New global exponential stability criteria for nonlinear delay differential systems with applications to BAM neural networks

Applied Mathematics and Computation 243 (2014) 899–910

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

New global exponential stability criteria for nonlinear delay differential systems with applications to BAM neural networks q Leonid Berezansky a, Elena Braverman b,⇑, Lev Idels c a

Dept. of Math, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Dept. of Math & Stats, University of Calgary, 2500 University Dr. NW, Calgary, AB T2N1N4, Canada c Dept. of Math, Vancouver Island University (VIU), 900 Fifth St., Nanaimo, BC V9S5J5, Canada b

a r t i c l e

i n f o

a b s t r a c t We consider a nonlinear non-autonomous system with time-varying delays

Keywords: Systems of nonlinear delay differential equations Time-varying delays Global stability M-matrix BAM neural network Leakage delays

x_i ðtÞ ¼ ai ðtÞxi ðhi ðtÞÞ þ

m X   F ij t; xj ðg ij ðtÞÞ ;

i ¼ 1; . . . ; m

j¼1

which has a large number of applications in the theory of artificial neural networks. Via the M-matrix method, easily verifiable sufficient stability conditions for the nonlinear system and its linear version are obtained. Application of the main theorem requires just to check whether a matrix, which is explicitly constructed using the system’s parameters, is an Mmatrix. Comparison with the tests obtained by Gopalsamy (2007) and Liu (2013) for BAM neural networks illustrates novelty of the stability theorems. Some open problems conclude the paper. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction One of the main motivations to study the nonlinear delay differential system

x_i ðtÞ ¼ ai ðtÞxi ðhi ðtÞÞ þ

m X   F ij t; xj ðg ij ðtÞÞ ;

t P 0;

i ¼ 1; . . . ; m

ð1:1Þ

j¼1

and its linear version

x_i ðtÞ ¼

m X aij ðtÞxj ðg ij ðtÞÞ;

i ¼ 1; . . . ; m

ð1:2Þ

j¼1

is their importance in the study of artificial neural network models [12,14]. Here m is the number of units in a neural network, xi ðtÞ 2 Rm corresponds to the state vector of the ith unit at the time t, and ai ðtÞ represents passive decay rates, F ij are the q

The research of the second author was partially supported by NSERC.

⇑ Corresponding author.

E-mail addresses: [email protected], [email protected] (E. Braverman). http://dx.doi.org/10.1016/j.amc.2014.06.060 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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output transfer functions, hi ðtÞ denote the leakage (self-inflicted) delays, and g ij ðtÞ are transmission delays for the ith unit along the axon of the jth unit; for more details see, for example, [12,14]. For linear system (1.2) several very interesting stability results were obtained in [7,11,26,27]. In [11] system (1.2) with constant coefficients aij was examined; in [27] the proofs were based on the assumption that aij and g ij are continuous functions and jaij ðtÞj 6 bij aii ðtÞ. Most of the results for system (1.1) were obtained in the case hi ðtÞ  t (see, for example, [21]). Also the requirement that all the functions involved in the system are continuous seem unduly restrictive, and we relax this assumption. In the present paper, we consider the so-called pure-delay case hi ðtÞXt, assuming that all parameters are measurable functions, and F ij ðt; uÞ are Carathéodory functions. Via M-matrix method we obtain novel stability results for nonlinear non-autonomous system (1.1) and linear non-autonomous system (1.2). It is to be emphasized that our technique does not require a long sequence of other theorems or conditions that must be proven or cited before the main result is justified. Gopalsamy in [10] studied a model of networks known as Bidirectional Associative Memory (BAM) with leakage delays: n   X    ð1Þ ð2Þ x_ i ðtÞ ¼ ai xi t  si aij fj yj t  rj þ þ Ii j¼1 n   X    ð2Þ ð1Þ bij g j xj t  rj þ þ Ji y_ i ðtÞ ¼ bi yi t  si

i ¼ 1; . . . ; n:

ð1:3Þ

j¼1

This model can be regarded as a network with two layers, xi ðtÞ is the state vector of the ith neuron in the first layer, whereas yi ðtÞ refers to the second layer; ai and bi denote the neuron charging time constants and passive decay rates, respectively; ðkÞ ðkÞ aij ; bij are synaptic connection strength; si ; rj ðk ¼ 1; 2Þ are the leakage and the transmission delays accordingly, and Ii ; J i are the exogenous inputs [21]. The paper [10] contains sufficient conditions for the existence of a unique equilibrium of (1.3) and its global stability. Some interesting results for system (1.3) were obtained via the construction of Lyapunov functionals in [6,18,22,23,28,29]. To extend and improve the results obtained in [10,18,20], we apply our main theorem to the non-autonomous system

"

n   X    ð1Þ ð2Þ aij fj yj lj ðtÞ þ Ii x_ i ðtÞ ¼ ri ðtÞ ai xi hi ðtÞ þ

#

j¼1

"

# n   X    ð2Þ ð1Þ bij g j xj lj ðtÞ þ J i : y_ i ðtÞ ¼ pi ðtÞ bi yi hi ðtÞ þ

ð1:4Þ

j¼1

Let us quickly sketch what we accomplish here. Section 2 incorporates the main result of the paper: if a certain matrix which is explicitly constructed from the functions and the coefficients of the system is an M-matrix, then the system is globally exponentially stable. It is demonstrated that the stability condition for a nonlinear system of two equations with constant delays improves the test obtained in [10]. In Section 3 we examine stability of BAM models and obtain stability results that for a nonlinear BAM systems generalize the main theorem in [18]. Finally, Section 4 contains discussion and outlines some open problems. 2. Main results Consider for any t 0 P 0 the system of delay differential equations

x_i ðtÞ ¼ ai ðtÞxi ðhi ðtÞÞ þ

m   X F ij t; xj ðg ij ðtÞÞ ;

t P t0 ;

i ¼ 1; . . . ; m;

ð2:1Þ

j¼1

with the initial conditions

xi ðtÞ ¼ ui ðtÞ; where

t0  r 6 t < t0 ;

xi ðt 0 Þ ¼ x0i ;

ð2:2Þ

r > 0 is denoted below in (a4), under the following assumptions:

