðKÞ þ lkIN  Ak N p p  INn INn aðL þ RÞ  CÞ where K ¼ ðIN  CÞ ðIN  DÞ and l ¼ max1  i  n fli g. Then the pinning controlled inertial MNNs (22) is globally exponentially synchronized for p ¼ 1; 2; 1; x. Proof The upper-right Dini derivative of the synchronization error, keðtÞkp with respect to t is calculated as follows,

Dþ keðtÞkp ¼ limþ h!0

¼ limþ

~~ Þ [ the assumption of Theorem 3, ðlp ðKÞ þ lkIN  Ak p ~~ [ 0. We have the result as, lkI  Bk N

p

keðtÞkp 

sup t0 s  v  t0

keðvÞkp erðtt0 Þ ;

where r is a bound on the exponential convergence rate with ~~ Þ  lkI  Bk ~~ ers : r ¼ k1  k2 ers ¼ ðlp ðKÞ þ lkIN  Ak N p p

_ þ oðhÞkp  keðtÞkp keðt þ hÞkp  keðtÞkp keðtÞ þ heðtÞ ¼ limþ h!0 h h       _1 ðtÞ e eðtÞ þ h þ oðhÞ  keðtÞkp  e_2 ðtÞ p

h       e ðtÞ þ e2 ðtÞ 1 eðtÞ þ h þ oðhÞ  keðtÞkp  k p ¼ limþ h!0 h       INn INn eðtÞ þ h eðtÞ  keðtÞkp  ðIN  CÞ ðIN  DÞ  aðL þ RÞ  C p  limþ h!0 h ~~ ðy ðtÞÞ  f ðpðtÞÞÞk þ kðIN  AÞðf k h!0

p

~~ ðy ðt  sðtÞÞÞ  f ðpðt  sðtÞÞÞÞk þ kðIN  BÞðf k p       I I Nn Nn eðtÞ þ h eðtÞ  keðtÞkp  ðIN  CÞ ðIN  DÞ  aðL þ RÞ  C p  limþ h!0 h ~~ ky ðtÞ  pðtÞk þ lkI  Bk ~~ ky ðt  sðtÞÞ  pðt  sðtÞÞk þ lkIN  Ak N p k p p k p kIn þ hðKÞkp  1 ~~ ke ðtÞk keðtÞkp þ lkIN  Ak p 1 p h!0 h ~~ ke ðt  sðtÞÞk þ lkIN  Bk p 1 p

 limþ

~~ keðtÞk þ lkI  Bk ~~ keðt  sðtÞÞk ¼ lp ðKÞkeðtÞkp þ lkIN  Ak N p p p p ~ ~ ~ ~  ðl ðKÞ þ lkI  Ak ÞkeðtÞk þ lkI  Bk sup keðvÞk p

N

p

p

N

p

ts  v  t

~~ ÞkeðtÞk þ lkI  Bk ~~ ¼ ðlp ðKÞ  lkIN  Ak N p p p

where k ¼ ðIN  CÞe1 ðtÞ  ðIN  DÞe2 ðtÞ þ ðIN  ~ ~~ ðy ðt  sðtÞÞÞ f ðpðt  ~ AÞðf ðyk ðtÞÞ  f ðpðtÞÞÞ þ ðIN  BÞðf k sðtÞÞÞÞ  aððLþ RÞ  CÞe2 ðtÞ. Using Lemma 4 by con~~ Þ, k ¼ lkI  Bk ~~ and sidering k1 ¼ ðlp ðKÞþ lkIN  Ak 2 N p p

123

sup ts  v  t

p

keðvÞkp ;

Thus, we conclude that e(t) converges exponentially to zero with a convergence rate r. It implies that the global synchronization of the system (22) is achieved. This completes the proof. h

Cogn Neurodyn (2016) 10:437–451

447

Theorem 4 Under Assumption 1, if there exists a matrix measure lp ðÞðp ¼ 1; 1; xÞ such that ~~ [ 0; ð24Þ ðlp ðKÞ þ lp ðAL Þ [ lkIN  Bk   p 0 0 , L ¼ diagfl1 ; l2 ; . . .; ln ; 1; where A ¼ 0 IN  A~~ 1; . . .; 1g. Then the pinning controlled coupled inertial MNNs (22) is globally exponentially synchronized. Proof

