RAZUMIKHIN-TYPE THEOREMS ON STABILITY

OF STOCHASTIC NEURAL NETWORKS WITH DELAYS

Steve Blythe, Xuerong Mao1 and Anita Shah2

Department of Statistics and Modelling Science

University of Strathclyde

Glasgow G1 1XH, Scotland, U.K.

ABSTRACT

Although the stability of neural networks has been studied by many authors,the problem of stochastic effects on the stability has not been investigateduntil recently by Liao and Mao [7, 8]. In this paper we shall investigatethe stability problem for stochastic neural networks with time-varying delay.The main technique employed in this paper is the well-known Razumikhinargument, which is completely different from those used in Liao and Mao [7,8].

1. INTRODUCTION

Theoretical understanding of neural-network dynamics has advanced greatly

in the past ten years (cf. Coben and Crosshery [1], Denker [2], Hopfield

[4,5], Hopfield and Tank [6], Quezz et al. [13]). In many networks, time

1For any correspondence regarding this paper please address it to this author.

2Supported by the RD fund of Strathclyde University.

delays can not be avoided. For example, in electronic neural networks, time

delays will be present due to the finite switching speed of amplifiers. Marcus

and Westervelt [11] proposed, in a similar way as Hopfield [4], a model for a

network with delay as follows:

Cixi(t) = − 1Ri

xi(t) +n∑

j=1

Tijgj(xj(t− τ)), 1 ≤ i ≤ n, (1.1)

on t ≥ 0. The variable xi(t) represents the voltage on the input of the ith

neuron. Each neuron is characterized by an input capacitance Ci, a time

delay τ and a transfer function gi(·). The connection matrix element Tij has

a value +1/Rij when the noninverting output of the jth neuron is connected

to the input of the ith neuron through a resistance Rij , and a value −1/Rij

when the inverting output of the jth neuron is connected to the input of the

ith neuron through a resistance Rij . The parallel resistance at the input of

each neuron is defined Ri = (∑n

j=1 |Tij |)−1. The nonlinear transfer function

gi(u) is sigmoidal, saturating at ±1 with maximum slope at u = 0. That is,

in term of mathematics, gi(u) is nondecreasing and

|gi(u)| ≤ 1 ∧ βi|u| for all −∞ < u < ∞, (1.2)

where βi is the finite slope of gi(u) at u = 0. By defining

bi =1

CiRi, aij =

Tij

Ci

equation (1.1) can be re-written as

xi(t) = −bixi(t) +n∑

j=1

aijgj(xj(t− τ)), 1 ≤ i ≤ n, (1.3)

or, in form,

x(t) = −Bx(t) + Ag(x(t− τ)), t ≥ 0, (1.4)

where

x(t) = (x1(t), · · · , xn(t))T , g(x) = (g1(x1), · · · , gn(xn))T ,

A = (aij)n×n, B = diag.(b1, · · · , bn),

with

bi =n∑

j=1

|aij |, 1 ≤ i ≤ n. (1.5)

It is clear that for any given initial data x(θ) = ξ(θ) on −τ ≤ θ ≤ 0, which

is in C([−τ, 0];Rn), equation (1.4) has a unique global solution on t ≥ 0.

Suppose that there exists a stochastic perturbation to the neural net-

work and the stochastically perturbed network may be described by a sto-

chastic differential delay equation

dx(t) = [−Bx(t) + Ag(x(t− τ))]dt + σ(x(t), x(t− τ))dw(t). (1.6)

Moreover, under even closer scrutiny, it turns out that the time delay is

often time-dependent rather than constant. Then, a more realistic model for

a stochastic neural network would be

dx(t) = [−Bx(t) + Ag(x(t− δ(t)))]dt + σ(x(t), x(t− δ(t)))dw(t). (1.7)

Liao and Mao [7, 8] have investigated the exponential stability of equation

(1.6) via the method of Lyapunov functionals. However, their technique does

not work for the more general equation (1.7). The aim of this paper is to

develop the Razumikhin argument (cf Razumikhin [14, 15]) in the study of

stability of deterministic differential delay equation to cope with the stability

of the stochastic delay neural network (1.7).