(a1) ai are Lebesgue measurable essentially bounded on ½0; 1Þ functions, 0 < ai 6 ai ðtÞ 6 Ai almost everywhere (a.e.); (a2) F ij ðt; Þ are continuous functions, F ij ð; uÞ are measurable locally essentially bounded functions, jF ij ðt; uÞj 6 Lij juj for a.e. t P 0; (a3) hi ; g ij are Lebesgue measurable functions, 0 6 t  hi ðtÞ 6 si ; 0 6 t  g ij ðtÞ 6 rij ; (a4) ui are continuous functions on ½t 0  r; t 0 , where r ¼ maxfsk ; rij jk; i; j ¼ 1; . . . ; mg. Further on we assume that all the inequalities for Lebesgue measurable functions (coefficients and delays) are satisfied a.e. Definition 2.1. An absolutely continuous in ½t0 ; 1Þ vector-function X ¼ ðx1 ; . . . ; xm ÞT : R ! Rm is called a solution of problem (2.1)–(2.2) if it satisfies Eq. (2.1) for almost all t 2 ðt0 ; 1Þ and conditions (2.2) for t 6 t0 .

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Henceforth assume that conditions (a1)–(a4) hold for problem (2.1)–(2.2) and its modifications, and the problem has a unique solution. m We will use some traditional notations. A matrix B ¼ ðbij Þi;j¼1 is nonnegative if bij P 0 and positive if bij > 0; i; j ¼ 1; . . . ; m; kak is an arbitrary fixed norm of a column vector a ¼ ða1 ; . . . ; am ÞT in Rm ; kBk is the corresponding m matrix norm of a matrix B; jaj ¼ ðja1 j; . . . ; jam jÞT and jBj ¼ ðjbij jÞi;j¼1 . Problem (2.1)–(2.2) has a unique global solution on ½t 0 ; 1Þ if, for example, we assume along with (a1)–(a4) that the functions F ij ðt; uÞ are globally Lipschitz in u. The following classical definition of an M-matrix will be used. m

Definition 2.2 [5]. A matrix B ¼ ðbij Þi;j¼1 is called a (non-singular) M-matrix if bij 6 0; i – j and one of the following equivalent conditions holds: 1. there exists a positive inverse matrix B1 > 0; 2. the principal minors of matrix B are positive.

Lemma 2.3 [5]. B is an M-matrix if bij 6 0; i – j and at least one of the following conditions holds: 1. 2. 3. 4.

P bii > j – i jbij j; i ¼ 1; . . . ; m; P bjj > i – j jbij j; j ¼ 1; . . . ; m; P there exist positive numbers ni ; i ¼ 1; . . . ; m such that ni bii > j–i nj jbij j; i ¼ 1; . . . ; m; P there exist positive numbers ni ; i ¼ 1; . . . ; m such that nj bjj > i–j ni jbij j; j ¼ 1; . . . ; m.

Definition 2.4. System (2.1) is globally exponentially stable if there exist constants M > 0 and k > 0 such that for any solution XðtÞ of problem (2.1)–(2.2) the inequality

! kXðtÞk 6 Mekðtt0 Þ kXðt 0 Þk þ supkuðtÞk t
holds, where uðtÞ ¼ ðu1 ðtÞ; . . . ; um ðtÞÞT ; M and k do not depend on t 0 . We define the matrix C as follows

C ¼ ðcij Þm i;j¼1 ;

cii ¼ 1 

Ai ðAi þ Lii Þsi þ Lii

ai

;

cij ¼ 

Ai Lij si þ Lij

ai

;

i – j:

ð2:3Þ

Theorem 2.5. Suppose C defined by (2.3) is an M-matrix. Then system (2.1) is globally exponentially stable. ~ i . The solution XðtÞ ¼ ðx1 ðtÞ; . . . ; xm ðtÞÞT of Proof. We extend ui to ½t0 ; 1Þ by ui ðtÞ ¼ 0 and denote the extended function by u problem (2.1)–(2.2) is also a solution of the problem

x_i ðtÞ ¼ ai ðtÞxi ðhi ðtÞÞ þ

m   X ~ j ðg ij ðtÞÞ  ai ðtÞu ~ i ðhi ðtÞÞ; F ij t; xj ðg ij ðtÞÞ þ u

t P t0

ð2:4Þ

j¼1

and xi ðtÞ ¼ 0; t < t 0 ; xi ðt0 Þ ¼ x0i ; i ¼ 1; . . . ; m. Let 0 < k < mini ai . The substitution xi ðtÞ ¼ ekðtt0 Þ yi ðtÞ for t P t0 into system (2.4) yields

y_i ðtÞ ¼ kyi ðtÞ  ekðthi ðtÞÞ ai ðtÞyi ðhi ðtÞÞ þ

m   X ~ j ðg ij ðtÞÞ  ekðtt0 Þ ai ðtÞu ~ i ðhi ðtÞÞ: ekðtt0 Þ F ij t; ekðgij ðtÞt0 Þ yj ðg ij ðtÞÞ þ u

ð2:5Þ

j¼1

Let

li ðtÞ :¼ ekðthi ðtÞÞ ai ðtÞ  k, then y_i ðtÞ ¼ li ðtÞyi ðtÞ þ ekðthi ðtÞÞ ai ðtÞ

Z

t

hi ðtÞ

y_i ðsÞds þ

m   X ~ j ðg ij ðtÞÞ  ekðtt0 Þ ai ðtÞu ~ i ðhi ðtÞÞ: ekðtt0 Þ F ij t; ekðgij ðtÞt0 Þ yj ðg ij ðtÞÞ þ u j¼1

For the function y_i ðsÞ in the second term, we substitute the right-hand side of Eq. (2.5) and obtain

y_i ðtÞ ¼ li ðtÞyi ðtÞ þ ekðthi ðtÞÞ ai ðtÞ # Z t " m   X kðshi ðsÞÞ kðst 0 Þ kðg ij ðsÞt 0 Þ kðst Þ 0 ~ j ðg ij ðsÞÞ  e ~ i ðhi ðsÞÞ ds  kyi ðsÞ  e ai ðsÞyi ðhi ðsÞÞ þ e F ij s; e yj ðg ij ðsÞ þ u ai ðtÞu hi ðtÞ

j¼1

m   X ~ j ðg ij ðtÞÞ  ekðtt0 Þ ai ðtÞu ~ i ðhi ðtÞÞ: ekðtt0 Þ F ij t; ekðgij ðtÞt0 Þ yj ðg ij ðtÞÞ þ u þ j¼1

