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Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays
Information Sciences 232 (2013) 386–396
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Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays Guodong Zhang, Yi Shen ⇑, Quan Yin, Junwei Sun Department of Control Science and Engineering, Huazhong University of Science and Technology, The Key Laboratory of Education Ministry of Image Processing and Intelligent Control, Wuhan 430074, China
a r t i c l e
i n f o
Article history: Received 3 June 2012 Received in revised form 6 September 2012 Accepted 24 November 2012 Available online 13 December 2012 Keywords: Periodic solution Exponential stability Memristor Recurrent neural network Time delay
a b s t r a c t The paper presents theoretical results on the global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays. The dynamic analysis in the paper employs the theory of differential equations with discontinuous right-hand side as introduced by Filippov. By using the inequality techniques and a useful Lyapunov functional, some new testable algebraic criteria are obtained for ensuring the existence and global exponential stability of periodic solution of the system. The model based on the memristor widens the application scope for the design of neural networks, and the new effective results also enrich the toolbox for the qualitative analysis of neural networks. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The first memristor (as a contraction of memory and resistor), originally theorized by Dr. Chua in 1971, was identified by a team at HP Labs in 2008. The memristor is a two-terminal passive device whose value depends on the magnitude and polarity of the voltage applied to it and the length of time that the voltage has been applied. From the previous work [3,6,7,17,22,25,27–29,31], we know that the potential applications of this device is in next generation computer and powerful brain-like neural computer. As we know, the recurrent neural networks are very important nonlinear circuit networks because of their wide applications in associative memory, pattern recognition, signal processing and so on, for reference, see [4,5,9,11,13–16,18– 21,23,24,26,32–34]. And the Hopfield neural network model can be implemented in a circuit where the connection weights are implemented by resistors, motivated by these facts, recently, by using memristors instead of resistors, Bao and Zeng [3], Wu et al. [27–29] and Zhang et al. [31] have studied a new model where the connection weights change according to its state, i.e., a state-dependent switching recurrent neural networks. Different from the previous works [3,27–29,31], in this paper, we will deal with the problem of existence and global exponential stability of periodic solution for a class of memristor-based recurrent neural networks with multiple delays as follows: n n X X dxi ðtÞ aij ðxi ðtÞÞfj ðxj ðtÞÞ þ bij ðxi ðtÞÞg j ðxj ðt sij ÞÞ þ Ii ðtÞ; ¼ di ðxi ðtÞÞxi ðtÞ þ dt j¼1 j¼1
⇑ Corresponding author. Tel.: +86 27 87543630; fax: +86 27 87543130. E-mail addresses: [email protected] (G. Zhang), [email protected] (Y. Shen). 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2012.11.023
t P 0;
i ¼ 1; 2; . . . ; n;
ð1Þ
387
G. Zhang et al. / Information Sciences 232 (2013) 386–396
where
( di ðxi ðtÞÞ ¼
di ;
jxi ðtÞj < T i ;
di ;
jxi ðtÞj > T i ;
( bij ðxi ðtÞÞ ¼
bij ;
( aij ðxi ðtÞÞ ¼
aij ;
jxi ðtÞj < T i ;
a jxi ðtÞj > T i ; ij ;
jxi ðtÞj < T i ;
bij ; jxi ðtÞj > T i ;
in which switching jumps T i > 0; di > 0; di > 0; aij ; a ij ; bij ; bij ; i; j ¼ 1; 2; . . . ; n, are all constant numbers, fj and g j : R ! R are bounded continuous functions, Ii ðtÞ is a continuous x-periodic external input function. The organization of this paper is as follows. Some preliminaries are introduced in Section 2. The main results are given in Section 3. And then, numerical simulations are given to demonstrate the effectiveness of the proposed approach in Section 4. Finally, this paper ends by a conclusion. Remark 1. The authors in [27,29] have given a clear exposition about the relation between memristances and the coefficients of switching system (1), so readers can consult [27,29] to get more explanation.
2. Preliminaries
Let
Throughout this paper, solutions of all the systems considered in the following are intended in Filippov’s sense (see [12]). P s ¼ max16i;j6n fsij g, we define kwk ¼ sups6t60 ½ ni¼1 jwi ðtÞjp 1=p ; p P 1, for 8w ¼ ðw1 ðtÞ; w2 ðtÞ; . . . ; wn ðtÞÞ 2 Cð½s; 0; Rn Þ,
co½ni ; ni denotes the convex hull of ½ni ; ni . For a continuous function kðtÞ : R ! R; Dþ kðtÞ is called the upper right dini derivative and defined as Dþ kðtÞ ¼ limh!0þ 1h ðkðt þ hÞ kðtÞÞ. System (1) has the following form initial conditions: xðsÞ ¼ /ðsÞ ¼ ð/1 ðsÞ; /2 ðsÞ; . . . ; /n ðsÞÞT 2 Cð½s; 0; Rn Þ. For convenience, now, we first introduce the following Definitions about set-valued map and differential inclusion in [1,2,12]. Definition 1. Let E Rn ; x # FðxÞ is called a set-valued map from E,!Rn , if to each point x of a set E Rn , there corresponds a nonempty set FðxÞ Rn . Definition 2. A set-valued map F with nonempty values is said to be upper semi-continuous at x0 2 E Rn if, for any open set N containing Fðx0 Þ, there exists a neighborhood M of x0 such that FðMÞ N. FðxÞ is said to have a closed (convex, compact) image if for each x 2 E; FðxÞ is closed (convex, compact). Definition 3. For the system
UðxÞ ¼
\ \
dx dt
¼ f ðxÞ; x 2 Rn , with discontinuous right-hand sides, a set-valued map is defined as
co½f ðBðx; dÞ n NÞ;
d>0lðNÞ¼0
where co½E is the closure of the convex hull of set E; Bðx; dÞ ¼ fy : ky xk 6 dg, and lðNÞ is Lebesgue measure of set N. A solution in Filippov’s sense [12] of the Cauchy problem for this system with initial condition xð0Þ ¼ x0 is an absolutely continuous function xðtÞ; t 2 ½0; T, which satisfies xð0Þ ¼ x0 and differential inclusion:
dx 2 UðxÞ; dt
for a:e: t 2 ½0; T:
By applying the above theories of set-valued maps and differential inclusions [1,2,8,12], the memristor-based recurrent neural networks (1) can be written as the following differential inclusion: n n X X dxi ðtÞ 2 co½di ; di xi ðtÞ þ co½aij ; aij fj ðxj ðtÞÞ þ co½bij ; bij g j ðxj ðt sij ÞÞ þ Ii ðtÞ; dt j¼1 j¼1
for a:e: t P 0;
i ¼ 1; 2; . . . ; n;
ð2Þ
where
di ¼ minfdi ; di g; aij ¼ maxfaij ; a ij g;
di ¼ maxfdi ; di g; aij ¼ minfaij ; a ij g; bij ¼
minfbij ; bij g;
bij ¼ maxfbij ; bij g;
or equivalently, there exist b d i 2 co½di ; di ; b a ij 2 co½aij ; aij ; b b ij 2 co½bij ; bij , such that n n X X dxi ðtÞ b b b ij g j ðxj ðt sij ÞÞ þ Ii ðtÞ; a ij fj ðxj ðtÞÞ þ ¼ b d i xi ðtÞ þ dt j¼1 j¼1
for t P 0;
i ¼ 1; 2; . . . ; n:
ð3Þ
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G. Zhang et al. / Information Sciences 232 (2013) 386–396
Definition 4. A function (in Filippov’s sense) x ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT is a solution of system (1), with initial conditions /ðsÞ ¼ ð/1 ðsÞ; . . . ; /n ðsÞÞT 2 Cð½s; 0; Rn Þ, if x ðtÞ is an absolutely continuous function and satisfies the differential inclusion n n X X dxi ðtÞ co½aij ; aij fj ðxj ðtÞÞ þ co½bij ; bij g j ðxj ðt sij ÞÞ þ Ii ðtÞ; 2 co½di ; di xi ðtÞ þ dt j¼1 j¼1
for a:e: t P 0;
i ¼ 1; 2; . . . ; n; ð4Þ
or equivalently, there exist b d i 2 co½di ; di ; b a ij 2 co½aij ; aij ; b b ij 2 co½bij ; bij , such that n n X X dxi ðtÞ b b b ij g j ðxj ðt sij ÞÞ þ Ii ðtÞ; a ij fj ðxj ðtÞÞ þ ¼ b d i xi ðtÞ þ dt j¼1 j¼1
for t P 0;
i ¼ 1; 2; . . . ; n:
ð5Þ
Definition 5. Let x ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT is a solution of system (1), with initial conditions /ðsÞ ¼ ð/1 ðsÞ; /2 ðsÞ; . . . ; /n ðsÞÞT 2 Cð½s; 0; Rn Þ, the solution x ðtÞ of system (1) is said to be globally exponential stability, if there exist positive constants e and b P 1 such that for any solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT of system (1) with initial conditions wðsÞ ¼ ðw1 ðsÞ; w2 ðsÞ; . . . ; wn ðsÞÞT 2 Cð½s; 0; Rn Þ satisfies
" #1=p n X xi ðtÞ x ðtÞp 6 beet kw /k; i
for 8t P 0:
ð6Þ
i¼1
Now we do the following assumptions for the system (1): (H1) For i ¼ 1; 2; :; n; 8s1 ; s2 2 R; s1 – s2 , the neuron activation functions fi ; g i are bounded and satisfy condition
jfi ðs1 Þ fi ðs2 Þj 6 ri js1 s2 j;
jg i ðs1 Þ g i ðs2 Þj 6 qi js1 s2 j;
where ri > 0; qi > 0. (H10 ) For i ¼ 1; 2; :; n; 8s1 ; s2 2 R; s1 – s2 , the neuron activation functions fi ; g i are bounded and satisfy condition
0 < ri <
fi ðs1 Þ fi ðs2 Þ 6 ri ; s1 s2
jg i ðs1 Þ g i ðs2 Þj 6 qi js1 s2 j;
where ri > 0; ri > 0; qi > 0. (H2) For the differential inclusion (2), the following conditions hold
co½di ; di yi ðtÞ co½di ; di xi ðtÞ # co½di ; di ðyi ðtÞ xi ðtÞÞ; co½aij ; aij fj ðyj ðtÞÞ co½aij ; aij fj ðxj ðtÞÞ # co½aij ; aij ðfj ðyj ðtÞÞ fj ðxj ðtÞÞÞ; co½bij ; bij g j ðyj ðtÞÞ co½bij ; bij g j ðxj ðtÞÞ # co½bij ; bij ðg j ðyj ðtÞÞ g j ðxj ðtÞÞÞ; where xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT ; yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞ; . . . ; yn ðtÞÞ are two solutions of system (1) with initial conditions /ðsÞ ¼ ð/1 ðsÞ; . . . ; /n ðsÞÞT ; wðsÞ ¼ ðw1 ðsÞ; . . . ; wn ðsÞÞT 2 Cð½s; 0; Rn Þ, respectively.