2. MAINRESULTS

Throughout this paper, unless otherwise specified, we let τ > 0 and δ : R+ →

[0, τ ] be a continuous function. Denote by C([−τ, 0];Rn) the family of con-

tinuous functions ϕ from [−τ, 0] to Rn with the norm ||ϕ|| = supτ≤θ≤0 |ϕ(θ)|,

where | · | is the Euclidean norm in Rn. If A is a vector or matrix, its trans-

pose is denoted by AT . If A is a matrix, its operator norm ||A|| is defined by

||A|| = sup|Ax| : |x| = 1 (without any confusion with ||ϕ||). Moreover, let

w(t) = (w1(t), · · · , wm(t))T be an m-dimensional Brownian motion defined

on a complete probability space (Ω,F , P ) with the natural filtration Ftt≥0

(i.e. Ft = σw(s) : 0 ≤ s ≤ t). For every t ≥ 0, denote by L2Ft

([−τ, 0];Rn)

the family of all Ft-measurable C([−τ, 0];Rn)-valued random variables φ =

φ(θ) : −τ ≤ θ ≤ 0 such that ||φ||2L2 := sup−τ≤θ≤0 E|φ(θ)|2 < ∞. Let

L2(Ω;Rn) denote the family of all Rn-valued random variables X such that

E|X|2 < ∞.

Let σ : Rn × Rn → Rn×m (i.e. σ(x, y) = (σij(x, y))n×m) be locally

Lipschitz continuous and satisfy the linear growth condition (cf. Mao [9,

10] or Mohammed [12]). Let ξ = ξ(θ) : −τ ≤ θ ≤ 0 ∈ L2F0

([−τ, 0];Rn).

Consider the stochastic delay neural network (1.7), namely

dx(t) = [−Bx(t) + Ag(x(t− δ(t)))]dt + σ(x(t), x(t− δ(t)))dw(t) (2.1)

on t ≥ 0 with initial data x(θ) = ξ(θ) for θ ∈ [−τ, 0], where B,A, g have

been defined before. It is well-known (cf. Mao [9, 10] or Mohammed [12])

that equation (2.1) has a unique global solution on t ≥ 0, which is denoted

by x(t; ξ) in this paper, and that the second moment of the solution is con-

tinuous. We will also assume σ(0, 0) = 0 for the stability purpose of this

paper. So equation (2.1) has the solution x(t; 0) ≡ 0 corresponding to the

initial data x(θ) = 0 on [−τ, 0]. This solution is called the trivial solution or

equilibrium position.

Let C2(Rn;R+) denote the family of all C2-functions from Rn to R+.

If V ∈ C2(Rn;R+), define an operator LV : Rn ×Rn → R by

LV (x, y) = Vx(x)[−Bx + Ag(y)] +12trace

[σT (x, y)Vxx(x)σ(x, y)

],

where

Vx(x) =(∂V (x)

∂x1, · · · , ∂V (x)

∂xn

)and Vxx(x) =

(∂2V (x)∂xi∂xj

)n×n

.

The main idea of the Razumikhin technique is to use a Lyapunov function,

rather than a functional, to investigate the stability of the delay system:

Applying the well-known Ito formula to eλtV (x(t)) we have

eλtEV (x(t)) = EV (ξ(0)) +∫ t

0

eλs[λV (x(s)) + LV (x(s), x(s− δ(s)))

]ds.

Exponential stability in mean square required that

ELV (x(t), x(t− δ(t))) ≤ −λEV (x(t)) (2.2)

for all t ≥ 0. As a result, one would be forced to impose very severe re-

strictions on the functions g and σ to the extent that the state x(t) plays a

dominant role but the history x(t − δ(t)) has little impact. Therefore, the

results will apply only to networks that are very similar to non-delay ones.

Fortunately, by the Razumikhin argument, one needs (2.2) to hold only for

those t ≥ 0 for which

EV (x(t− δ(t)) ≤ qEV (x(t)),

where q > 1 is a constant, but not necessarily for all t ≥ 0. Hence the restric-

tions on the functions can be much weakened. This is the basic idea used in

this paper. Let us now start to establish the Razumikhin-type theorems.