Similar to the proof of Theorem 2.

h

Remark 6 Compared with the results on pinning synchronization of the coupled inertial delayed neural networks of Hu et al. (2015), our results on pinning synchronization are with discontinuous right-hand side. So the results in this paper are more superior than in Hu et al. (2015). Remark 7 Recently, many of the researchers have developed synchronization results by constructing suitable Lyapunov–Krasovskii functional and by using linear matrix inequality techniques. Several effective methods such as delay decomposition approach, convex combination, free weighting matrix approach and inequalities technique have been explored and developed in the literature, see for examples Balasubramaniam et al. (2011), Balasubramaniam and Vembarasan (2012). Moreover, in Wang and Shen (2015), authors have concerned the synchronization of memristor-based neural networks with time-varying delays is investigated by employing the Newton–Leibniz formulation and inequality technique. In Bao and Cao (2015), authors deal with the problem of projective synchronization of fractional-order memristorbased neural networks in the sense of Caputo’s fractional derivative. However, to the best of our knowledge, upto now the memristor based neural networks have been investigated by the Lyapunov–Krasovskii method but the result discussed in this paper is more superior than the results that are obtained by constructing a Lyapunov– Krasovskii functional. Another important feature of the derived results is because of the usage of matrix measure and Halanay inequality. Remark 8 In the real world, resistors are used to model connection weight to emulate the synapses in analog

implementation of neural networks. By making use of the memristor which has memory and behaviour more like biological synapses we can able to develop memristor based neural network models. The memristive neural networks have characteristics of complex brain networks such as node degree, distribution and assortativity both at the whole-brain scale of human neuroimaging. With the development of application as well as many integrated technologies, memristive neural networks have proven as a promising architecture in neuromorphic systems for the high-density, non-volatility, and unique memristive characteristic. Due to the promising applications in wide areas, various memristive materials, such as ferroelectric materials, chalcogenide materials, metal oxides have attracted great attention. Further, several physical mechanisms have been proposed to illustrate the memristive behaviors, such as electronic barrier modulation from migration of oxygen vacancies, voltage-controlled domain configuration, formation and annihilation of conducting filaments via diffusion of oxygen vacancies, trapping of charge carriers and metal ions from electrodes. Motivated by the aforementioned applications, in this paper we investigate the problem of stability and synchronization analysis of inertial memristive neural networks with time delays.

Numerical simulation In this section, numerical examples are presented to illustrate the effectiveness and usefulness of the theoretical result. Example 1

Consider the following inertial MNNs: n X d si ðtÞ dsi ðtÞ  c ¼ d s ðtÞ þ aij ðsi ðtÞÞfj ðsj ðtÞÞ i i i dt2 dt i¼1 2

þ

n X

ð25Þ

bij ðsi ðtÞÞfj ðsj ðt  sðtÞÞÞ þ Ii ;

i¼1

where the activation function is given as, fj ðxÞ ¼ sinðjxj  1Þ for j ¼ 1; 2, d1 ¼ c1 ¼ 6, d2 ¼ c2 ¼ 8, I1 ¼ I2 ¼ 6, and et sðtÞ ¼ 1þe t . Also the memristive weights of the system (25) is given as,

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448

pffiffiffi 8 0:01p 2 > > < ; js1 ðtÞj  1; 8 pffiffiffi a11 ðs1 ðtÞÞ ¼ > > : 0:07p 2 ; js1 ðtÞj [ 1; 8  0:06; js1 ðtÞj  1; a12 ðs1 ðtÞÞ ¼ 0:04; js1 ðtÞj [ 1; pffiffiffi 8 0:01p 2 > > ; js2 ðtÞj  1; < 8 pffiffiffi a21 ðs2 ðtÞÞ ¼ > > : 0:07p 2 ; js2 ðtÞj [ 1; 8  0:06; js2 ðtÞj  1; a22 ðs2 ðtÞÞ ¼ 0:04; js2 ðtÞj [ 1;