Lemma 1. Each solution xðtÞ with an initial condition in Cð½s; 0; Rn Þ of system (1) is bounded and defined on ½s; þ1Þ. And when IðtÞ is a constant external input, system (1) exists at least one equilibrium point. 0
Proof. Under assumption (H1) or (H1 ), the neuron activation functions fi ; g i in (1) are bounded and Lipschitz continuous. And di > 0; di > 0; aij ; a ij ; bij ; bij ; Ii ðtÞ are all bounded, then, we have the local existence of a solution xðtÞ with initial condition /ðsÞ ¼ ð/1 ðsÞ; /2 ðsÞ; . . . ; /n ðsÞÞT 2 Cð½s; 0; Rn Þ of system (1) is a straightforward consequence of [12], and there exists a constant L0i > 0 such that n n X X jaij ðxi ðtÞÞfj ðxj ðtÞÞj þ jbij ðxi ðtÞÞg j ðxj ðt sij ÞÞj þ jIi ðtÞj 6 L0i ; j¼1
j¼1
Now, from di 6 j b d i j and system (3), we have
jxi ðtÞj 6 ðjxi ð0Þj L0i Þedi t þ L0i ! L0i ;
djxi ðtÞj dt
þ di jxi ðtÞj 6 L0i , that is,
for t # þ 1;
then, we obtain that all the solutions xðtÞ of system (1) are bounded for a.e. t P 0, therefore, by the Continuation Theorem [12, Th2, p. 78] and [8], we know that each solution with an initial condition /ðsÞ ¼ ð/1 ðsÞ; . . . ; /n ðsÞÞT 2 Cð½s; 0; Rn Þ of system (1) is defined on ½s; þ1Þ. And when IðtÞ is a constant input, system (3) exists at least one equilibrium point, then, system (1) exists at least one equilibrium point. This completes the proof. h
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G. Zhang et al. / Information Sciences 232 (2013) 386–396 1c
Lemma 2. Assume that a > 0; b > 0; 0 < c < 1, then the inequality: ac b
6 ca þ ð1 cÞb holds.
Proof. Consider the function qðxÞ ¼ xc cx ð1 cÞ, obviously, the maximum of qðxÞ on ð0; þ1Þ is qð1Þ ¼ 0, then, we obtain c for 8x > 0; qðxÞ 6 qð1Þ, that is, xc cx ð1 cÞ 6 0. Now, taking x ¼ ba > 0, we get ðab Þc c ba ð1 cÞ 6 0, i.e., abc 6 c ba þ ð1 cÞ, 1c multiplied the inequality above by b, we have ac b 6 ca þ ð1 cÞb. The inequality is the transmutation of the Young inequality. This completes the proof. h Lemma 3 [10]. If X is a Banach space, P kðcÞ ðRn Þ denote the collection of all nonempty, compact and convex subset of Rn , C # X is nonempty convex with 0 2 C and G : C ! PkðcÞ ðCÞ is an upper semi-continuous which maps bounded sets into relatively compact sets, then one of the following statements is true: (i) the set C ¼ fx 2 C : x 2 kGðxÞ; k 2 ð0; 1Þg is unbounded; (ii) the GðÞ has a fixed point in C, i.e., there exists x 2 C such that x 2 GðxÞ. In next section, the paper aims to analysis the existence and global exponential stability of periodic solution for system (1). 3. Main results Theorem 1. Under assumption (H1) and (H2), if there exists constants nij ; nij ; fij ; fij 2 R; ci > 0 and p P 1, such that
n pn pf 1=ðp1Þ pn pf 1=ðp1Þ X n f n f ðp 1Þ Aij ij rj ij þ cj =ci Ajiji ri ji þ ðp 1Þ Bij ij qj ij þ ðcj =ci ÞBjiji qi ji < pdi ;
ð7Þ
j¼1
where Aij ¼ maxfjaij j; jaij jg, Bij ¼ maxfjbij j; jbij jg; i; j ¼ 1; 2; . . . ; n. Then, system (1) exists exactly one x-periodic solution and all other solutions of system (1) converge exponentially to it as t ! þ1. P Proof. For 8w ¼ ðw1 ðtÞ; w2 ðtÞ; . . . ; wn ðtÞÞ 2 Cð½s; 0; Rn Þ, we define the norm kwk ¼ sups6t60 ½ ni¼1 jwi ðtÞjp 1=p , then, Cð½s; 0; Rn Þ n is a Banach space. For all / 2 Cð½s; 0; R Þ, we denote the solutions of system (1) through ð0; /Þ as xðt; /Þ ¼ ðx1 ðt; /Þ; x2 ðt; /Þ; . . . ; xn ðt; /ÞÞT . Now, under assumption (H1) and (H2), we prove that the system (1) exists at least one x-periodic solution. Let L1 ðT 0 ; Rn Þ denotes the Banach space of all functions x : T 0 ¼ ½0; x ! Rn which are Bochner integrable. Let n 1;1 W 1;1 ðT 0 ; Rn Þ ¼ fx 2 L1 ðT 0 ; Rn Þ : x_ 2 L1 ðT 0 ; Rn Þg, W 1;1 ðT 0 ; Rn Þ : xð0Þ ¼ xðxÞg and LðxÞ ¼ x_ þ x for all p ðT 0 ; R Þ ¼ fx 2 W n 1;1 n 1 n x 2 W 1;1 ðT ; R Þ, then by the literature [10,21,30], we know L : W ðT 0 0 ; R Þ ! L ðT 0 ; R Þ is a linear operator and bijective p p 1 (one to one and surjective), and L is completely continuous. For any x 2 L1 ðT 0 ; Rn Þ, define
F i ðt; xÞ ¼ co½di ; di xi ðtÞ þ
n n X X co½aij ; aij fj ðxj ðtÞÞ þ co½bij ; bij g j ðxj ðt sij ÞÞ þ Ii ðtÞ; j¼1
i ¼ 1; 2; . . . ; n;
j¼1
NðxÞ ¼ fv 2 L1 ðT 0 ; Rn Þ : v 2 Fðt; xÞ ¼ ðF 1 ðt; xÞ; . . . ; F n ðt; xÞÞT ;
a:e: on T 0 g:
And from Ref. [26], we get NðÞ is upper semi-continuous and a bounded operator from L1 ðT 0 ; Rn Þ to L1 ðT 0 ; Rn Þw (denote weak topology), then, we obtain that L1 N : L1 ðT 0 ; Rn Þ ! P kðcÞ ðL1 ðT 0 ; Rn ÞÞ is upper semi-continuous and maps bounded sets into reln 1 atively compact sets. Let C ¼ fx 2 W 1;1 p ðT 0 ; R Þ : x 2 aL NðxÞ; a 2 ð0; 1Þg, and as the same proof of Lemma 1, we can easy get the x-periodic solutions of the following system are bounded,
(
) n n X X x_ i ðtÞ 2 ð1 aÞxi ðtÞ a co½di ; di xi ðtÞ þ co½aij ; aij fj ðxj ðtÞÞ þ co½bij ; bij g j ðxj ðt sij ÞÞ þ Ii ðtÞ ; j¼1
j¼1
or, there exist b d i 2 co½di ; di ; b a ij 2 co½aij ; aij ; b b ij 2 co½bij ; bij , such that
"
d i xi ðtÞ þ x_ i ðtÞ ¼ ð1 aÞxi ðtÞ a b
n X j¼1
b a ij fj ðxj ðtÞÞ þ
n X
# b b ij g j ðxj ðt sij ÞÞ þ Ii ðtÞ ;
j¼1
where a 2 ð0; 1Þ; i ¼ 1; 2; . . . ; n. That is, the set C is bounded. By the Lemma 3 (Leray–Schauder alternative theorem) and n 1 [12,21,26], there exists x 2 W 1;1 p ðT 0 ; R Þ such that x 2 L NðxÞ. Clearly, x is an x-periodic solution of system (1).