Theorem 2.1 Let λ, c1, c2 all be positive numbers, and q > 1. Assume that

there exists a function V ∈ C2(Rn;R+) such that

c1|x|2 ≤ V (x) ≤ c2|x|2 for all x ∈ Rn (2.3)

and, moreover, that

ELV (X, Y ) ≤ −λEV (X) (2.4)

for those X, Y ∈ L2(Ω;Rn) which satisfy EV (Y ) ≤ qEV (X). Then, for all

ξ ∈ L2F0

([−τ, 0];Rn), we have

E|x(t; ξ)|2 ≤ c2

c1||ξ||2L2 e−γt on t ≥ 0, (2.5)

where γ = minλ, log(q)/τ. In other words, the trivial solution of equation

(2.1) is exponentially stable in mean square.

Proof. Fix any initial data ξ ∈ L2F0

([−τ, 0];Rn) and write x(t; ξ) = x(t) to

simplify notation. Without loss of generality, we may assume that ||ξ||2L2 > 0;

otherwise ξ = 0 a.s., hence x(t) ≡ 0, and (2.5) holds already. Therefore,

by condition (2.3), we see that sup−τ≤θ≤0 V (ξ(θ)) > 0. Let γ ∈ (0, γ) be

arbitrary. It is easy to show that

0 < γ < λ and q > eγτ . (2.6)

We now claim that

eγtEV (x(t)) ≤ sup−τ≤θ≤0

V (ξ(θ)) for all t ≥ 0. (2.7)

Suppose this is not true. Then there is a number ρ ≥ 0 such that

eγtEV (x(t)) ≤ eγρEV (x(ρ)) = sup−τ≤θ≤0

V (ξ(θ)) (2.8)

for all 0 ≤ t ≤ ρ and, further, there is a sequence of tkk≥1 such that tk ↓ 0

and

eγtkEV (x(tk)) > eγρEV (x(ρ)). (2.9)

Noting that (2.8) holds for −τ ≤ t ≤ 0 as well, we find

eγ(ρ−δ(ρ))EV (x(ρ− δ(ρ))) ≤ eγρEV (x(ρ)).

Therefore, by (2.6),

EV (x(ρ− δ(ρ))) ≤ eγδ(ρ)EV (x(ρ)) ≤ eγτEV (x(ρ)) ≤ qEV (x(ρ)).

By condition (2.4),

ELV (x(ρ), x(ρ− δ(ρ))) ≤ −λEV (x(ρ)).

Recall the fact that γ < λ and EV (x(ρ)) > 0. Using the continuity of the

solution and of the functions V, δ etc., one then sees that for all sufficiently

small h > 0,

ELV (x(t), x(t− δ(t)) ≤ −γEV (x(t)) if ρ ≤ t ≤ ρ + h.

Now, by Ito’s formula, we can derive that, for all sufficiently small h > 0,

eγ(ρ+h)EV (x(ρ + h))− eγρEV (x(ρ))

=∫ ρ+h

ρ

eγt[ELV (x(t), x(t− δ(t)) + γEV (x(t))

]dt ≤ 0.

But this is in contradiction with (2.9) so (2.7) must hold. Finally, we obtain

from (2.7) and condition (2.3) that

E|x(t)|2 ≤ c2

c1||ξ||2L2 e−γt,

and the desired assertion (2.5) follows by letting γ → γ. The proof is com-

plete.

Corollary 2.2 Assume that there exists a symmetric positive-definite n×n-

matrix Q, and constants λ > 0, q > 1, such that

E(2XT Q[−BX + Ag(Y )] + trace[σT (X, Y )Qσ(X, Y )]

)≤ −λE(XT QX)

for all of those X, Y ∈ L2(Ω;Rn) satisfying E(Y T QY ) ≤ qE(XT QX).