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Hence 

yi ðtÞ ¼ e

þ

Rt t0

Z

li ðsÞds

t

e



xi ðt 0 Þ

Rt s

li ðfÞ df

ekðshi ðsÞÞ ai ðsÞ

t0

e

"

s hi ðsÞ

#

kðft 0 Þ

Z

~ i ðhi ðfÞÞ df þ ai ðfÞu

m X

kyi ðfÞ  ekðfhi ðfÞÞ ai ðfÞyi ðhi ðfÞÞ þ

m X   ~ j ðg ij ðfÞÞ ekðft0 Þ F ij f; ekðgij ðfÞt0 Þ yj ðg ij ðfÞÞ þ u j¼1

kðst 0 Þ

e



kðg ij ðsÞt 0 Þ

F ij s; e

!  kðst Þ 0 ~ j ðg ij ðsÞÞ  e ~ i ðhi ðsÞÞ ds: yj ðg ij ðsÞÞ þ u ai ðsÞu

j¼1

We have

 kðtt Þ  kðg ðtÞt Þ    0 0 e ~ j ðg ij ðtÞÞ  6 ekðtt0 Þ Lij ekðgij ðtÞt0 Þ jyj ðg ij ðtÞÞj þ ju ~ j ðg ij ðtÞÞj F ij t; e ij yj ðg ij ðtÞÞ þ u   ~ j ðg ij ðtÞÞj : ¼ Lij ekðtgij ðtÞÞ jyj ðg ij ðtÞÞj þ ekðtt0 Þ ju ~ j ðg ij ðtÞÞ ¼ 0 for t P t þ r, the inequalities Since t  g ij ðtÞ 6 r and u

~ j ðg ij ðtÞÞj 6 ekr ju ~ j ðg ij ðtÞÞj; ekðtt0 Þ ju ~ i ðhi ðtÞÞj 6 eksi ju ~ i ðhi ðtÞÞj ekðtt0 Þ ju hold. Then

jyi ðtÞj 6 jxi ðt 0 Þj þ

Z

t



e

Rt s

li ðfÞdf

t0

li ðsÞ Ai e

ksi

Z

"

s

kjyi ðfÞj þ eksi Ai jyi ðhi ðfÞÞj þ

hi ðsÞ

m X ekr Lij jyj ðg ij ðfÞÞj j¼1

! # ! !, m m m X X X Lij ekr þ eksi Ai U df þ ekr Lij jyj ðg ij ðsÞÞj þ Lij þ eksi Ai U li ðsÞ ds; þ j¼1

j¼1

j¼1

i ¼ maxt0 6t6b jyi ðtÞj, ~ i ðtÞj ¼ maxi supt0 r6t6t0 jui ðtÞj. Let us fix some b > t 0 and denote y where U ¼ maxi supt0 r6t6t0 ju T    Y ¼ ðy1 ; . . . ; ym Þ . Then

P m i 6 jxi ðt 0 Þj þ y

j¼1 Lij e

kr

 þ eksi Ai ðAi eksi si þ 1Þ

ai  k

" ksi

U þ Ai s i e

i þ Ai eksi y i þ ky

m X j ekr Lij y

!

m X j þ ekr Lij y

j¼1

#, ðai  kÞ:

j¼1

We define the matrix CðkÞ ¼ ðcij ðkÞÞm with the entries i;j¼1

Ai eksi ðk þ Ai eksi þ ekr Lii Þsi þ ekr Lii ; ai  k Ai eksi ekr Lij si þ ekr Lij cij ðkÞ ¼  ; i – j: ai  k

cii ðkÞ ¼ 1 

 6 jXðt0 Þj þ UM 1 ðkÞ is valid for t 0 6 t 6 b, where Clearly, the vector inequality CðkÞY

M1 ðkÞ ¼

 8P m kr < þ A1 eks1 ðA1 eks1 s1 þ 1Þ j¼1 L1j e :

a1  k

P ;...;

m kr j¼1 Lmj e

 9T þ Am eksm ðAm eksm sm þ 1Þ=

am  k

;

and we have limk!0 CðkÞ ¼ Cð0Þ ¼ C. By the assumption of the theorem, Cð0Þ is an M-matrix. For 0 < k < mini ai the entries of the matrix CðkÞ are continuous functions; therefore, the determinant of this matrix is a continuous function. For some small k > 0 all the principal minors of CðkÞ are positive; the latter along with cij ðkÞ 6 0; i – j implies that CðkÞ is an M-matrix for  there is an a priori estimate small k. If we fix such parameter k ¼ k0 , then for Y

 6 MðkXðt 0 Þk þ UÞ; kYk

M ¼ kC 1 ðk0 Þk maxf1; kM 1 ðk0 Þkg;

where M does not depend on b and t 0 . Finally, XðtÞ ¼ ek0 ðtt0 Þ YðtÞ, hence

kXðtÞk 6 M ðkXðt 0 Þk þ UÞek0 ðtt0 Þ ; which completes the proof.

h

Consider the system with off-diagonal nonlinearities

x_i ðtÞ ¼ ai ðtÞxi ðhi ðtÞÞ þ

X   F ij t; xj ðg ij ðtÞÞ ; j–i

t P 0;

i ¼ 1; . . . ; m:

ð2:6Þ

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Corollary 1. Suppose that the matrix C defined by

C ¼ ðcij Þm i;j¼1 ;

A2i si

cii ¼ 1 

ai

cij ¼ 

;

Ai Lij si þ Lij

ai

;

i–j

ð2:7Þ

is an M-matrix. Then system (2.6) is globally exponentially stable. The next corollary examines the system with a non-delay linear term

x_i ðtÞ ¼ ai ðtÞxi ðtÞ þ

m X   F ij t; xj ðg ij ðtÞÞ ;

t P 0;

i ¼ 1; . . . ; m:

ð2:8Þ

j¼1

Denote m

B ¼ ðbij Þi;j¼1 ;

Lii

bii ¼ 1 

ai

bij ¼ 

;

Lij

ai

i – j:

;

ð2:9Þ

Corollary 2. Suppose B defined by (2.9) is an M-matrix. Then system (2.8) is globally exponentially stable.