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G. Zhang et al. / Information Sciences 232 (2013) 386–396
And now, Let xðt; u Þ is an x-periodic solution of system (1). Under assumption (H1) and (H2), it follows from (2) or (3) that
½xi ðt; /Þ xi ðt; u Þ0 2 co½di ; di ðxi ðt; /Þ xi ðt; u ÞÞ þ
n n X X co½aij ; aij ½ fj ðxj ðt; /ÞÞ fj ðxj ðt; u ÞÞ þ co½bij ; bij j¼1
j¼1
½g j ðxj ðt sij ; /ÞÞ g j ðxj ðt sij ; u ÞÞ; P 0;
for a:e: t
i ¼ 1; 2; . . . ; n:
For i; j ¼ 1; 2; . . . ; n, we can choose a small
ð8Þ
e > 0 such that
n X
pn pf 1=ðp1Þ pn pf 1=ðp1Þ n f n f ðp 1Þ Aij ij rj ij þ ðcj =ci ÞAjiji ri ji þ ðp 1Þ Bij ij qj ij þ epes ðcj =ci ÞBjiji qi ji < pðdi eÞ:
ð9Þ
j¼1
Now, we consider a Lyapunov functional as
" Z n n X X f VðtÞ ¼ ci jxi ðt; /Þ xi ðt; u Þjp epet þ jb b ij jnij qj ij i¼1
#
t
p peðsþsij Þ
jxj ðt; /Þ xj ðt; u Þj e
ds ;
ð10Þ
tsij
j¼1
where b b ij 2 co½bij ; bij . By calculating the upper right derivation Dþ VðtÞ of VðtÞ along the solution to (2) or (3), we obtain Dþ VðtÞjð2Þ or ð3Þ ¼ epet
n n X X pci jxi ðt;/Þ xi ðt; u Þjp1 sgnðxi ðt;/Þ xi ðt; u ÞÞ ½xi ðt;/Þ xi ðt; u Þ0 þ epet peci jxi ðt;/Þ i¼1
i¼1
n X n X f ci j b b ij jnij qj ij ½epesij jxj ðt;/Þ xj ðt; u Þjp xi ðt; u Þj þ e
p
pet
i¼1 j¼1 p
jxj ðt sij ;/Þ xj ðt sij ; u Þj 2 epet
n X p1 pci jxi ðt;/Þ xi ðt; u Þj sgnðxi ðt;/Þ xi ðt; u ÞÞ i¼1
(
n X co½di ;di ðxi ðt;/Þ xi ðt; u ÞÞ þ co½aij ;aij ½fj ðxj ðt;/ÞÞ fj ðxj ðt; u ÞÞ
j¼1
) n n X X
þ co½bij ;bij g j ðxj ðt sij ;/ÞÞ g j ðxj ðt sij ; u ÞÞ þ epet peci jxi ðt;/Þ xi ðt; u Þjp j¼1
i¼1
n X n X f þ epet ci j b b ij jnij qj ij ½epesij jxj ðt;/Þ xj ðt; u Þjp jxj ðt sij ;/Þ xj ðt sij ; u Þjp i¼1 j¼1
( n n X X pet p1 6e pci jxi ðt;/Þ xi ðt; u Þj ðe di Þjxi ðt;/Þ xi ðt; u Þj þ Aij jfj ðxj ðt;/ÞÞ fj ðxj ðt; u ÞÞj i¼1
j¼1
) n n X n X X n f þ Bij jg j ðxj ðt sij ;/ÞÞ g j ðxj ðt sij ; u ÞÞj þ epet ci Bijij qj ij epes jxj ðt;/Þ xj ðt; u Þjp j¼1
i¼1 j¼1
p
jxj ðt sij ;/Þ xj ðt sij ; u Þj :
ð11Þ
Now, under assumption (H1) and (H2), and by using Lemma 2, we obtain the following estimate for the right-hand side of (11) Dþ VðtÞjð2Þ or ð3Þ 6 epet
n X
(" pci ðe di Þ þ epes
i¼1
# n n X X n f cj Bjiji qi ji jxi ðt;/Þ xi ðt; u Þjp þ Aij pci rj jxj ðt;/Þ xj ðt; u Þjjxi ðt;/Þ j¼1
p1
xi ðt; u Þj
þ
n X
j¼1
Bij pci qj jxi ðt;/Þ xi ðt; u Þj
j¼1 n X xj ðt sij ; u Þjp ¼ epet
(" pci ðe di Þ þ epes
i¼1
þ pci
n X
pn ½ðAij ij
r
p1
n X
þ pci
pnij
Bij
pfij 1=ðp1Þ
qj
p 11=p
jxi ðt;/Þ xi ðt; u Þjp
j¼1
j¼1 n
f
io
f
n
f
Bijij qj ij jxj ðt sij ;/Þ
j¼1
#
cj Bjiji qi ji jxi ðt;/Þ xi ðt; u Þjp
u Þj
n X n f ci Bijij qj ij jxj ðt sij ;/Þ xj ðt sij ; u Þjp
þ epes cj Bjiji qi ji
n
n X
j¼1
pfij 1=ðp1Þ Þ jxi ðt;/Þ xi ðt; j
j¼1
n X
jxj ðt sij ;/Þ xj ðt sij ; u Þj ci
1=p n f Aijij rj ij jxj ðt;/Þ xj ðt; u Þjp
11=p
) 6 epet
n X i¼1
jxi ðt;/Þ xi ðt; u Þjp 6 0:
n f 1=p Bijij qj ij jxj ðt sij ;/Þ xj ðt sij ; u Þjp
( pci ðe di Þ þ
n h X
pnij
ci ðp 1ÞðAij
n f pn pf ij 1=ðp1Þ rpf Þ þ cj Ajiji ri ji þ ci ðp 1ÞðBij ij qj ij Þ1=ðp1Þ j
j¼1
ð12Þ
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G. Zhang et al. / Information Sciences 232 (2013) 386–396
Thus,
VðtÞ 6 Vð0Þ;
for t P 0:
ð13Þ
Since
Vð0Þ ¼
" Z n n X X f ci j/i ð0Þ ui ð0Þjp þ jb b ij jnij qj ij i¼1
j¼1
"
6 maxci þ sepes 16i6n
0
# jxj ðs; /Þ xj ðs; u Þjp epeðsþsij Þ ds
sij
# n n f X ji ji cj max Bji qi j/i ðtÞ ui ðtÞjp ; sup
n X j¼1
16i6n
ð14Þ
s6t60 i¼1
And from (10), for t P 0, we have
VðtÞ P
X n n X ci epet jxi ðt; /Þ xi ðt; u Þjp P min ci epet jxi ðt; /Þ xi ðt; u Þjp : 16i6n
i¼1
ð15Þ
i¼1
Then, from (13)–(15), we get
" #1=p " #1=p n n X X p p et jxi ðt; /Þ xi ðt; u Þj 6 be sup j/i ðtÞ ui ðtÞj ; s6t60
i¼1
i¼1
that is,
kxðt; /Þ xðt; u Þk 6 beet k/ u k;
ð16Þ
where
n f 31=p 2 Pn max16i6n ci þ sepes j¼1 cj max16i6n Bjiji qi ji 5 P 1: b¼4 min16i6n ci From (16), we know that all other solutions of system (1) converge exponentially to the x-periodic solution xðt; u Þ as t ! þ1 and it is also to get xðt; u Þ is the unique x-periodic solution of system (1). This completes the proof. h Theorem 2. Under assumption (H1) and (H2), if n X ½Aij rj þ Aji ri þ Bij qj þ Bji qi < 2di ;
ð17Þ
j¼1
where Aij ¼ maxfjaij j; jaij jg, Bij ¼ maxfjbij j; jbij jg; i ¼ 1; 2; . . . ; n. Then, system (1) exists exactly one x-periodic solution and all other solutions of system (1) converge exponentially to it as t ! þ1. nij ¼ fij ¼ fij ¼ 1; i; j ¼ 1; 2; . . . ; n. h Proof. We can obtain Theorem 2 directly from Theorem 1 by taking p ¼ 2; ci ¼ 1; nij ¼ Now, from Theorem 1 and 2, we can get the following Corollary 1. Corollary 1. Under assumption (H1) and (H2), if Ii ðtÞ ¼ Ii and (7) or (17) hold. Then, the equilibrium point x of system (1) is globally exponentially stable.
Theorem 3. Under assumption (H10 ) and (H2), if there exists constants ci > 0 and p P 1, aii < 0; a ii < 0, such that
pdi þ pri Aii þ
n X
½ðp 1ÞAij þ ðcj =ci ÞAji rpi þ
j¼1;j–i
n X ½ðp 1ÞBij þ ðcj =ci ÞBji qpi < 0;
ð18Þ
j¼1
where Aii ¼ aii ; Aij ¼ maxfjaij j; jaij jgði – jÞ, Bij ¼ maxfjbij j; jbij jg; i; j ¼ 1; 2; . . . ; n. Then, system (1) exists exactly one x-periodic solution and all other solutions of system (1) converge exponentially to it as t ! þ1. Proof. For i; j ¼ 1; 2; . . . ; n, we can choose a small
ðe pdi Þ þ pri Aii þ
n X j¼1;j–i
e > 0 such that
ðp 1ÞAij þ ðcj =ci ÞAji rpi þ
n X ½ðp 1ÞBij þ ðcj =ci Þees Bji qpi < 0;
ð19Þ
j¼1
And now, as the proof in Theorem 1, we know system (1) exists at least one x-periodic solution. Let xðt; u Þ is an x-periodic solution of system (1). Here, we consider a Lyapunov functional as
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G. Zhang et al. / Information Sciences 232 (2013) 386–396
VðtÞ ¼
Z n n X X ci ½jxi ðt; /Þ xi ðt; u Þjp eet þ jb b ij j i¼1
t
tsij
j¼1
jg j ðxj ðt; /ÞÞ g j ðxj ðt; u ÞÞjp eeðsþsij Þ ds;
ð20Þ
where b b ij 2 co½bij ; bij . As the proof of Theorem 1, by calculating the upper right derivation Dþ VðtÞ of VðtÞ along the solution to (2) or (3), we obtain " n n X X p p D VðtÞjð2Þ or ð3Þ 6 e ci ðe pdi Þjxi ðt;/Þ xi ðt; u Þj þ pri Aii jxi ðt;/Þ xi ðt; u Þj þ pAij jxi ðt;/Þ et
þ
i¼1
j¼1;j–i
xi ðt; u Þj
p1
n X jfj ðxj ðt;/ÞÞ fj ðxj ðt; u ÞÞj þ pBij jxi ðt;/Þ xi ðt; u Þjp1 jg j ðxj ðt sij ;/ÞÞ j¼1
# n n X X g j ðxj ðt sij ; u ÞÞj þ e Bij jg j ðxj ðt;/ÞÞ g j ðxj ðt; u ÞÞjp Bij jg j ðxj ðt sij ;/ÞÞ g j ðxj ðt sij ; u ÞÞjp : es
j¼1
ð21Þ
j¼1
Now, under assumption (H10 ) and (H2) and by using Lemma 2, we obtain the following estimate for the right-hand side of (21) Dþ VðtÞjð2Þ
or ð3Þ
6 ee t
" n n X X p1 1 ci ðe pdi þ pAii ri Þjxi ðt; /Þ xi ðt; u Þjp þ pAij ðjxi ðt; /Þ xi ðt; u Þjp Þ p ðjfj ðxj ðt; /ÞÞ fj ðxj ðt; u ÞÞjp Þp i¼1
j¼1;j–i
n n X X p1 1 pBij ðjxi ðt; /Þ xi ðt; u Þjp Þ p ðjg j ðxj ðt sij ; /ÞÞ g j ðxj ðt sij ; u ÞÞjp Þp þ ees Bij jg j ðxj ðt; /ÞÞ g j ðxj ðt; u ÞÞjp þ j¼1
j¼1
# n n X X Bij jg j ðxj ðt sij ; /ÞÞ g j ðxj ðt sij ; u ÞÞjp 6 eet ci e pdi þ pAii ri jxi ðt; /Þ xi ðt; u Þjp j¼1
i¼1
n X
þ
p
p j jxj ðt; /Þ
Aij ½ðp 1Þjxi ðt; /Þ xi ðt; u Þj þ r
j¼1;j–i n X þ ees Bij qpj jxj ðt; /Þ xj ðt; u Þjp j¼1
)
xj ðt; u Þjp þ
n X Bij ðp 1Þjxi ðt; /Þ xi ðt; u Þjp j¼1
( n n X X et 6e ci ðe pdi þ pAii ri Þ þ ½ci Aij ðp 1Þ þ cj Aji rpi i¼1
j¼1;j–i
) n X ½ci Bij ðp 1Þ þ ees cj Bji qpi jxi ðt; /Þ xi ðt; u Þjp 6 0: þ
ð22Þ
j¼1
Then, as the proof of Theorem 1, we can have e
kxðt; /Þ xðt; u Þk 6 b ept k/ u k;
ð23Þ
where
" b ¼
#1=p P max16i6n ci þ sees nj¼1 cj max16i6n ðBji qpi Þ P 1: min16i6n ci
Then, we know that all other solutions of system (1) converge exponentially to a unique x-periodic solution as t ! þ1. This completes the proof. h Theorem 4. Under assumption (H10 ) and (H2), if there exists constants ci > 0 and p P 1, aii < 0; a ii < 0; and one of the following conditions hold
di þ ri Aii þ
n X
Aji ri þ
j¼1;j–i
pdi þ pri Aii þ
n X Bji qi < 0;
ð24Þ
j¼1
n h X
pnij
ðp 1ÞðAij
i
n f ij 1=ðp1Þ rpf Þ þ ðcj =ci ÞAjiji ri ji þ j
j¼1;j–i
n h X
pnij
ðp 1ÞðBij
pfij 1=ðp1Þ
qj
Þ
n f i þ ðcj =ci ÞBjiji qi ji < 0;
j¼1
ð25Þ
2di þ 2ri Aii þ
n X
½Aij rj þ Aji ri þ
j¼1;j–i
n X ½Bij qj þ Bji qi < 0;
ð26Þ
j¼1
where Aii ¼ aii ; Aij ¼ maxfjaij j; jaij jgði – jÞ, Bij ¼ maxfjbij j; jbij jg; i; j ¼ 1; 2; . . . ; n. Then, system (1) exists exactly one x-periodic solution and all other solutions of system (1) converge exponentially to it as t ! þ1.