Then, for all ξ ∈ L2F0

([−τ, 0];Rn), we have

E|x(t; ξ)|2 ≤ λmax(Q)λmin(Q)

||ξ||2L2 e−γt on t ≥ 0,

where γ = minλ, log(q)/τ. Here λmax(Q) and λmin(Q) denote the largest

and smallest eigenvalue of Q, respectively.

This corollary follows from Theorem 2.1 by using V (x) = xT Qx. We

now establish a theorem on the almost sure exponential stability of the

stochastic delay neural network (2.1)

Theorem 2.3 Assume that there is a positive constant K such that

trace[σT (x, y)σ(x, y)] ≤ K(|x|2 + |y|2) for all x, y ∈ Rn. (2.10)

Then (2.5) implies that

lim supt→∞

1t

log |x(t; ξ)| ≤ −γ

2a.s. (2.11)

In particular, if all of the assumptions of Theorem 2.1 are fulfilled and in

addition (2.10) holds, then the trivial solution of equation (2.1) is almost

surely exponentially stable.

This theorem can be proved in the same way as Lemma 4.6 of Liao and

Mao [8] so the details are omitted.

3. USEFULCOROLLARIES

We shall now employ the main results obtained in the previous section to

establish a number of useful corollaries.

Corollary 3.1 Let λ1 > λ2 > 0 and c2 ≥ c1 > 0. Assume that there exists

a function V ∈ C2(Rn;R+) such that

c1|x|2 ≤ V (x) ≤ c2|x|2 for all x ∈ Rn (3.1)

and

LV (x, y) ≤ −λ1V (x) + λ2V (y) for all (x, y) ∈ Rn ×Rn. (3.2)

Then, for all ξ ∈ L2F0

([−τ, 0];Rn), we have

E|x(t; ξ)|2 ≤ c2

c1||ξ||2L2 e−(λ1−qλ2)t on t ≥ 0, (3.3)

where q ∈ (1, λ1/λ2) is the unique root to the equation λ1− qλ2 = log(q)/τ .

Proof. For any pair of X, Y ∈ L2(Ω;Rn) with EV (Y ) ≤ qEV (X), we have

from condition (3.2) that

ELV (X, Y ) ≤ −λ1EV (x) + λ2EV (y) ≤ −(λ1 − qλ2)EV (X).

An application of Theorem 2.1 (with λ = λ1−qλ2) yields the desired assertion

(3.3).

Corollary 3.2 Assume that there exists a symmetric positive-definite n×n-

matrix Q, and constants λ1 > λ2 > 0, such that

2xT Q[−Bx+Ag(y)]+trace[σT (x, y)Qσ(x, y)] ≤ −λ1xT Qx+λ2y

T Qy (3.4)

for all (x, y) ∈ Rn ×Rn. Then, for all ξ ∈ L2F0

([−τ, 0];Rn),

E|x(t; ξ)|2 ≤ λmax(Q)λmin(Q)

||ξ||2L2 e−(λ1−qλ2)t on t ≥ 0, (3.5)

where q ∈ (1, λ1/λ2) is as defined in Corollary 3.1.

This corollary follows from Corollary 3.1 by using V (x) = xT Qx. So far

we have not used conditions (1.2) and (1.5) explicitly but we shall do so from

now on. To make the statements more clear, we will mention these conditions

when they are used explicitly although they are the standing hypotheses.

Corollary 3.3 Let (1.2) hold. Assume that there are nonnegative constants

νi, µi, 1 ≤ i ≤ n, such that

trace[σT (x, y)σ(x, y)

]≤

n∑i=1

(νix2i +µiy

2i ) for all (x, y) ∈ Rn×Rn. (3.6)

If, furthermore, there is a matrix (εij)n×n with all the elements positive such

that

λ1 := min1≤i≤n

(2bi − νi −

n∑j=1

|aij |βjεij

)> λ2 := max

1≤j≤n

(µj +

n∑i=1

|aij |βj

εij

), (3.7)

then the trivial solution of equation (2.1) is exponentially stable in mean

square and is also almost surely exponentially stable.