Remark 1. System (2.8) represents the so-called pure-delay case. Interesting and significant results for neural networks models with pure delays were recently obtained in [8,19]. Note that Corollary 2 is a partial case of the results of the paper [19]. For the delay linear system

x_i ðtÞ ¼

m X aij ðtÞxj ðg ij ðtÞÞ;

i ¼ 1; . . . ; m

ð2:10Þ

j¼1

assume that aij are essentially bounded on ½0; 1Þ, 0 < ai 6 aii ðtÞ 6 Ai ; jaij ðtÞj 6 Aij ; i – j, g ij are Lebesgue measurable functions, 0 6 t  g ij ðtÞ 6 rij . Denote m

D ¼ ðdij Þi;j¼1 ;

dii ¼ 1 

A2i rii

ai

dij ¼ 

;

Ai Aij rii þ Aij

ai

;

i – j:

ð2:11Þ

Corollary 3. Suppose D defined by (2.11) is an M-matrix. Then system (2.10) is exponentially stable. The same result holds for the linear system with non-delay diagonal terms

x_i ðtÞ ¼ ai ðtÞxi ðtÞ þ

X

aij ðtÞxj ðg ij ðtÞÞ;

i ¼ 1; . . . ; m;

ð2:12Þ

j–i

where aij are essentially bounded on ½0; 1Þ functions, 0 < ai 6 ai ðtÞ 6 Ai ; jaij ðtÞj 6 Aij ; i – j, g ij are measurable functions, 0 6 t  g ij ðtÞ 6 r. Denote m

F ¼ ðfij Þi;j¼1 ;

f ii ¼ 1; fij ¼ 

Aij

ai

;

i – j:

ð2:13Þ

Corollary 4. Suppose F defined by (2.13) is an M-matrix. Then system (2.12) is exponentially stable. Corollary 5. Suppose m ¼ 2, A1 ðA1 þ L11 Þs1 þ L11 < a1 and

ða1  A1 ðA1 þ L11 Þs1  L11 Þða2  A2 ðA2 þ L22 Þs2  L22 Þ > L12 L21 ð1 þ A1 s1 Þð1 þ A2 s2 Þ:

ð2:14Þ

Then system (2.1) is globally exponentially stable. Proof. For m ¼ 2 the matrix C denoted by (2.3) has the form

0 C¼@

1  A1 ðA1 þLa111 Þs1 þL11

 A1 L12as11 þL12

 A2 L21as22 þL21

1  A2 ðA2 þLa222 Þs2 þL22

1 A:

The off-diagonal entries are negative, by the assumption of the corollary the principal minors are positive, so C is an Mmatrix. h Corollary 6. Suppose m ¼ 2; L11 < a1 and ða1  L11 Þða2  L22 Þ > L12 L21 . Then system (2.8) is globally exponentially stable.

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Corollary 7. Suppose m ¼ 2;

r11 < a1 =A21 and the inequality

ða1  A21 r11 Þða2  A22 r22 Þ > A12 A21 ð1 þ A1 r11 Þð1 þ A2 r22 Þ holds. Then system (2.10) is exponentially stable. Corollary 8. Suppose m ¼ 2 and a1 a2 > A12 A21 . Then system (2.12) is exponentially stable. Remark 2. By Corollary 8, the linear system

_ xðtÞ ¼ a11 xðtÞ þ a12 yðtÞ _ yðtÞ ¼ a21 xðtÞ  a22 yðtÞ

ð2:15Þ

with the coefficients aii > 0 is exponentially stable if

a11 a22 > a12 a21 :

ð2:16Þ

Condition (2.16) is necessary and sufficient for exponential stability of system (2.15); therefore, for system (2.15), which is a partial case of (2.1), Theorem 2.5 gives necessary and sufficient exponential stability conditions. In the paper [10] Gopalsamy considered autonomous system (1.3). For n ¼ 1 it has the form

_ xðtÞ ¼ a1 xðt  s1 Þ þ a12 f1 ðyðt  r1 ÞÞ _yðtÞ ¼ a2 yðt  s2 Þ þ a21 f2 ðxðt  r2 ÞÞ where ai > 0; aij > 0; ai si < 1 and

ð2:17Þ

si P 0; ri P 0; jfi ðuÞj 6 Li juj; i ¼ 1; 2. In [10] the following global attractivity result was obtained: if

1  a1 s1 a12 L1 1  a2 s2 a21 L2 > ; > 1 þ a1 s1 a1 1 þ a2 s2 a2

ð2:18Þ

then any solution of system (2.17) tends to zero. By Corollary 5 Eq. (2.17) is exponentially stable if ai si < 1 and

ð1  a1 s1 Þð1  a2 s2 Þ a12 a21 L1 L2 > : ð1 þ a1 s1 Þð1 þ a2 s2 Þ a1 a2

ð2:19Þ

Obviously condition (2.18) implies (2.19). Example 1. Consider system (2.17) where a1 ¼ 0:8; a2 ¼ 0:5; a12 ¼ a21 ¼ 1; L1 ¼ 0:5; L2 ¼ 0:2; ri P 0. Here the first inequality in (2.18) does not hold since

s1 ¼ 0:5; s2 ¼ 0:4, jfi ðuÞj 6 Li juj with

1  a1 s1 3 5 a12 L1 ¼ < ¼ 1 þ a1 s1 7 8 a1 and therefore the result of [10] cannot be applied. However, a1 s1 ¼ 0:4 < 1 and inequality (2.19)