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G. Zhang et al. / Information Sciences 232 (2013) 386–396 0.25 0.2 0.15 0.1
States
0.05 0
−0.05 −0.1 −0.15 −0.2 −0.25
0
5
10
15
20
25
30
35
40
45
50
time − T Fig. 1. State trajectory xi ðtÞ ði ¼ 1; 2Þ of the DRNN system (27) with eternal input I1 ðtÞ ¼ cos t; I2 ðtÞ ¼ 0:5 sin 2t.
Proof. We can obtain (24) directly from Theorem 3 by taking p ¼ 1; ci ¼ 1; i ¼ 1; 2; . . . ; n. And if (25) hold, we can get the result by the same proof of Theorem 1 and 3. We can obtain (26) directly from (25) by taking p ¼ 2; ci ¼ 1; nij ¼ nij ¼ fij ¼ fij ¼ 1; i; j ¼ 1; 2; . . . ; n. This completes the proof. h Now, from Theorem 3 and 4 above, we can get the following Corollary 2. Corollary 2. Under assumption (H10 ) and (H2), if Ii ðtÞ ¼ Ii , aii < 0; aii < 0, and (18) or (24) or (25) or (26) old. Then, the equilibrium point x of system (1) is globally exponentially stable. Remark 2. The proposed results are different from those in [3,27–29,31] and the references cited therein. And the results of this paper also complement previously known results. In addition, these conditions, which can be directly derived from the parameters of the neural networks, are very easily verified.
4. Numerical simulations Now, we perform some numerical simulations to illustrate our analysis by using MATLAB (7.0) programming. System 1. Consider two-dimensional memristor-based neural networks
dx1 ðtÞ ¼ d1 ðx1 ðtÞÞx1 ðtÞ þ a11 ðx1 ðtÞÞf1 ðx1 ðtÞÞ þ a12 ðx1 ðtÞÞf2 ðx2 ðtÞÞ þ b11 ðx1 ðtÞÞg 1 ðx1 ðt s11 ÞÞ þ b12 ðx1 ðtÞÞg 2 ðx2 ðt s12 ÞÞ dt þ I1 ðtÞ; dx2 ðtÞ ¼ d2 ðx2 ðtÞÞx2 ðtÞ þ a21 ðx2 ðtÞÞf1 ðx1 ðtÞÞ þ a22 ðx2 ðtÞÞf2 ðx2 ðtÞÞ þ b21 ðx2 ðtÞÞg 1 ðx1 ðt s21 ÞÞ þ b22 ðx2 ðtÞÞg 2 ðx2 ðt dt s22 ÞÞ þ I2 ðtÞ; where
d1 ðx1 ðtÞÞ ¼
d2 ðx2 ðtÞÞ ¼
4;
jx1 ðtÞj < 1;
4:5; jx1 ðtÞj > 1;
5; jx2 ðtÞj < 1; 4; jx2 ðtÞj > 1;
a21 ðx2 ðtÞÞ ¼
1; jx2 ðtÞj < 1; 1;
jx2 ðtÞj > 1;
a11 ðx1 ðtÞÞ ¼
a12 ðx1 ðtÞÞ ¼
1; jx1 ðtÞj < 1; 1;
1;
jx1 ðtÞj < 1;
1; jx1 ðtÞj > 1; (
b11 ðx1 ðtÞÞ ¼
jx1 ðtÞj > 1;
1 ; 2
jx1 ðtÞj < 1; 12 ; jx1 ðtÞj > 1;
ð27Þ
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G. Zhang et al. / Information Sciences 232 (2013) 386–396
a22 ðx2 ðtÞÞ ¼
And
(
jx2 ðtÞj < 1;
b21 ðx2 ðtÞÞ ¼
2; jx2 ðtÞj > 1; (
b12 ðx1 ðtÞÞ ¼
2;
1 ; 4
(
12 ; jx1 ðtÞj < 1; 1 ; jx1 ðtÞj > 1; 2
b22 ðx2 ðtÞÞ ¼
jx2 ðtÞj < 1;
14 ; jx2 ðtÞj > 1; 14 ; jx2 ðtÞj < 1; 1 ; jx2 ðtÞj > 1: 4
s11 ¼ s22 ¼ 1:5; s12 ¼ s21 ¼ 2; IðtÞ ¼ ðI1 ðtÞ; I2 ðtÞÞT , take the activation function as follows: f1 ðxÞ ¼ f2 ðxÞ ¼ g 1 ðxÞ ¼ g 2 ðxÞ ¼ ðex ex Þ=ðex þ ex Þ:
Obviously, we have r1 ¼ r2 ¼ q1 ¼ q2 ¼ 1, d1 ¼ d2 ¼ 4; A11 ¼ A12 ¼ A21 ¼ 1; A22 ¼ 2; B11 ¼ B12 ¼ 12 ; B21 ¼ B22 ¼ 14. From (17) of Theorem 2, we get
0.25 0.2 0.15
X2 (t)
0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 −0.25 −0.2 −0.15 −0.1 −0.05 0
0.05 0.1 0.15 0.2
0.25
X1 (t) Fig. 2. The phase plot of system (27) corresponding to Fig. 1.
2.5 2 1.5 1
States
0.5 0 −0.5 −1 −1.5 −2 −2.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time − T Fig. 3. State trajectory xi ðtÞ ði ¼ 1; 2Þ of the DRNN system (27) with eternal input I1 ðtÞ ¼ 2:7; I2 ðtÞ ¼ 2:8.
G. Zhang et al. / Information Sciences 232 (2013) 386–396
395
23 < 2d1 ¼ 8; 4 29 < 2d2 ¼ 8: ¼ 4
2A11 þ A12 þ A21 þ 2B11 þ B12 þ B21 ¼ A12 þ A21 þ 2A22 þ B12 þ B21 þ 2B22
Thus, the conditions required in Theorem 2 are satisfied. Now, we choose the eternal input I1 ðtÞ ¼ cos t; I2 ðtÞ ¼ 0:5 sin 2t, by Theorem 2, we know that a unique x-periodic solution x ðtÞ of system (27) is globally exponentially stable, the simulation results of network (27) with 15 random initial values are depicted in Fig. 1. The phase plot of system (27) corresponding to Fig. 1 are shown in Fig. 2, and here the memristor-based recurrent neural networks (27) exhibit clear 8-shaped curves. And when we choose the eternal input I1 ðtÞ ¼ 2:7; I2 ðtÞ ¼ 2:8, by Corollary 1, we know that there is a unique equilibrium point x of system (27) and x is globally exponentially stable, the simulation results of network (27) with 15 random initial values are depicted in Fig. 3. Some examples can also be given for other Theorems, here they are omitted. 5. Conclusion In this paper, under the framework of Filippov’s solution, we made an effort to deal with the problem of the existence and global exponential stability analysis of periodic solution for a class of memristor recurrent neural networks with multiple delays. Through building a useful Lyapunov functional and the inequality techniques, some new testable algebraic criteria are obtained for ensuring the existence and global exponential stability of periodic solution of the system. The model based on the memristor widens the application scope for the design of neural networks. In addition, the new effective results enrich the toolbox for the qualitative analysis of neural networks. Acknowledgements The authors gratefully acknowledge anonymous referees’ comments and patient work. This work is supported by the National Science Foundation of China under Grant No. 11271146, the Key Program of National Natural Science Foundation of China under Grant No. 61134012, and the 973 Program of China under Grant 2011CB710606. References [1] J.P. Aubin, A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, Germany, 1984. [2] J.P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990. [3] G. Bao, Z.G. Zeng, Multistability of periodic delayed recurrent neural network with memristors, Neural Comput. Appl., doi: 10.1007/s00521-012-0954x. [4] J.D. Cao, New results concerning exponential stability and periodic solutions of delayed cellular neural networks, Phys. lett. A 307 (2003) 136–147. [5] J. Cao, J. Wang, Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays, Neural Netw. 17 (3) (2004) 379–390. [6] F. Corinto, A. Ascoli, M. Gilli, Nonlinear dynamics of memristor oscillators, IEEE Trans. Circuits Syst. I, Reg. Pap. 58 (2011) 1323–1336. [7] L.O. Chua, Memristor-the missing circuit element, IEEE Trans. Circ. Theory CT-18 (1971) 507–519. [8] F.H. Clarke, Y.S. Ledyaev, R.J. Stem, R.R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998. [9] B.S. Chen, J. Wang, Global exponential periodicity of a class of recurrent neural networks with oscillating parameters and time-varying delays, IEEE Trans. Neural Netw. 16 (2005) 1440–1448. [10] J. Dugundji, A. Grranas, Fixed point theory, in: Monografie Matematyczne, vol. 123, PWN, 1986. [11] M.F. Dong, Global exponential stability and existence of periodic solutions of CNNs with delays, Phys. Lett. A 300 (2002) 49–57. [12] A.F. Filippov, Differential Equations with Discontinuous Right-hand Sides, Kluwer, Dordrecht, 1988. [13] M. Forti, P. Nistri, D. Papini, Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain, IEEE Trans. Neural Netw. 16 (2005) 1449–1463. [14] M. Forti, M. Grazzini, P. Nistri, L. Pancioni, Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations, Physica D 214 (2006) 88–99. [15] X. Hong, K.M. Yan, B.Y. Wang, Existence and global exponential stability of periodic solution for delayed high-order Hopfield-type neural networks, Phys. Lett. A 352 (2006) 341–349. [16] X. Hu, J. Wang, Design of general projection neural networks for solving monotone linear variational inequalities and linear and quadratic optimization problems, IEEE Trans, Syst. Man Cybernet. Part B 37 (2007) 1414–1421. [17] M. Itoh, L.O. Chua, Memristor oscillators, Int. J. Bifur. Chaos 18 (2008) 3183–3206. [18] M.Q. Liu, Global asymptotic stability analysis of discrete-time Cohen–Grossberg neural networks based on interval systems, Nonlinear Anal. 69 (2008) 2403–2411. [19] Y.R. Liu, Z.D. Wang, X.H. Liu, Global exponential stability of generalized recurrent neural networks with discrete and distributed delays, Neural Netw. 19 (2006) 667–675. [20] X.Y. Liu, J. Cao, On periodic solutions of neural networks via differential inclusions, Neural Netw. 22 (2009) 329–334. [21] D. Papini, V. Taddei, Global exponential stability of the periodic solution of a delayed neural network with discontinuous activations, Phys. Lett. A 343 (2005) 117–128. [22] D.B. Strukov, G.S. Snider, G.R. Stewart, R.S. Williams, The missing memristor found, Nature 453 (2008) 80–83. [23] Q.K. Song, Z.D. Wang, Neural networks with discrete and distributed time-varying delays: a general stability analysis, Chaos, Solitons Fractals 37 (2008) 1538–1547. [24] Y. Shen, J. Wang, An improved algebraic criterion for global exponential stability of recurrent neural networks with time-varying delays, IEEE Trans. Neural Netw. 19 (2008) 528–531. [25] M.D. Ventra, Y.V. Pershin, L.O. Chua, Circuit elements with memory: memristors, memcapacitors and meminductors, Proc. IEEE 97 (2009) 1717–1724. [26] J.F. Wang, L.H. Huang, Z.Y. Guo, Dynamical behavior of delayed Hopfield neural networks with discontinuous activations, Appl. Math. Model. 33 (2009) 1793–1802. [27] A.L. Wu, S.P. Wen, Z.G. Zeng, Synchronization control of a class of memristor-based recurrent neural networks, Inform. Sci. 183 (2012) 106–116.