Proof. Let V (x) = |x|2. Then

LV (x, y) = 2xT [−Bx + Ag(y)] + trace[σT (x, y)σ(x, y)

]≤ −2

n∑i=1

bix2i + 2

n∑i,j=1

xiaijgj(yj) +n∑

i=1

(νix2i + µiy

2i )

≤ −n∑

i=1

(2bi − νi)x2i + 2

n∑i,j=1

|aij |βj |xi||yj |+n∑

j=1

µjy2j .

Noting that

2|xi||yj | ≤ εijx2i +

y2j

εij,

we obtain that

LV (x, y) ≤ −n∑

i=1

(2bi − νi)x2i +

n∑i,j=1

|aij |βj

(εijx

2i +

y2j

εij

)+

n∑j=1

µjy2j

≤ −n∑

i=1

(2bi − νi −

n∑j=1

|aij |βjεij

)x2

i +n∑

j=1

(µj +

n∑i=1

|aij |βj

εij

)y2

j

≤ −λ1|x|2 + λ2|y|2.

By Corollary 3.1, we see that for all ξ ∈ L2F0

([−τ, 0];Rn),

E|x(t; ξ)|2 ≤ ||ξ||2L2 e−(λ1−qλ2)t on t ≥ 0, (3.8)

where q ∈ (1, λ1/λ2) is the unique root to the equation λ1 − qλ2 = log(q)/τ .

That is, the trivial solution is exponentially stable in mean square. Finally,

almost sure exponential stability follows from Theorem 2.3, (3.8) and condi-

tion (3.6).

The use of this corollary depends on the choice of the matrix (εij)n×n,

which should be selected based on the structure of the given stochastic delay

neural network. Although it can give better conditions on stability, it is

somewhat inconvenient in application when the dimension n of the network

is large. We shall now establish a number of easier-to-use criteria.

Corollary 3.4 Let (1.2), (1.5) and (3.6) hold. Assume that the network is

symmetric in the sense that

|aij | = |aji| for all 1 ≤ i, j ≤ n. (3.9)

If

max0<ε<2

[min

1≤i≤n

[(2− ε)bi − νi

]− max

1≤j≤n

(µj +

bjβ2j

ε

) ]> 0, (3.10)

then the trivial solution of equation (2.1) is exponentially stable in mean

square and is also almost surely exponentially stable. If in addition νi = ν

and µi = µ for all 1 ≤ i ≤ n, then (3.10) reduces to

κ >12

(ζ + ν + µ +

√ζ[ζ + 2(ν + µ)]

), (3.11)

where

κ = min1≤i≤n

bi and ζ = max1≤j≤n

bjβ2j .

Proof. By (3.10), one can find an ε ∈ (0, 2) for which

min1≤i≤n

[(2− ε)bi − νi

]> max

1≤j≤n

(µj +

bjβ2j

ε

). (3.12)

Set the elements of the matrix (εij) by εij = ε/βj . Then, using (1.5) and

(3.9), we have

λ1 := min1≤i≤n

(2bi − νi −

n∑j=1

|aij |βjεij

)

= min1≤i≤n

(2bi − νi − ε

n∑j=1

|aij |)

= min1≤i≤n

[(2− ε)bi − vi

],

and

λ2 := max1≤j≤n

(µj +

n∑i=1

|aij |βj

εij

)= max

1≤j≤n

(µj +

n∑i=1

|aij |β2

j

ε

)= max

1≤j≤n

(µj +

bjβ2j

ε

).

By (3.12), we see that λ1 > λ2. So the stability assertions follow from

Corollary 3.3. In the case νi = ν and µi = µ, we have

max0<ε<2

[min

1≤i≤n

[(2− ε)bi − νi

]− max

1≤j≤n

(µj +

bjβ2j

ε

) ]= max

0<ε<2

((2− ε)κ− ν − µ− ζ

ε

)= 2κ− ν − µ− min

0<ε<2

(εκ +

ζ

ε

)= 2κ− ν − µ− 2

√κζ,

where we have used the elementary property that εκ + ζ/ε reaches its mini-

mum 2√

κζ at ε =√

ζ/κ, bearing in mind that√

ζ/κ ∈ (0, 1) by condition

(3.11). So (3.10) becomes

2κ− ν − µ− 2√

κζ > 0.