ð1  a1 s1 Þð1  a2 s2 Þ 2 1 a12 a21 L1 L2 ¼ > ¼ ð1 þ a1 s1 Þð1 þ a2 s2 Þ 7 4 a1 a2 holds, thus Corollary 5 implies exponential stability, hence for n ¼ 1 (m ¼ 2) we obtained the result which is sharper than the relevant result in [10]. In the next section, we provide more in-depth analysis of systems with leakage delays which include (2.17) as a special case. 3. BAM Network with time-varying delays In [10] a class (1.3) of BAM neural networks with leakage (forgetting) delays was under study. Via Lyapunov functionals method, sufficient conditions for the existence of a unique equilibrium and its global stability for system (1.3) were obtained. To extend and improve the results of [10,18,20], we will focus on the non-autonomous BAM neural network model

x_ i ðtÞ ¼ ri ðtÞ

ð1Þ ai xi ðhi ðtÞÞ

n    X ð2Þ þ aij fj yj lj ðtÞ þ Ii

!

j¼1 ð2Þ y_ i ðtÞ ¼ pi ðtÞ bi yi ðhi ðtÞÞ þ

! n    X ð1Þ bij g j xj lj ðtÞ þ J i ;

ð3:1Þ

j¼1

i ¼ 1; . . . ; n; t P 0, with the initial conditions

xi ðtÞ ¼ ui ðtÞ;

yi ðtÞ ¼ uiþn ðtÞ;

t < 0;

i ¼ 1; . . . ; n:

ð3:2Þ

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L. Berezansky et al. / Applied Mathematics and Computation 243 (2014) 899–910

We will say that a norm in Rn is monotone if 0 6 a 6 b (a componentwise inequality, i.e. 0 6 ai 6 bi for all i) implies kak 6 kbk. The following auxiliary lemmas will be used. Lemma 3.1 [13, Chapter 2, Section 8.1], [9, Chapter 4, Problem 228]. For any linear operator L : Rm ! Rm and exists a norm in Rm such that for the spectral radius rðLÞ of L we have

e > 0 there

rðLÞ > kLk  e:

ð3:3Þ m

If the operator L is positive then the norm in R is monotone. Consider the algebraic system

ui ¼

m X F ij ðuj Þ;

i ¼ 1; . . . ; m;

ð3:4Þ

j¼1

where jF ij ðuÞ  F ij ðv Þj 6 Lij ju  v j. Lemma 3.2. Let rðLÞ be a spectral radius of the matrix L ¼ ðLij Þm . If rðLÞ < 1 then system (3.4) has a unique solution. i;j¼1 Proof. Consider the operator T : Rm ! Rm denoted by T

TðuÞ :¼ Tððu1 ; . . . ; um Þ Þ ¼

m X

!T m X F 1j ðuj Þ; . . . ; F mj ðuj Þ :

j¼1

j¼1

Then

jTðuÞ  Tðv Þj 6

m m X X L1j juj  v j j; . . . ; Lmj juj  v j j j¼1

!T ¼ Lju  v j;

j¼1

where the inequality is understood componentwise. Since rðLÞ < 1 and the operator L is positive, by Lemma 3.1 we can choose a monotone norm in Rm such that the corresponding norm kLk 6 q < 1. We fix now such a norm and have

kTðuÞ  Tðv Þk 6 kLju  v jk 6 kLkku  v k 6 qku  v k: By the Banach contraction principle the equation u ¼ TðuÞ has a unique solution.

h

Corollary 9. Suppose at least one of the following conditions holds: 1. max jkðLÞj < 1, where the maximum is taken over all eigenvalues of matrix L; P 2. maxi m j¼1 Lij < 1; P 3. maxj m i¼1 Lij < 1; Pm Pm 2 4. i¼1 j¼1 Lij < 1. Then system (3.4) has a unique solution. It should be noted that the proof of Lemma 3.2 is original and shorter than, for example, the recently published proof [25, Theorem 2.2]. Henceforth, assume that the following assumptions hold for (3.1)–(3.2): i 6 pi ðtÞ 6 P i ; (b1) ri ; pi are Lebesgue measurable essentially bounded on ½0; 1Þ functions, 0 < ri 6 r i ðtÞ 6 Ri ; 0 < p (b2) fj ðÞ; g j ðÞ are continuous functions; jfj ðuÞ  fj ðv Þj 6 Lfj ju  v j; jg j ðuÞ  g j ðv Þj 6 Lgj ju  v j; ð1Þ

ð2Þ

ð1Þ

ð2Þ

ð1Þ

ð1Þ

ð2Þ

ð2Þ

ð1Þ

ð1Þ

(b3) hi ; hi ; lj ; lj are Lebesgue measurable functions, 0 6 t  hi ðtÞ 6 si ; 0 6 t  hi ðtÞ 6 si , 0 6 t  li ðtÞ 6 ri ; ð2Þ ð2Þ 0 6 t  li ðtÞ 6 ri ; (b4) ui are continuous functions.   Let ðx ; y Þ ¼ x1 ; . . . ; xn ; y1 ; . . . ; yn be a solution of the system

ai xi ¼

n X aij fj ðyj Þ þ Ii j¼1

n X bi yi ¼ bij g j ðxj Þ þ J i : j¼1

Let us transform system (3.5) into the following one

ð3:5Þ

906

L. Berezansky et al. / Applied Mathematics and Computation 243 (2014) 899–910

xi ¼

n X aij j¼1

yi ¼

ai

n X bij j¼1

bi

fj ðyj Þ þ

Ii ai

g j ðxj Þ þ

ð3:6Þ

Ji : bi

Denoting uj ¼ xj ; j ¼ 1; . . . ; n; uj ¼ yjn ; j ¼ n þ 1; . . . ; 2n,

8 ai;jn f ðuÞ þ aIi ; i ¼ 1; . . . ; n; j ¼ n þ 1; . . . ; 2n; > > i < ai jn F ij ðuÞ ¼ bin;j g j ðuÞ þ Jin ; i ¼ n þ 1; . . . ; 2n; j ¼ 1; . . . ; n; b bin > > : in 0; otherwise;

we can rewrite system (3.6) in the form of (3.4) with m ¼ 2n; jF ij ðuÞ  F ij ðv Þj 6 Lij ju  v j. We introduce the matrix A ¼ ðLij Þ2n : i;j¼1

0

f

0

B B. B. B. B B B0 A¼B B jb11 jLg B 1 B b1 B B. B .. @ g jbn1 jL1 bn

... 0

ja11 jL1 a1

.. .