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[28] A.L. Wu, J. Zhang, Z.G. Zeng, Dynamic behaviors of a class of memristor-based Hopfield networks, Phys. Lett. A 375 (2011) 1661–1665. [29] A.L. Wu, Z.G. Zeng, X.S. Zhu, J. Zhang, Exponential synchronization of memristor-based recurrent neural networks with time delays, Neurocomputing 74 (2011) 3043–3050. [30] X.P. Xue, J.F. Yu, Periodic solutions for semi-linear evolution inclusions, J. Math. Anal. Appl. 331 (2007) 1246–1262. [31] G.D. Zhang, Y. Shen, J.W. Sun, Global exponential stability of a class of memristor-based recurrent neural networks with time varying delays, Neurocomputing 97 (2012) 149–154. [32] J. Zhang, Z.J. Gui, Periodic solutions of nonautonomous cellular neural networks with impulses and delays, Nonlinear Anal. 10 (2009) 1891–1903. [33] T.J. Zhou, A. Chen, Y.Y. Zhou, Existence and global exponential stability of periodic solution to BAM neural networks with periodic coefficients and continuously distributed delays, Phys. Lett. A 343 (2005) 336–350. [34] H.Q. Zhao, X.P. Zeng, J.S. Zhang, T.R. Li, Y.G. Liu, D. Ruan, Pipelined functional link artificial recurrent neural network with the decision feedback structure for nonlinear channel equalization, Inform. Sci. 181 (2011) 3677–3692.
Contents lists available at SciVerse ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays Guodong Zhang, Yi Shen ⇑, Quan Yin, Junwei Sun Department of Control Science and Engineering, Huazhong University of Science and Technology, The Key Laboratory of Education Ministry of Image Processing and Intelligent Control, Wuhan 430074, China
a r t i c l e
i n f o
Article history: Received 3 June 2012 Received in revised form 6 September 2012 Accepted 24 November 2012 Available online 13 December 2012 Keywords: Periodic solution Exponential stability Memristor Recurrent neural network Time delay
a b s t r a c t The paper presents theoretical results on the global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays. The dynamic analysis in the paper employs the theory of differential equations with discontinuous right-hand side as introduced by Filippov. By using the inequality techniques and a useful Lyapunov functional, some new testable algebraic criteria are obtained for ensuring the existence and global exponential stability of periodic solution of the system. The model based on the memristor widens the application scope for the design of neural networks, and the new effective results also enrich the toolbox for the qualitative analysis of neural networks. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The first memristor (as a contraction of memory and resistor), originally theorized by Dr. Chua in 1971, was identified by a team at HP Labs in 2008. The memristor is a two-terminal passive device whose value depends on the magnitude and polarity of the voltage applied to it and the length of time that the voltage has been applied. From the previous work [3,6,7,17,22,25,27–29,31], we know that the potential applications of this device is in next generation computer and powerful brain-like neural computer. As we know, the recurrent neural networks are very important nonlinear circuit networks because of their wide applications in associative memory, pattern recognition, signal processing and so on, for reference, see [4,5,9,11,13–16,18– 21,23,24,26,32–34]. And the Hopfield neural network model can be implemented in a circuit where the connection weights are implemented by resistors, motivated by these facts, recently, by using memristors instead of resistors, Bao and Zeng [3], Wu et al. [27–29] and Zhang et al. [31] have studied a new model where the connection weights change according to its state, i.e., a state-dependent switching recurrent neural networks. Different from the previous works [3,27–29,31], in this paper, we will deal with the problem of existence and global exponential stability of periodic solution for a class of memristor-based recurrent neural networks with multiple delays as follows: n n X X dxi ðtÞ aij ðxi ðtÞÞfj ðxj ðtÞÞ þ bij ðxi ðtÞÞg j ðxj ðt sij ÞÞ þ Ii ðtÞ; ¼ di ðxi ðtÞÞxi ðtÞ þ dt j¼1 j¼1
⇑ Corresponding author. Tel.: +86 27 87543630; fax: +86 27 87543130. E-mail addresses: [email protected] (G. Zhang), [email protected] (Y. Shen). 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2012.11.023
t P 0;
i ¼ 1; 2; . . . ; n;
ð1Þ
387
G. Zhang et al. / Information Sciences 232 (2013) 386–396
where
( di ðxi ðtÞÞ ¼
di ;
jxi ðtÞj < T i ;
di ;
jxi ðtÞj > T i ;
( bij ðxi ðtÞÞ ¼
bij ;
( aij ðxi ðtÞÞ ¼
aij ;
jxi ðtÞj < T i ;
a jxi ðtÞj > T i ; ij ;
jxi ðtÞj < T i ;
bij ; jxi ðtÞj > T i ;
in which switching jumps T i > 0; di > 0; di > 0; aij ; a ij ; bij ; bij ; i; j ¼ 1; 2; . . . ; n, are all constant numbers, fj and g j : R ! R are bounded continuous functions, Ii ðtÞ is a continuous x-periodic external input function. The organization of this paper is as follows. Some preliminaries are introduced in Section 2. The main results are given in Section 3. And then, numerical simulations are given to demonstrate the effectiveness of the proposed approach in Section 4. Finally, this paper ends by a conclusion. Remark 1. The authors in [27,29] have given a clear exposition about the relation between memristances and the coefficients of switching system (1), so readers can consult [27,29] to get more explanation.
2. Preliminaries
Let
Throughout this paper, solutions of all the systems considered in the following are intended in Filippov’s sense (see [12]). P s ¼ max16i;j6n fsij g, we define kwk ¼ sups6t60 ½ ni¼1 jwi ðtÞjp 1=p ; p P 1, for 8w ¼ ðw1 ðtÞ; w2 ðtÞ; . . . ; wn ðtÞÞ 2 Cð½s; 0; Rn Þ,
co½ni ; ni denotes the convex hull of ½ni ; ni . For a continuous function kðtÞ : R ! R; Dþ kðtÞ is called the upper right dini derivative and defined as Dþ kðtÞ ¼ limh!0þ 1h ðkðt þ hÞ kðtÞÞ. System (1) has the following form initial conditions: xðsÞ ¼ /ðsÞ ¼ ð/1 ðsÞ; /2 ðsÞ; . . . ; /n ðsÞÞT 2 Cð½s; 0; Rn Þ. For convenience, now, we first introduce the following Definitions about set-valued map and differential inclusion in [1,2,12]. Definition 1. Let E Rn ; x # FðxÞ is called a set-valued map from E,!Rn , if to each point x of a set E Rn , there corresponds a nonempty set FðxÞ Rn . Definition 2. A set-valued map F with nonempty values is said to be upper semi-continuous at x0 2 E Rn if, for any open set N containing Fðx0 Þ, there exists a neighborhood M of x0 such that FðMÞ N. FðxÞ is said to have a closed (convex, compact) image if for each x 2 E; FðxÞ is closed (convex, compact). Definition 3. For the system
UðxÞ ¼
\ \
dx dt
¼ f ðxÞ; x 2 Rn , with discontinuous right-hand sides, a set-valued map is defined as
co½f ðBðx; dÞ n NÞ;
d>0lðNÞ¼0
where co½E is the closure of the convex hull of set E; Bðx; dÞ ¼ fy : ky xk 6 dg, and lðNÞ is Lebesgue measure of set N. A solution in Filippov’s sense [12] of the Cauchy problem for this system with initial condition xð0Þ ¼ x0 is an absolutely continuous function xðtÞ; t 2 ½0; T, which satisfies xð0Þ ¼ x0 and differential inclusion:
dx 2 UðxÞ; dt
for a:e: t 2 ½0; T:
By applying the above theories of set-valued maps and differential inclusions [1,2,8,12], the memristor-based recurrent neural networks (1) can be written as the following differential inclusion: n n X X dxi ðtÞ 2 co½di ; di xi ðtÞ þ co½aij ; aij fj ðxj ðtÞÞ þ co½bij ; bij g j ðxj ðt sij ÞÞ þ Ii ðtÞ; dt j¼1 j¼1
for a:e: t P 0;
i ¼ 1; 2; . . . ; n;
ð2Þ
where
di ¼ minfdi ; di g; aij ¼ maxfaij ; a ij g;
di ¼ maxfdi ; di g; aij ¼ minfaij ; a ij g; bij ¼
minfbij ; bij g;
bij ¼ maxfbij ; bij g;
or equivalently, there exist b d i 2 co½di ; di ; b a ij 2 co½aij ; aij ; b b ij 2 co½bij ; bij , such that n n X X dxi ðtÞ b b b ij g j ðxj ðt sij ÞÞ þ Ii ðtÞ; a ij fj ðxj ðtÞÞ þ ¼ b d i xi ðtÞ þ dt j¼1 j¼1
for t P 0;
i ¼ 1; 2; . . . ; n:
ð3Þ
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Definition 4. A function (in Filippov’s sense) x ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT is a solution of system (1), with initial conditions /ðsÞ ¼ ð/1 ðsÞ; . . . ; /n ðsÞÞT 2 Cð½s; 0; Rn Þ, if x ðtÞ is an absolutely continuous function and satisfies the differential inclusion n n X X dxi ðtÞ co½aij ; aij fj ðxj ðtÞÞ þ co½bij ; bij g j ðxj ðt sij ÞÞ þ Ii ðtÞ; 2 co½di ; di xi ðtÞ þ dt j¼1 j¼1
for a:e: t P 0;
i ¼ 1; 2; . . . ; n; ð4Þ
or equivalently, there exist b d i 2 co½di ; di ; b a ij 2 co½aij ; aij ; b b ij 2 co½bij ; bij , such that n n X X dxi ðtÞ b b b ij g j ðxj ðt sij ÞÞ þ Ii ðtÞ; a ij fj ðxj ðtÞÞ þ ¼ b d i xi ðtÞ þ dt j¼1 j¼1
for t P 0;
i ¼ 1; 2; . . . ; n:
ð5Þ
Definition 5. Let x ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT is a solution of system (1), with initial conditions /ðsÞ ¼ ð/1 ðsÞ; /2 ðsÞ; . . . ; /n ðsÞÞT 2 Cð½s; 0; Rn Þ, the solution x ðtÞ of system (1) is said to be globally exponential stability, if there exist positive constants e and b P 1 such that for any solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT of system (1) with initial conditions wðsÞ ¼ ðw1 ðsÞ; w2 ðsÞ; . . . ; wn ðsÞÞT 2 Cð½s; 0; Rn Þ satisfies
" #1=p n X xi ðtÞ x ðtÞp 6 beet kw /k; i
for 8t P 0:
ð6Þ
i¼1
Now we do the following assumptions for the system (1): (H1) For i ¼ 1; 2; :; n; 8s1 ; s2 2 R; s1 – s2 , the neuron activation functions fi ; g i are bounded and satisfy condition
jfi ðs1 Þ fi ðs2 Þj 6 ri js1 s2 j;
jg i ðs1 Þ g i ðs2 Þj 6 qi js1 s2 j;
where ri > 0; qi > 0. (H10 ) For i ¼ 1; 2; :; n; 8s1 ; s2 2 R; s1 – s2 , the neuron activation functions fi ; g i are bounded and satisfy condition
0 < ri <
fi ðs1 Þ fi ðs2 Þ 6 ri ; s1 s2
jg i ðs1 Þ g i ðs2 Þj 6 qi js1 s2 j;
where ri > 0; ri > 0; qi > 0. (H2) For the differential inclusion (2), the following conditions hold
co½di ; di yi ðtÞ co½di ; di xi ðtÞ # co½di ; di ðyi ðtÞ xi ðtÞÞ; co½aij ; aij fj ðyj ðtÞÞ co½aij ; aij fj ðxj ðtÞÞ # co½aij ; aij ðfj ðyj ðtÞÞ fj ðxj ðtÞÞÞ; co½bij ; bij g j ðyj ðtÞÞ co½bij ; bij g j ðxj ðtÞÞ # co½bij ; bij ðg j ðyj ðtÞÞ g j ðxj ðtÞÞÞ; where xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT ; yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞ; . . . ; yn ðtÞÞ are two solutions of system (1) with initial conditions /ðsÞ ¼ ð/1 ðsÞ; . . . ; /n ðsÞÞT ; wðsÞ ¼ ðw1 ðsÞ; . . . ; wn ðsÞÞT 2 Cð½s; 0; Rn Þ, respectively.