Cogn Neurodyn (2016) 10:437–451

 b11 ðs1 ðtÞÞ ¼

8 p > < 0:04 þ ; 16 b12 ðs1 ðtÞÞ ¼ p > : 0:01 þ ; 16  b21 ðs2 ðtÞÞ ¼

0:02; 0:06;

8 p > < 0:09 þ ; 16 b22 ðs2 ðtÞÞ ¼ p > : 1:01 þ ; 16

In order to show that the system is globally exponen~ Þ tially stable, we have to prove ðlp ðGÞ þ lkAk p   I I n n ~ p [ 0, where G ¼ [ lkBk . C D From the given example we calculate the following,   1 0 ; C ¼ C þ In  D ¼ 0 1   5 0 ; D ¼ D  In ¼ 0 7 p ffiffi ffi 0 1 0:04p 2 0:05 B C 8 pffiffiffi C; A~ ¼ B @ A 0:04p 2 0:05 8 0 p1 0:03 0:025 þ B 16 C B~ ¼ @ p A 0:04 0:55 þ 16 and li ¼ 1 for i ¼ 1; 2. ~ ¼ When p ¼ 2, we have l2 ðGÞ ¼ 1; kAk 2 ~ 0:0774; kBk2 ¼ 0:7799. Hence we have 0:9226 [ 0:7763 [ 0. By choosing n1 ¼ n2 ¼ 2:1 and p ¼ 1 we ~ ~ have, l1 ðGÞ ¼ 1:1, kAk 1 ¼ 0:0722, kBk1 ¼ 0:7863. Hence we obtain 1:0278 [ 0:7863 [ 0. Figure 1a, b, depicts the trajectories of system (25) for different initial

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0:01; 0:05;

js1 ðtÞj  1; js1 ðtÞj [ 1; js1 ðtÞj  1; js1 ðtÞj [ 1; js2 ðtÞj  1; js2 ðtÞj [ 1; js2 ðtÞj  1; js2 ðtÞj [ 1:

values. It is clear that the equilibrium is globally exponentially stable. Hence, the Theorem 1 is verified. Example 2 Consider the following pinned inertial MNNs with 10 nodes: d2 xk ðtÞ dxk ðtÞ  Cxk ðtÞ þ Aðxk ðtÞÞf ðxk ðtÞÞ ¼ D 2 dt dt þ Bðxk ðtÞÞf ðxk ðt  sðtÞÞÞ   N X dxl ðtÞ þ xl ðtÞ ; þIþa Wkl C dt l¼1

ð26Þ

where k ¼ 1; 2; . . .; 10. xk ðtÞ ¼ ðxk1 ðtÞ; xk2 ðtÞÞT 2 R2 is the state variable of kth node. The isolated node of MNNs (26) is given as, d2 sðtÞ dsðtÞ  CsðtÞ þ AðsðtÞÞf ðsðtÞÞ ¼ D 2 dt dt þ BðsðtÞÞf ðsðt  sðtÞÞÞ þ I;

ð27Þ

where sðtÞ ¼ ðs1 ðtÞ; s2 ðtÞÞT 2 R2 . Let N ¼ 10, T f ðxk ðtÞÞ ¼ ðtanhðjxk1 ðtÞj  1Þ; tanhðjxk2 ðtÞj  1ÞÞ , I¼ T 0:15et ð0:2; 0:6Þ and sðtÞ ¼ 1þet . So, it is easy to get lk ¼ 1 and q ¼ 0:0375. The coefficient matrices are given as,     0:9 0 0:2 0 , and ,D¼ C¼ 0 0:8 0 0:1

Cogn Neurodyn (2016) 10:437–451

(a)

(b)

20

100 80

0

i

−20

−40

−60

i

−80

u1

−100

−120 0

u (t) and du i (t)/dt, i=1,2

60

u (t) and du i (t)/dt, i=1,2

Fig. 1 a, b The state trajectories of inertial memristive neural networks of Example 1 with different initial values

449

t

20 0 −20 −40

u1

du1 /dt

−60

du 1 /dt

u2

−80

u2

du2 /dt 20

40

−100

40

du 2 /dt 0

20

40

t

Fig. 4 Synchronization error for Example 2 Fig. 2 Digraph with 10 nodes Fig. 3 a State trajectories xk ðtÞ of system (26) and b objective state trajectory s(t) of system (27)