.. .

.. . g

...

jb1n jLn b1

0

.. .

.. .

.. .

...

jbnn jLn bn

g

ja1n jLfn a1

1

C C C C C f C jann jLn C ... an C C: C ... 0 C C C .. .. C . . A ... 0

.. .

jan1 jLf1 an

... 0

...

0

By the same token we can rewrite (3.5) in the form (3.4), where

8   u > > > ai;jn fjn bjn þ Ii ; i ¼ 1; . . . ; n; j ¼ n þ 1; . . . ; 2n; <   F ij ðuÞ ¼ b g u þ J ; i ¼ n þ 1; . . . ; 2n; j ¼ 1; . . . ; n; in;j j a in > j > > : 0; otherwise;

with m ¼ 2n; jF ij ðuÞ  F ij ðv Þj 6 Lij ju  v j, and introduce the matrix B ¼ ðLij Þ2n i;j¼1 :

0

0 B B. B. B. B B B0 B¼B B jb11 jLg B 1 B a1 B B. B .. @ g jbn1 jL1 a1

... 0

ja11 jLf1 b1

.. .

.. .

.. .

f

jan1 jL1 b1

... 0 ...

jb1n jLgn an

0

.. .

.. .

.. .

...

jbnn jLgn an

0

f

...

ja1n jLn bn

.. .

.. .

1

C C C C C f C jann jLn C ... bn C C: C ... 0 C C C .. .. C . . A ... 0

In the following theorem we apply Lemma 3.2 to systems (3.5) and (3.6) with L ¼ A and L ¼ B, and obtain conditions 1–4 and 5–8, respectively. Theorem 3.3. Suppose at least one of the following conditions holds: 1. max jkðAÞj < 1, where maximum is taken on all eigenvalues of matrix A. P jaij jLf P jbij jLg 2. maxi nj¼1 a j < 1; maxi nj¼1 b j < 1. i i P jaij jLf P jbij jLg 3. maxj ni¼1 ai j < 1; maxj ni¼1 bi j < 1. " 2  g 2 # f Pn Pn jaij jLj jbij jLj þ 4. < 1. i¼1 j¼1 a b i

i

5. max jkðBÞj < 1, where maximum is taken on all eigenvalues of matrix B. P jaij jLf P jbij jLg 6. maxi nj¼1 b j < 1; maxi nj¼1 a j < 1. j j P jaij jLf P jbij jLg 7. maxj ni¼1 bj j < 1; maxj ni¼1 aj j < 1. " 2  g 2 # f Pn Pn jaij jLj jbij jLj 8. þ < 1. i¼1 j¼1 b a j

j

Then system (3.5) has a unique solution and thus system (3.1) has a unique equilibrium.

L. Berezansky et al. / Applied Mathematics and Computation 243 (2014) 899–910

907

Remark 3. Note that the conclusion of Theorem 3.3 under condition 7 was obtained in paper [10]. Below we assume everywhere that system (3.1) has a unique equilibrium ðx ; y Þ. To obtain a global stability condition for this equilibrium, consider the matrix C BAM ¼ ðcij Þ2n , where i;j¼1

(

cii ¼

ð1Þ 1  ai R2i si =r i ;

1

i ¼ 1; . . . ; n;

ð2Þ in ; bin P2in in =p

s

ð3:7Þ

i ¼ n þ 1; . . . ; 2n;

8 ð1Þ f  > > < jai;jn jRi Ljn ðai Ri si þ 1Þ=ðri ai Þ; i ¼ 1; . . . ; n; j ¼ n þ 1; . . . ; 2n; ð2Þ g cij ¼ jbin;j jPin L ðbin Pin s þ 1Þ=ðp in bin Þ; i ¼ n þ 1; . . . ; 2n; j ¼ 1; . . . ; n; in j > > : 0; otherwise:

ð3:8Þ

Theorem 3.4. Suppose matrix C BAM is an M-matrix. Then the equilibrium ðx ; y Þ of system (3.1) is globally exponentially stable. Proof. After the substitution xi ðtÞ ¼ ui ðtÞ þ xi ; yi ðtÞ ¼ v i ðtÞ þ yi , system (3.1) has the form ð1Þ u_i ðtÞ ¼ r i ðtÞai ui ðhi ðtÞÞ þ

n   X ð2Þ aij ri ðtÞ fj ðv j ðlj ðtÞÞ þ yj Þ  fj ðyj Þ j¼1

v_ i ðtÞ ¼ pi ðtÞbi v

ð2Þ i ðhi ðtÞÞ

n   X ð1Þ þ bij pi ðtÞ g j ðuj ðlj ðtÞÞ þ xj Þ  g j ðxj Þ ;

ð3:9Þ

j¼1

where

xi ðtÞ ¼ ( hi ðtÞ ¼



ui ðtÞ;

i ¼ 1; . . . ; n;

v in ðtÞ;

i ¼ n þ 1; . . . ; 2n;

ð1Þ

hi ðtÞ;

i ¼ 1; . . . ; n;

ð2Þ hin ðtÞ;

i ¼ n þ 1; . . . ; 2n;

ai ðtÞ ¼

r i ðtÞai ; i ¼ 1; . . . ; n; in ðtÞbin ; i ¼ n þ 1; . . . ; 2n; p

8 ð2Þ > > < ljn ðtÞ; i ¼ 1; . . . ; n; j ¼ n þ 1; . . . ; 2n; g i;j ðtÞ ¼ lð1Þ ðtÞ; i ¼ n þ 1; . . . ; 2n; j ¼ 1; . . . ; n; j > > : 0; otherwise; 8   > ai;jn r i ðtÞ fjn ðx þ yjn Þ  fjn ðyjn Þ ; i ¼ 1; . . . ; n; j ¼ n þ 1; . . . ; 2n; > > <   F i;j ðt; xÞ ¼ b p ðtÞ g ðx þ x Þ  g ðx Þ ; i ¼ n þ 1; . . . ; 2n; j ¼ 1; . . . ; n; j j j > in;j in j > > : 0; otherwise: We have 0 < ci 6 ai ðtÞ 6 Ai , where

ci ¼



r i ai ; i ¼ 1; . . . ; n; in bin ; i ¼ n þ 1; . . . ; 2n; p

Ai ¼



Ri ai ;

i ¼ 1; . . . ; n;