Lemma 1. Each solution xðtÞ with an initial condition in Cð½s; 0; Rn Þ of system (1) is bounded and defined on ½s; þ1Þ. And when IðtÞ is a constant external input, system (1) exists at least one equilibrium point. 0
Proof. Under assumption (H1) or (H1 ), the neuron activation functions fi ; g i in (1) are bounded and Lipschitz continuous. And di > 0; di > 0; aij ; a ij ; bij ; bij ; Ii ðtÞ are all bounded, then, we have the local existence of a solution xðtÞ with initial condition /ðsÞ ¼ ð/1 ðsÞ; /2 ðsÞ; . . . ; /n ðsÞÞT 2 Cð½s; 0; Rn Þ of system (1) is a straightforward consequence of [12], and there exists a constant L0i > 0 such that n n X X jaij ðxi ðtÞÞfj ðxj ðtÞÞj þ jbij ðxi ðtÞÞg j ðxj ðt sij ÞÞj þ jIi ðtÞj 6 L0i ; j¼1
j¼1
Now, from di 6 j b d i j and system (3), we have
jxi ðtÞj 6 ðjxi ð0Þj L0i Þedi t þ L0i ! L0i ;
djxi ðtÞj dt
þ di jxi ðtÞj 6 L0i , that is,
for t # þ 1;
then, we obtain that all the solutions xðtÞ of system (1) are bounded for a.e. t P 0, therefore, by the Continuation Theorem [12, Th2, p. 78] and [8], we know that each solution with an initial condition /ðsÞ ¼ ð/1 ðsÞ; . . . ; /n ðsÞÞT 2 Cð½s; 0; Rn Þ of system (1) is defined on ½s; þ1Þ. And when IðtÞ is a constant input, system (3) exists at least one equilibrium point, then, system (1) exists at least one equilibrium point. This completes the proof. h
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G. Zhang et al. / Information Sciences 232 (2013) 386–396 1c
Lemma 2. Assume that a > 0; b > 0; 0 < c < 1, then the inequality: ac b
6 ca þ ð1 cÞb holds.
Proof. Consider the function qðxÞ ¼ xc cx ð1 cÞ, obviously, the maximum of qðxÞ on ð0; þ1Þ is qð1Þ ¼ 0, then, we obtain c for 8x > 0; qðxÞ 6 qð1Þ, that is, xc cx ð1 cÞ 6 0. Now, taking x ¼ ba > 0, we get ðab Þc c ba ð1 cÞ 6 0, i.e., abc 6 c ba þ ð1 cÞ, 1c multiplied the inequality above by b, we have ac b 6 ca þ ð1 cÞb. The inequality is the transmutation of the Young inequality. This completes the proof. h Lemma 3 [10]. If X is a Banach space, P kðcÞ ðRn Þ denote the collection of all nonempty, compact and convex subset of Rn , C # X is nonempty convex with 0 2 C and G : C ! PkðcÞ ðCÞ is an upper semi-continuous which maps bounded sets into relatively compact sets, then one of the following statements is true: (i) the set C ¼ fx 2 C : x 2 kGðxÞ; k 2 ð0; 1Þg is unbounded; (ii) the GðÞ has a fixed point in C, i.e., there exists x 2 C such that x 2 GðxÞ. In next section, the paper aims to analysis the existence and global exponential stability of periodic solution for system (1). 3. Main results Theorem 1. Under assumption (H1) and (H2), if there exists constants nij ; nij ; fij ; fij 2 R; ci > 0 and p P 1, such that
n pn pf 1=ðp1Þ pn pf 1=ðp1Þ X n f n f ðp 1Þ Aij ij rj ij þ cj =ci Ajiji ri ji þ ðp 1Þ Bij ij qj ij þ ðcj =ci ÞBjiji qi ji < pdi ;
ð7Þ
j¼1
where Aij ¼ maxfjaij j; jaij jg, Bij ¼ maxfjbij j; jbij jg; i; j ¼ 1; 2; . . . ; n. Then, system (1) exists exactly one x-periodic solution and all other solutions of system (1) converge exponentially to it as t ! þ1. P Proof. For 8w ¼ ðw1 ðtÞ; w2 ðtÞ; . . . ; wn ðtÞÞ 2 Cð½s; 0; Rn Þ, we define the norm kwk ¼ sups6t60 ½ ni¼1 jwi ðtÞjp 1=p , then, Cð½s; 0; Rn Þ n is a Banach space. For all / 2 Cð½s; 0; R Þ, we denote the solutions of system (1) through ð0; /Þ as xðt; /Þ ¼ ðx1 ðt; /Þ; x2 ðt; /Þ; . . . ; xn ðt; /ÞÞT . Now, under assumption (H1) and (H2), we prove that the system (1) exists at least one x-periodic solution. Let L1 ðT 0 ; Rn Þ denotes the Banach space of all functions x : T 0 ¼ ½0; x ! Rn which are Bochner integrable. Let n 1;1 W 1;1 ðT 0 ; Rn Þ ¼ fx 2 L1 ðT 0 ; Rn Þ : x_ 2 L1 ðT 0 ; Rn Þg, W 1;1 ðT 0 ; Rn Þ : xð0Þ ¼ xðxÞg and LðxÞ ¼ x_ þ x for all p ðT 0 ; R Þ ¼ fx 2 W n 1;1 n 1 n x 2 W 1;1 ðT ; R Þ, then by the literature [10,21,30], we know L : W ðT 0 0 ; R Þ ! L ðT 0 ; R Þ is a linear operator and bijective p p 1 (one to one and surjective), and L is completely continuous. For any x 2 L1 ðT 0 ; Rn Þ, define
F i ðt; xÞ ¼ co½di ; di xi ðtÞ þ
n n X X co½aij ; aij fj ðxj ðtÞÞ þ co½bij ; bij g j ðxj ðt sij ÞÞ þ Ii ðtÞ; j¼1
i ¼ 1; 2; . . . ; n;
j¼1
NðxÞ ¼ fv 2 L1 ðT 0 ; Rn Þ : v 2 Fðt; xÞ ¼ ðF 1 ðt; xÞ; . . . ; F n ðt; xÞÞT ;
a:e: on T 0 g:
And from Ref. [26], we get NðÞ is upper semi-continuous and a bounded operator from L1 ðT 0 ; Rn Þ to L1 ðT 0 ; Rn Þw (denote weak topology), then, we obtain that L1 N : L1 ðT 0 ; Rn Þ ! P kðcÞ ðL1 ðT 0 ; Rn ÞÞ is upper semi-continuous and maps bounded sets into reln 1 atively compact sets. Let C ¼ fx 2 W 1;1 p ðT 0 ; R Þ : x 2 aL NðxÞ; a 2 ð0; 1Þg, and as the same proof of Lemma 1, we can easy get the x-periodic solutions of the following system are bounded,
(
) n n X X x_ i ðtÞ 2 ð1 aÞxi ðtÞ a co½di ; di xi ðtÞ þ co½aij ; aij fj ðxj ðtÞÞ þ co½bij ; bij g j ðxj ðt sij ÞÞ þ Ii ðtÞ ; j¼1
j¼1
or, there exist b d i 2 co½di ; di ; b a ij 2 co½aij ; aij ; b b ij 2 co½bij ; bij , such that
"
d i xi ðtÞ þ x_ i ðtÞ ¼ ð1 aÞxi ðtÞ a b
n X j¼1
b a ij fj ðxj ðtÞÞ þ
n X
# b b ij g j ðxj ðt sij ÞÞ þ Ii ðtÞ ;
j¼1
where a 2 ð0; 1Þ; i ¼ 1; 2; . . . ; n. That is, the set C is bounded. By the Lemma 3 (Leray–Schauder alternative theorem) and n 1 [12,21,26], there exists x 2 W 1;1 p ðT 0 ; R Þ such that x 2 L NðxÞ. Clearly, x is an x-periodic solution of system (1).