But this is equivalent to

√κ >

12

[√ζ +

√ζ + 2(ν + µ)

]and (3.11) follows. The proof is complete.

Corollary 3.5 Let (1.2) and (3.6) hold. If

min1≤i≤n

(2bi − νi) > minε>0

max1≤j≤n

(ε + µj +

1ε||A||2β2

j

), (3.13)

then the trivial solution of equation (2.1) is exponentially stable in mean

square and is also almost surely exponentially stable. If in addition µj = µ

for 1 ≤ j ≤ n, then (3.13) reduces to

min1≤i≤n

(2bi − νi) > µ + 2||A|| max1≤j≤n

βj . (3.14)

Proof. By (3.13), one can find an ε > 0 for which λ1 > λ2, where

λ1 := min1≤i≤n

(2bi − νi)− ε

and

λ2 := max1≤j≤n

(µj +

1ε||A||2β2

j

).

Let V (x) = |x|2. Then

LV (x, y) = 2xT [−Bx + Ag(y)] + trace[σT (x, y)σ(x, y)

]≤ −2

n∑i=1

bix2i + 2|x| ||A|| |g(y)|+

n∑i=1

(νix2i + µiy

2i )

≤ −n∑

i=1

(2bi − νi)x2i + ε|x|2 +

1ε||A|| |g(y)|2 +

n∑j=1

µjy2j .

≤ −n∑

i=1

(2bi − νi − ε)x2i +

n∑j=1

(µj +

1ε||A||β2

j

)y2

j

≤ −λ1|x|2 + λ2|y|2. (3.15)

Therefore mean square exponential stability follows from Corollary 3.1, while

almost sure exponential stability follows from Theorem 2.3. In the case

µj = µ for all 1 ≤ j ≤ n, we have

minε>0

max1≤j≤n

(ε + µj +

1ε||A||2β2

j

)= µ + min

ε>0

[ε +

1e||A||2 max

1≤j≤nβ2

j

]= µ + 2||A|| max

1≤j≤nβj .

So (3.13) becomes

min1≤i≤n

(2bi − νi) > µ + 2||A|| max1≤j≤n

βj

as required. The proof is complete.

Before a discussion of examples, let us establish one more corollary.

Corollary 3.6 Let (1.2) hold. Assume that there is a symmetric nonnegative

definite n× n matrix H, and nonnegative constants µi, 1 ≤ i ≤ n, such that

trace[σT (x, y)σ(x, y)

]≤ xT Hx +

n∑i=1

µiy2i for all (x, y) ∈ Rn ×Rn.

(3.16)

If there is an ε > 0 such that

λmin(2B −H − εAAT ) > max1≤i≤n

(β2i

ε+ µi

), (3.17)

then the trivial solution of equation (2.1) is exponentially stable in mean

square and is also almost surely exponentially stable.

Proof. Let V (x) = |x|2. Then

LV (x, y) = 2xT [−Bx + Ag(y)] + trace[σT (x, y)σ(x, y)

]≤ −2xT Bx + 2xT Ag(y) + xT Hx +

n∑i=1

µiy2i

≤ −xT (2B −H)x + εxT AAT x +1ε|g(y)|2 +

n∑i=1

µiy2i

≤ −xT (2B −H − εAAT )x +n∑

i=1

(β2i

ε+ µi

)y2

i

≤ −λmin(2B −H − εAAT )|x|2 + max1≤i≤n

(β2i

ε+ µi

)|y|2.

Therefore the conclusions follow from Corollary 3.1 and Theorem 2.3.

4. EXAMPLES

In this section we shall discuss a number of examples in order to illustrate

the theory. We shall not mention the initial data since they always belong

to L2F0

([−τ, 0];Rn).

Example 4.1 Let us first consider a 3-dimensional symmetric stochastic

delay neural network

dx(t) = [−Bx(t) + Ag(x(t− δ(t)))]dt + σ(x(t), x(t− δ(t)))dw(t). (4.1)

Here we choose

B = diag.(4, 4, 4), A =

1 2 12 1 11 1 2

gi(xi) = (0.5xi ∧ 1) ∨ (−1), g(x) = (g1(x1), g2(x2), g3(x3))T .