Pin bin ; i ¼ n þ 1; . . . ; 2n

and jF ij ðt; xÞj 6 Lij jxj with the constant

8 f > < jai;jn jRi Ljn ; i ¼ 1; . . . ; n; j ¼ n þ 1; . . . ; 2n; Li;j ¼ jbin;j jPin Lgj ; i ¼ n þ 1; . . . ; 2n; j ¼ 1; . . . ; n; > : 0; otherwise: System (3.9) with m ¼ 2n has form (2.6), where matrix C BAM corresponds to matrix C defined by (2.7). All conditions of Corollary 1 hold; therefore, the trivial solution of system (3.9) is globally exponentially stable; hence the equilibrium ðx ; y Þ of system (3.1) is globally exponentially stable. h Corollary 10. Suppose at least one of the following conditions holds: 1.

ð1Þ Xn jaij jRi Lfj ðai Ri sð1Þ þ 1Þ ai R2i si i <1 , j¼1 r i r i ai ð2Þ

g Pn jbij jPi Lj ðbi Pi si j¼1 i bi p

þ 1Þ

ð2Þ

<1

bi P 2i si , i ¼ 1; . . . ; n. i p

908

2.

L. Berezansky et al. / Applied Mathematics and Computation 243 (2014) 899–910

Pn

ð1Þ jaij jRi Lfj ðai Ri i

Pn

ð2Þ jbij jPi Lgj ðbi Pi i

s þ 1Þ

i¼1

ri ai

s þ 1Þ

i¼1

i bi p

ð2Þ

<1

bj P2j sj , j p

<1

aj R2j sj ; j ¼ 1; . . . ; n. r j

ð1Þ

3. There exist positive numbers Pn

l

ð1Þ f jþn jaij jRi Lj ðai Ri i

l

ð2Þ g j jbij jP i Lj ðbi P i i

s þ 1Þ

j¼1

Pn

ri ai

s þ 1Þ

j¼1

i bi p

lk ; k ¼ 1; . . . ; 2n such that ð1Þ

< li

ai R2i si 1 r i

!

,

! ð2Þ

< liþn 1 

bi P 2i si i p

; i ¼ 1; . . . ; n.

lk ; k ¼ 1; . . . ; 2n such that ! ð2Þ s þ 1Þ bj P 2j sj < lj 1  , 

4. There exist positive numbers Pn

l

ð1Þ f iþn jaij jRi Lj ðai Ri i

l

ð2Þ g j jbij jP i Lj ðbi P i i

i¼1

Pn

r i ai

pj

s þ 1Þ

i¼1

i bi p

ð1Þ

< ljþn

aj R2j sj 1 r j

! , j ¼ 1; . . . ; n.

Then the equilibrium ðx ; y Þ of system (3.1) is globally exponentially stable.

Proof. By Lemma 2.3 any of the conditions 1–4 implies that C BAM is an M-matrix.

h

Remark 4. Part 3 of Corollary 10 coincides with [18, Theorem 3.1] in the case when ri ðtÞ and pi ðtÞ are constants. In addition to being more general than [18, Theorem 3.1], the result of Theorem 3.4 does not require to find some positive constants, i.e., the check of the signs of principal minors will immediately indicate whether such constants exist or not. In the following statement consider system (3.1) without delays in the leakage terms. ð1Þ

ð2Þ

Corollary 11. Suppose hi ðtÞ  t; hi ðtÞ  t, and at least one of the following conditions holds: f Xn jbij jP i Lgj Pn jaij jRi Lj < 1; < 1, i ¼ 1; . . . ; n. j¼1 j¼1 r i ai i bi p f Xn jbij jPi Lgj Pn jaij jRi Lj < 1; < 1; j ¼ 1; . . . ; n. 2. i¼1 i¼1 r i ai i bi p 3. There exist positive numbers lk ; k ¼ 1; . . . ; 2n such that

1.

ljþn jaij jRi Lfj

Pn

j¼1

r i ai

< li ;

Xn j¼1

4. There exist positive numbers Pn

liþn jaij jRi Lfj

i¼1

ri ai

< lj ;

Xn i¼1

lj jbij jPi Lgj i bi p

< liþn ; i ¼ 1; . . . ; n.

lk ; k ¼ 1; . . . ; 2n such that li jbij jPi Lgj i bi p

< ljþn ; j ¼ 1; . . . ; n.

Then the equilibrium ðx ; y Þ of system (3.1) is globally exponentially stable. Consider system (3.1) with n ¼ 1:

_ xðtÞ ¼ rðtÞðaxðh1 ðtÞÞ þ Af ðyðl2 ðtÞÞÞ þ IÞ; _yðtÞ ¼ pðtÞðayðh2 ðtÞÞ þ Bf ðxðl1 ðtÞÞÞ þ J Þ;

ð3:10Þ

where

a > 0;

b > 0;

0 < r 6 rðtÞ 6 R; g

jgðuÞ  gðv Þj 6 L ju  v j; Corollary 12. Suppose

aR2 s1 r

 6 pðtÞ 6 P; 0
0 6 t  hi ðtÞ 6 si ;

jf ðuÞ  f ðv Þj 6 Lf ju  v j;

0 6 t  li ðtÞ 6 ri ;

i ¼ 1; 2:

< 1, and

ABRPLf Lg ðaRs1 þ 1ÞðaPs2 þ 1Þ <  ab r p

1

aR2 s1 r

! 1

! 2 bP s2 :  p

Then the equilibrium ðx ; y Þof system (3.10) is globally exponentially stable.