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G. Zhang et al. / Information Sciences 232 (2013) 386–396
And now, Let xðt; u Þ is an x-periodic solution of system (1). Under assumption (H1) and (H2), it follows from (2) or (3) that
½xi ðt; /Þ xi ðt; u Þ0 2 co½di ; di ðxi ðt; /Þ xi ðt; u ÞÞ þ
n n X X co½aij ; aij ½ fj ðxj ðt; /ÞÞ fj ðxj ðt; u ÞÞ þ co½bij ; bij j¼1
j¼1
½g j ðxj ðt sij ; /ÞÞ g j ðxj ðt sij ; u ÞÞ; P 0;
for a:e: t
i ¼ 1; 2; . . . ; n:
For i; j ¼ 1; 2; . . . ; n, we can choose a small
ð8Þ
e > 0 such that
n X
pn pf 1=ðp1Þ pn pf 1=ðp1Þ n f n f ðp 1Þ Aij ij rj ij þ ðcj =ci ÞAjiji ri ji þ ðp 1Þ Bij ij qj ij þ epes ðcj =ci ÞBjiji qi ji < pðdi eÞ:
ð9Þ
j¼1
Now, we consider a Lyapunov functional as
" Z n n X X f VðtÞ ¼ ci jxi ðt; /Þ xi ðt; u Þjp epet þ jb b ij jnij qj ij i¼1
#
t
p peðsþsij Þ
jxj ðt; /Þ xj ðt; u Þj e
ds ;
ð10Þ
tsij
j¼1
where b b ij 2 co½bij ; bij . By calculating the upper right derivation Dþ VðtÞ of VðtÞ along the solution to (2) or (3), we obtain Dþ VðtÞjð2Þ or ð3Þ ¼ epet
n n X X pci jxi ðt;/Þ xi ðt; u Þjp1 sgnðxi ðt;/Þ xi ðt; u ÞÞ ½xi ðt;/Þ xi ðt; u Þ0 þ epet peci jxi ðt;/Þ i¼1
i¼1
n X n X f ci j b b ij jnij qj ij ½epesij jxj ðt;/Þ xj ðt; u Þjp xi ðt; u Þj þ e
p
pet
i¼1 j¼1 p
jxj ðt sij ;/Þ xj ðt sij ; u Þj 2 epet
n X p1 pci jxi ðt;/Þ xi ðt; u Þj sgnðxi ðt;/Þ xi ðt; u ÞÞ i¼1
(
n X co½di ;di ðxi ðt;/Þ xi ðt; u ÞÞ þ co½aij ;aij ½fj ðxj ðt;/ÞÞ fj ðxj ðt; u ÞÞ
j¼1
) n n X X
þ co½bij ;bij g j ðxj ðt sij ;/ÞÞ g j ðxj ðt sij ; u ÞÞ þ epet peci jxi ðt;/Þ xi ðt; u Þjp j¼1
i¼1
n X n X f þ epet ci j b b ij jnij qj ij ½epesij jxj ðt;/Þ xj ðt; u Þjp jxj ðt sij ;/Þ xj ðt sij ; u Þjp i¼1 j¼1
( n n X X pet p1 6e pci jxi ðt;/Þ xi ðt; u Þj ðe di Þjxi ðt;/Þ xi ðt; u Þj þ Aij jfj ðxj ðt;/ÞÞ fj ðxj ðt; u ÞÞj i¼1
j¼1
) n n X n X X n f þ Bij jg j ðxj ðt sij ;/ÞÞ g j ðxj ðt sij ; u ÞÞj þ epet ci Bijij qj ij epes jxj ðt;/Þ xj ðt; u Þjp j¼1
i¼1 j¼1
p
jxj ðt sij ;/Þ xj ðt sij ; u Þj :
ð11Þ
Now, under assumption (H1) and (H2), and by using Lemma 2, we obtain the following estimate for the right-hand side of (11) Dþ VðtÞjð2Þ or ð3Þ 6 epet
n X
(" pci ðe di Þ þ epes
i¼1
# n n X X n f cj Bjiji qi ji jxi ðt;/Þ xi ðt; u Þjp þ Aij pci rj jxj ðt;/Þ xj ðt; u Þjjxi ðt;/Þ j¼1
p1
xi ðt; u Þj
þ
n X
j¼1
Bij pci qj jxi ðt;/Þ xi ðt; u Þj
j¼1 n X xj ðt sij ; u Þjp ¼ epet
(" pci ðe di Þ þ epes
i¼1
þ pci
n X
pn ½ðAij ij
r
p1
n X
þ pci
pnij
Bij
pfij 1=ðp1Þ
qj
p 11=p
jxi ðt;/Þ xi ðt; u Þjp
j¼1
j¼1 n
f
io
f
n
f
Bijij qj ij jxj ðt sij ;/Þ
j¼1
#
cj Bjiji qi ji jxi ðt;/Þ xi ðt; u Þjp
u Þj
n X n f ci Bijij qj ij jxj ðt sij ;/Þ xj ðt sij ; u Þjp
þ epes cj Bjiji qi ji
n
n X
j¼1
pfij 1=ðp1Þ Þ jxi ðt;/Þ xi ðt; j
j¼1
n X
jxj ðt sij ;/Þ xj ðt sij ; u Þj ci
1=p n f Aijij rj ij jxj ðt;/Þ xj ðt; u Þjp
11=p
) 6 epet
n X i¼1
jxi ðt;/Þ xi ðt; u Þjp 6 0:
n f 1=p Bijij qj ij jxj ðt sij ;/Þ xj ðt sij ; u Þjp
( pci ðe di Þ þ
n h X
pnij
ci ðp 1ÞðAij
n f pn pf ij 1=ðp1Þ rpf Þ þ cj Ajiji ri ji þ ci ðp 1ÞðBij ij qj ij Þ1=ðp1Þ j
j¼1
ð12Þ
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Thus,
VðtÞ 6 Vð0Þ;
for t P 0:
ð13Þ
Since
Vð0Þ ¼
" Z n n X X f ci j/i ð0Þ ui ð0Þjp þ jb b ij jnij qj ij i¼1
j¼1
"
6 maxci þ sepes 16i6n
0
# jxj ðs; /Þ xj ðs; u Þjp epeðsþsij Þ ds
sij
# n n f X ji ji cj max Bji qi j/i ðtÞ ui ðtÞjp ; sup
n X j¼1
16i6n
ð14Þ
s6t60 i¼1
And from (10), for t P 0, we have
VðtÞ P
X n n X ci epet jxi ðt; /Þ xi ðt; u Þjp P min ci epet jxi ðt; /Þ xi ðt; u Þjp : 16i6n
i¼1
ð15Þ
i¼1
Then, from (13)–(15), we get
" #1=p " #1=p n n X X p p et jxi ðt; /Þ xi ðt; u Þj 6 be sup j/i ðtÞ ui ðtÞj ; s6t60
i¼1
i¼1
that is,
kxðt; /Þ xðt; u Þk 6 beet k/ u k;
ð16Þ
where
n f 31=p 2 Pn max16i6n ci þ sepes j¼1 cj max16i6n Bjiji qi ji 5 P 1: b¼4 min16i6n ci From (16), we know that all other solutions of system (1) converge exponentially to the x-periodic solution xðt; u Þ as t ! þ1 and it is also to get xðt; u Þ is the unique x-periodic solution of system (1). This completes the proof. h Theorem 2. Under assumption (H1) and (H2), if n X ½Aij rj þ Aji ri þ Bij qj þ Bji qi < 2di ;
ð17Þ
j¼1
where Aij ¼ maxfjaij j; jaij jg, Bij ¼ maxfjbij j; jbij jg; i ¼ 1; 2; . . . ; n. Then, system (1) exists exactly one x-periodic solution and all other solutions of system (1) converge exponentially to it as t ! þ1. nij ¼ fij ¼ fij ¼ 1; i; j ¼ 1; 2; . . . ; n. h Proof. We can obtain Theorem 2 directly from Theorem 1 by taking p ¼ 2; ci ¼ 1; nij ¼ Now, from Theorem 1 and 2, we can get the following Corollary 1. Corollary 1. Under assumption (H1) and (H2), if Ii ðtÞ ¼ Ii and (7) or (17) hold. Then, the equilibrium point x of system (1) is globally exponentially stable.