(a)

(b)

123

450

Cogn Neurodyn (2016) 10:437–451



a11 ðxk ðtÞÞ ¼ b11 ðxk ðtÞÞ ¼

0:08; 0:95;  0:03; 

a12 ðxk ðtÞÞ ¼

0:05; 0:05;

0:28; 0:005; b12 ðxk ðtÞÞ ¼ 0:015;  0:95; a21 ðxk ðtÞÞ ¼ 0:05;  0:075; b21 ðxk ðtÞÞ ¼ 0:095;  0:06; a22 ðxk ðtÞÞ ¼ 0:28;  0:02; b22 ðxk ðtÞÞ ¼ 0:08; 

researches that most of the authors have attempted to find a proper controller such that the considered coupled complex network can achieve synchronization, i.e., the synchronization error dynamical network is asymptotically stable. However, controlling all nodes in a network may result in a high cost, and is hard to implement. To tackle these issues, the pinning control scheme has been adopted to study synchronization of coupled complex networks, in which only some selected nodes need to be controlled.

jxk ðtÞj  1; jxk ðtÞj [ 1; jxk ðtÞj  1; jxk ðtÞj [ 1; jxk ðtÞj  1; jxk ðtÞj [ 1; jxk ðtÞj  1; jxk ðtÞj [ 1; jxk ðtÞj  1;

Remark 10 The condition of Theorem 3 is difficult to verify for the case p ¼ 1; 1; x in our example. In Example 1, by introducing the coefficient matrix ni in variable transformation we verified the stability condition for p ¼ 1. But this transformation is not applicable in our synchronization case.

jxk ðtÞj [ 1; jxk ðtÞj  1; jxk ðtÞj [ 1; jxk ðtÞj  1; jxk ðtÞj [ 1; jxk ðtÞj  1; jxk ðtÞj [ 1:

From the given example we calculate the following,   0:3 0 C ¼ C þ In  D ¼ ; 0 0:3   0:1 0 D ¼ D  In ¼ ; 0 0:2   0:5 0:2 and A~~ ¼ 0:5 0:1   0:04 0:01 B~~ ¼ : 0:08 0:05 The coupling matrix W is determined by the directed topology given in Fig. 2. From Fig. 2, it is clear that the pinning node is 1 and 9. The inner coupling matrix C is given as, C ¼ diagf6; 4g and the coupling strength is chosen as a ¼ 20. In order to show that the system is ~~ Þ [ lkBk ~~ [ 0. Choossynchronizable, ðlp ðKÞ þ lkAk p

In this paper, we have discussed the global exponential stability of inertial memristive neural networks and global exponential pinning synchronization of coupled inertial memristive neural networks by using matrix measure strategy and Halanay inequality. Firstly, matrix measure strategies are utilized to analyze the closed-loop error system, under which two sufficient criteria have been established such that the exponential synchronization can be achieved. The model based on the memristor widens the application scope for the design of neural networks. Finally, numerical simulations have been given to demonstrate the effectiveness of our theoretical results. Our future works may involve the stability and synchronization analysis of inertial memristive neural networks under different control schemes.

p

~~ ¼ 0:7616 and ing p ¼ 2 we have, l2 ðKÞ ¼ 0:9805; kAk p ~ ~ kBk ¼ 0:1023. Hence we have, 0:2189 [ 0:1023 [ 0. p

Based on the conclusion of Theorem 3 the coupled inertial memristive neural networks can be exponentially synchronized. The synchronization state trajectories xk ðtÞ; k ¼ 1; 2; . . .; 10 and objective state trajectory s(t) are given in Fig. 3a, b respectively. The synchronization of error system ek ðtÞ; k ¼ 1; 2; . . .; 10 is given in Fig. 4. Hence, the Theorem 3 is verified. Remark 9 Note that coupled complex dynamical networks can be represented by a large number of interconnected nodes, in which each node consists of a dynamical system. Owing to their applications in real life, many researchers have studied coupled complex dynamical networks recently. In addition, it can be seen from previous

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Conclusion

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