L. Berezansky et al. / Applied Mathematics and Computation 243 (2014) 899–910

909

Example 2. Consider the particular case of BAM network described by (3.10)

    1 þ j sin tj 1  2 _ þ y t  3 sin ðtÞ þ 10000 xðtÞ ¼ ð20 þ l sin tÞ x t  2000 720

    1 þ j cos tj 1  2 _ þ y t  2 sin ðtÞ þ 20000 yðtÞ ¼ ð40 þ l cos tÞ x t  2000 200

for

l P 0. Here Lf ¼ Lg ¼ a ¼ b ¼ 1; r ¼ 20  l; R ¼ 20 þ l; p ¼ 42  l; P ¼ 40 þ l, ð20þlÞ By Corollary 12, system (3.11) is exponentially stable if 1000ð20 lÞ < 1 and l l ð20þ þ 1Þð40þ þ 1Þ 1000 1000 < 720  200ð20  lÞð40  lÞ

1

ð20 þ lÞ2 1000ð20  lÞ

!

1

ð3:11Þ

1 1 1 s1 ¼ s2 ¼ 1000 ; A ¼ 720 ; B ¼ 200 .

! ð40 þ lÞ2 ; 1000ð40  lÞ

which is satisfied, for example, if 0 6 l 6 18. Note that for l ¼ 0 this example coincides with [18, Example 4.1]. It was also mentioned in [18] that exponential stability of (3.11) cannot be obtained using the results of [10,15,16,23], since leakage delays in (3.11) are time-variable. Therefore, our results for the case l > 0 and with time-varying coefficients and delays are new and applicable to more general models. 4. Discussion and open problems To obtain sufficient stability conditions for nonlinear delay systems, four different approaches might be used: construction of Lyapunov functionals, application of fixed point theory, either development of estimations for matrix or operator norms, or making use of some special matrix (M-matrix) properties, and the transformations of a given nonlinear system to an operator equation with a Volterra casual operator. While the Lyapunov direct method has been and remains the leading technique, numerous difficulties with the theory and applications to specific systems persist. One of the problems with using the fixed point techniques is the construction of an appropriate map (integral equation) that is sometimes quite difficult or impossible. The technique used in papers [15–17,24] is based on the construction of Lyapunov functionals. Perhaps it is worth mentioning that the method applied here to nonlinear system (2.1) is somehow related to the approach used in [11] for linear system (2.10); however, to the best of our knowledge, there are no similar results for (2.1). Remark 2, Example 1, Theorem 3.4 and its corollaries improve and extend results previously obtained for BAM neural networks in [6,10,18,23,28,29]. In the framework of this paper, we could not consider all the applications of the M-matrix method to specific models; therefore we outline some particular cases and extensions that might be of interest for scientists who plan to start future research in this field. 1. Find global stability conditions of system (2.1) for the special cases:

F ij ðt; xÞ ¼ tanh ðai xðtÞÞ and F ij ðt; xÞ ¼

1 : 1 þ ai exðtÞ

2. Study global stability for a more general than (2.1) model:

x_i ðtÞ ¼ ai ðtÞxi ðhi ðtÞÞ þ

m X s X   F ijk t; xj ðg ijk ðtÞÞ ;

t P 0;

i ¼ 1; . . . ; m:

j¼1 k¼1

3. Derive sufficient stability tests for the equation with a distributed transmission delay

x_i ðtÞ ¼ ai ðtÞxi ðhi ðtÞÞ þ

m Z X j¼1

t

tsj

  K ij ðt; sÞF ij s; xj ðg ij ðsÞÞ ds;

t P 0; i ¼ 1; . . . ; m:

4. Investigate stability of the system with distributed delays in all the terms

x_i ðtÞ ¼ ai ðtÞ

Z

t

xi ðhi ðsÞÞds Ri ðt; sÞ þ

tri

m Z X j¼1

t

tsj

  F ij s; xj ðg ij ðsÞÞ ds T ij ðt; sÞ;

t P 0; i ¼ 1; . . . ; m:

5. Obtain sufficient stability conditions for the system with an infinite distributed delay

x_i ðtÞ ¼ ai ðtÞxi ðhi ðtÞÞ þ

m Z X j¼1

t

  K ij ðt; sÞF ij s; xj ðg ij ðsÞÞ ds;

t P 0; i ¼ 1; . . . ; m;

1

where jK ij ðt; sÞj 6 MemðtsÞ . Generalize this result to the case of exponentially decaying infinite leakage delays as well. 6. Analyze global asymptotic stability conditions of (2.1) when condition (a3) is not satisfied but limt!1 hi ðtÞ ¼ 1; limt!1 g ij ðtÞ ¼ 1, e.g., the pantograph-type delays hi ðtÞ ¼ ki t for 0 < ki < 1. Is it possible to estimate the rate of convergence for some classes of delays?

910

L. Berezansky et al. / Applied Mathematics and Computation 243 (2014) 899–910

7. Under which conditions will solutions of BAM system (3.1) with Ii > 0; J i > 0 and positive initial functions be permanent (positive, bounded and separated from zero)? 8. Apply the M-matrix method to the following generalization of (2.1)

x_i ðtÞ ¼ ai ðtÞxi ðhi ðtÞÞ þ

m   X ð1Þ ðmÞ F ij t; x1 ðg ij ðtÞÞ; . . . ; xm ðg ij ðtÞÞ ;

t P 0; i ¼ 1; . . . ; m:

j¼1

9. Conjecture: If C defined by (2.3) has negative off-diagonal entries aij 6 0; i – j, and its Moore-Penrose pseudoinverse matrix is nonnegative then system (2.1) is stable.

Remark 5. To apply the results of the present paper, first use [4, Theorem 9] to reduce exponential stability of equations with distributed delays to exponential stability of equations with concentrated delays. For some other methods see recent papers [1–3] and references therein. Acknowledgment The authors are very grateful to the anonymous reviewers and the editor whose thoughtful comments significantly contributed to the present form of the paper. References [1] L. Berezansky, E. Braverman, Stability and linearization for differential equations with a distributed delay, Funct. Diff. Equ. 19 (2012) 27–43. [2] L. Berezansky, E. Braverman, On nonoscillation and stability for systems of differential equations with a distributed delay, Automatica 48 (2012) 612– 618. [3] L. Berezansky, E. Braverman, On the existence of positive solutions for systems of differential equations with a distributed delay, Comput. Math. Appl. 63 (2012) 1256–1265. [4] L. Berezansky, E. 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