Theorem 3. Under assumption (H10 ) and (H2), if there exists constants ci > 0 and p P 1, aii < 0; a ii < 0, such that
pdi þ pri Aii þ
n X
½ðp 1ÞAij þ ðcj =ci ÞAji rpi þ
j¼1;j–i
n X ½ðp 1ÞBij þ ðcj =ci ÞBji qpi < 0;
ð18Þ
j¼1
where Aii ¼ aii ; Aij ¼ maxfjaij j; jaij jgði – jÞ, Bij ¼ maxfjbij j; jbij jg; i; j ¼ 1; 2; . . . ; n. Then, system (1) exists exactly one x-periodic solution and all other solutions of system (1) converge exponentially to it as t ! þ1. Proof. For i; j ¼ 1; 2; . . . ; n, we can choose a small
ðe pdi Þ þ pri Aii þ
n X j¼1;j–i
e > 0 such that
ðp 1ÞAij þ ðcj =ci ÞAji rpi þ
n X ½ðp 1ÞBij þ ðcj =ci Þees Bji qpi < 0;
ð19Þ
j¼1
And now, as the proof in Theorem 1, we know system (1) exists at least one x-periodic solution. Let xðt; u Þ is an x-periodic solution of system (1). Here, we consider a Lyapunov functional as
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G. Zhang et al. / Information Sciences 232 (2013) 386–396
VðtÞ ¼
Z n n X X ci ½jxi ðt; /Þ xi ðt; u Þjp eet þ jb b ij j i¼1
t
tsij
j¼1
jg j ðxj ðt; /ÞÞ g j ðxj ðt; u ÞÞjp eeðsþsij Þ ds;
ð20Þ
where b b ij 2 co½bij ; bij . As the proof of Theorem 1, by calculating the upper right derivation Dþ VðtÞ of VðtÞ along the solution to (2) or (3), we obtain " n n X X p p D VðtÞjð2Þ or ð3Þ 6 e ci ðe pdi Þjxi ðt;/Þ xi ðt; u Þj þ pri Aii jxi ðt;/Þ xi ðt; u Þj þ pAij jxi ðt;/Þ et
þ
i¼1
j¼1;j–i
xi ðt; u Þj
p1
n X jfj ðxj ðt;/ÞÞ fj ðxj ðt; u ÞÞj þ pBij jxi ðt;/Þ xi ðt; u Þjp1 jg j ðxj ðt sij ;/ÞÞ j¼1
# n n X X g j ðxj ðt sij ; u ÞÞj þ e Bij jg j ðxj ðt;/ÞÞ g j ðxj ðt; u ÞÞjp Bij jg j ðxj ðt sij ;/ÞÞ g j ðxj ðt sij ; u ÞÞjp : es
j¼1
ð21Þ
j¼1
Now, under assumption (H10 ) and (H2) and by using Lemma 2, we obtain the following estimate for the right-hand side of (21) Dþ VðtÞjð2Þ
or ð3Þ
6 ee t
" n n X X p1 1 ci ðe pdi þ pAii ri Þjxi ðt; /Þ xi ðt; u Þjp þ pAij ðjxi ðt; /Þ xi ðt; u Þjp Þ p ðjfj ðxj ðt; /ÞÞ fj ðxj ðt; u ÞÞjp Þp i¼1
j¼1;j–i
n n X X p1 1 pBij ðjxi ðt; /Þ xi ðt; u Þjp Þ p ðjg j ðxj ðt sij ; /ÞÞ g j ðxj ðt sij ; u ÞÞjp Þp þ ees Bij jg j ðxj ðt; /ÞÞ g j ðxj ðt; u ÞÞjp þ j¼1
j¼1
# n n X X Bij jg j ðxj ðt sij ; /ÞÞ g j ðxj ðt sij ; u ÞÞjp 6 eet ci e pdi þ pAii ri jxi ðt; /Þ xi ðt; u Þjp j¼1
i¼1
n X
þ
p
p j jxj ðt; /Þ
Aij ½ðp 1Þjxi ðt; /Þ xi ðt; u Þj þ r
j¼1;j–i n X þ ees Bij qpj jxj ðt; /Þ xj ðt; u Þjp j¼1
)
xj ðt; u Þjp þ
n X Bij ðp 1Þjxi ðt; /Þ xi ðt; u Þjp j¼1
( n n X X et 6e ci ðe pdi þ pAii ri Þ þ ½ci Aij ðp 1Þ þ cj Aji rpi i¼1
j¼1;j–i
) n X ½ci Bij ðp 1Þ þ ees cj Bji qpi jxi ðt; /Þ xi ðt; u Þjp 6 0: þ
ð22Þ
j¼1
Then, as the proof of Theorem 1, we can have e
kxðt; /Þ xðt; u Þk 6 b ept k/ u k;
ð23Þ
where
" b ¼
#1=p P max16i6n ci þ sees nj¼1 cj max16i6n ðBji qpi Þ P 1: min16i6n ci
Then, we know that all other solutions of system (1) converge exponentially to a unique x-periodic solution as t ! þ1. This completes the proof. h Theorem 4. Under assumption (H10 ) and (H2), if there exists constants ci > 0 and p P 1, aii < 0; a ii < 0; and one of the following conditions hold
di þ ri Aii þ
n X
Aji ri þ
j¼1;j–i
pdi þ pri Aii þ
n X Bji qi < 0;
ð24Þ
j¼1
n h X
pnij
ðp 1ÞðAij
i
n f ij 1=ðp1Þ rpf Þ þ ðcj =ci ÞAjiji ri ji þ j
j¼1;j–i
n h X
pnij
ðp 1ÞðBij
pfij 1=ðp1Þ
qj
Þ
n f i þ ðcj =ci ÞBjiji qi ji < 0;
j¼1
ð25Þ
2di þ 2ri Aii þ
n X
½Aij rj þ Aji ri þ
j¼1;j–i
n X ½Bij qj þ Bji qi < 0;
ð26Þ
j¼1
where Aii ¼ aii ; Aij ¼ maxfjaij j; jaij jgði – jÞ, Bij ¼ maxfjbij j; jbij jg; i; j ¼ 1; 2; . . . ; n. Then, system (1) exists exactly one x-periodic solution and all other solutions of system (1) converge exponentially to it as t ! þ1.
393
G. Zhang et al. / Information Sciences 232 (2013) 386–396 0.25 0.2 0.15 0.1
States
0.05 0
−0.05 −0.1 −0.15 −0.2 −0.25
0
5
10
15
20
25
30
35
40
45
50
time − T Fig. 1. State trajectory xi ðtÞ ði ¼ 1; 2Þ of the DRNN system (27) with eternal input I1 ðtÞ ¼ cos t; I2 ðtÞ ¼ 0:5 sin 2t.
Proof. We can obtain (24) directly from Theorem 3 by taking p ¼ 1; ci ¼ 1; i ¼ 1; 2; . . . ; n. And if (25) hold, we can get the result by the same proof of Theorem 1 and 3. We can obtain (26) directly from (25) by taking p ¼ 2; ci ¼ 1; nij ¼ nij ¼ fij ¼ fij ¼ 1; i; j ¼ 1; 2; . . . ; n. This completes the proof. h Now, from Theorem 3 and 4 above, we can get the following Corollary 2. Corollary 2. Under assumption (H10 ) and (H2), if Ii ðtÞ ¼ Ii , aii < 0; aii < 0, and (18) or (24) or (25) or (26) old. Then, the equilibrium point x of system (1) is globally exponentially stable. Remark 2. The proposed results are different from those in [3,27–29,31] and the references cited therein. And the results of this paper also complement previously known results. In addition, these conditions, which can be directly derived from the parameters of the neural networks, are very easily verified.
4. Numerical simulations Now, we perform some numerical simulations to illustrate our analysis by using MATLAB (7.0) programming. System 1. Consider two-dimensional memristor-based neural networks
dx1 ðtÞ ¼ d1 ðx1 ðtÞÞx1 ðtÞ þ a11 ðx1 ðtÞÞf1 ðx1 ðtÞÞ þ a12 ðx1 ðtÞÞf2 ðx2 ðtÞÞ þ b11 ðx1 ðtÞÞg 1 ðx1 ðt s11 ÞÞ þ b12 ðx1 ðtÞÞg 2 ðx2 ðt s12 ÞÞ dt þ I1 ðtÞ; dx2 ðtÞ ¼ d2 ðx2 ðtÞÞx2 ðtÞ þ a21 ðx2 ðtÞÞf1 ðx1 ðtÞÞ þ a22 ðx2 ðtÞÞf2 ðx2 ðtÞÞ þ b21 ðx2 ðtÞÞg 1 ðx1 ðt s21 ÞÞ þ b22 ðx2 ðtÞÞg 2 ðx2 ðt dt s22 ÞÞ þ I2 ðtÞ; where
d1 ðx1 ðtÞÞ ¼
d2 ðx2 ðtÞÞ ¼
4;
jx1 ðtÞj < 1;
4:5; jx1 ðtÞj > 1;
5; jx2 ðtÞj < 1; 4; jx2 ðtÞj > 1;
a21 ðx2 ðtÞÞ ¼
1; jx2 ðtÞj < 1; 1;
jx2 ðtÞj > 1;
a11 ðx1 ðtÞÞ ¼
a12 ðx1 ðtÞÞ ¼
1; jx1 ðtÞj < 1; 1;
1;
jx1 ðtÞj < 1;
1; jx1 ðtÞj > 1; (
b11 ðx1 ðtÞÞ ¼
jx1 ðtÞj > 1;
1 ; 2
jx1 ðtÞj < 1; 12 ; jx1 ðtÞj > 1;
ð27Þ
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G. Zhang et al. / Information Sciences 232 (2013) 386–396
a22 ðx2 ðtÞÞ ¼
And
(
jx2 ðtÞj < 1;
b21 ðx2 ðtÞÞ ¼
2; jx2 ðtÞj > 1; (
b12 ðx1 ðtÞÞ ¼
2;
1 ; 4
(
12 ; jx1 ðtÞj < 1; 1 ; jx1 ðtÞj > 1; 2
b22 ðx2 ðtÞÞ ¼
jx2 ðtÞj < 1;
14 ; jx2 ðtÞj > 1; 14 ; jx2 ðtÞj < 1; 1 ; jx2 ðtÞj > 1: 4
s11 ¼ s22 ¼ 1:5; s12 ¼ s21 ¼ 2; IðtÞ ¼ ðI1 ðtÞ; I2 ðtÞÞT , take the activation function as follows: f1 ðxÞ ¼ f2 ðxÞ ¼ g 1 ðxÞ ¼ g 2 ðxÞ ¼ ðex ex Þ=ðex þ ex Þ:
Obviously, we have r1 ¼ r2 ¼ q1 ¼ q2 ¼ 1, d1 ¼ d2 ¼ 4; A11 ¼ A12 ¼ A21 ¼ 1; A22 ¼ 2; B11 ¼ B12 ¼ 12 ; B21 ¼ B22 ¼ 14. From (17) of Theorem 2, we get
0.25 0.2 0.15
X2 (t)
0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 −0.25 −0.2 −0.15 −0.1 −0.05 0
0.05 0.1 0.15 0.2
0.25
X1 (t) Fig. 2. The phase plot of system (27) corresponding to Fig. 1.
2.5 2 1.5 1
States
0.5 0 −0.5 −1 −1.5 −2 −2.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time − T Fig. 3. State trajectory xi ðtÞ ði ¼ 1; 2Þ of the DRNN system (27) with eternal input I1 ðtÞ ¼ 2:7; I2 ðtÞ ¼ 2:8.
G. Zhang et al. / Information Sciences 232 (2013) 386–396
395
23 < 2d1 ¼ 8; 4 29 < 2d2 ¼ 8: ¼ 4
2A11 þ A12 þ A21 þ 2B11 þ B12 þ B21 ¼ A12 þ A21 þ 2A22 þ B12 þ B21 þ 2B22
Thus, the conditions required in Theorem 2 are satisfied. Now, we choose the eternal input I1 ðtÞ ¼ cos t; I2 ðtÞ ¼ 0:5 sin 2t, by Theorem 2, we know that a unique x-periodic solution x ðtÞ of system (27) is globally exponentially stable, the simulation results of network (27) with 15 random initial values are depicted in Fig. 1. The phase plot of system (27) corresponding to Fig. 1 are shown in Fig. 2, and here the memristor-based recurrent neural networks (27) exhibit clear 8-shaped curves. And when we choose the eternal input I1 ðtÞ ¼ 2:7; I2 ðtÞ ¼ 2:8, by Corollary 1, we know that there is a unique equilibrium point x of system (27) and x is globally exponentially stable, the simulation results of network (27) with 15 random initial values are depicted in Fig. 3. Some examples can also be given for other Theorems, here they are omitted. 5. Conclusion In this paper, under the framework of Filippov’s solution, we made an effort to deal with the problem of the existence and global exponential stability analysis of periodic solution for a class of memristor recurrent neural networks with multiple delays. Through building a useful Lyapunov functional and the inequality techniques, some new testable algebraic criteria are obtained for ensuring the existence and global exponential stability of periodic solution of the system. The model based on the memristor widens the application scope for the design of neural networks. In addition, the new effective results enrich the toolbox for the qualitative analysis of neural networks. Acknowledgements The authors gratefully acknowledge anonymous referees’ comments and patient work. 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