Moreover, σ : R3 ×R3 → R3×m satisfies

trace[σT (s, y)σ(x, y)] ≤ ν|x|2 + µ|y|2 (4.2)

for some nonnegative constants ν and µ. Note that (1.2) and (1.5) are

satisfied, and particularly, that b1 = b2 = b3 = 4 and β1 = β2 = β3 = 0.5. In

this case, criterion (3.11) becomes

4 >12

(1 + ν + µ +

√1 + 2(ν + µ)

),

which yields

ν + µ < 4. (4.3)

Therefore, by Corollary 3.4, we see that (4.3) is a sufficient condition for

mean square and a.s. exponential stability. On the other hand, it is easy to

show that ||A|| = 4. Thus criterion (3.14) becomes

2× 4− ν > µ + 2× 4× 0.5

which yields (4.3) too. In other words, Corollary 3.5 gives the same stability

condition as Corollary 3.4 in this example.

Example 4.2 Let us still consider network (4.1) but with different A and

B as follows

B = diag.(2, 3, 4), A =

0 1 11 1 11 1 2

.

Besides, condition (4.2) is replaced with

trace[σT (s, y)σ(x, y)] ≤ xT Hx + µ|y|2, (4.4)

where µ ≥ 0 and H is a symmetric nonnegative-definite 3× 3 matrix. Note

that the network is still symmetric and β1 = β2 = β3 = 0.5, but b1 =

2, b2 = 3, b3 = 4. It is easy to show that ||A|| = 3.215. If we apply

Corollary 3.5 we can show that a sufficient condition for exponential stability

is ||H|| + µ < 0.785. On the other hand, by Corollary 3.4, we can obtain a

better condition ||H||+ µ < 4− 2√

2 = 1.715. However, we shall now apply

Corollary 3.6 to obtain an even better condition. According to Corollary 3.6,

we need to seek an ε > 0 such that

λmin(2B −H − εAAT ) >0.25ε

+ µ. (4.5)

Since

λmin(2B −H − εAAT ) ≥ λmin(2B − εAAT )− ||H||,

it is enough to find an ε > 0 for

λmin(2B − εAAT )− ||H|| > 0.25ε

+ µ. (4.6)

Say we choose ε = 0.216 and compute λmin(2B − εAAT ) = 3.25792. Substi-

tuting these into (4.6) yields the stability condition

||H||+ µ < 2.1005, (4.7)

which improves the above conditon by 22%.

Let us further specify H as

H =

0 0 00 1.25 1.250 1.25 2.5

.

Compute ||H|| = 3.27254 and we therefore see condition (4.7) is not satisfied.

However we can still use (4.5) to show exponential stability if µ is sufficiently

small. For example, choose ε = 0.16 and compute λmin(2B −H − εAAT ) =

2.28382. Substituting these into (4.5) we see that a sufficient condition for

the stability in this case is now

µ < 0.72132. (4.8)

Note that ||H||+ µ < 3.99386 which improves (4.7) greatly, but of course H

is known in this case.

Example 4.3 Consider a 2-dimensional stochastic delay neural network

dx(t) = [−Bx(t) + Ag(x(t− δ(t)))]dt + σ(x(t), x(t− δ(t)))dw(t). (4.9)

Here w(t) is an m-dimensional Brownian motion,

B =(

5 00 3

)A =

(1 42 1

)gi(xi) =

0.5(exi − e−xi)exi + e−xi

, g(x) = (g1(x1), g2(x2))T

and, moreover, σ : R2 ×R2 → R2×m satisfies

trace[σT (x, y)σ(x, y)] ≤ ν1x21 + ν2x

22 + 3y2

1 + y22 .

We shall apply our results to obtain restrictions on parameters ν1 and ν2 in

order to have required exponential stability. First of all, note that β1 = β2 =

0.5. By Corollary 3.6, we need to seek an ε > 0 such that

λmin(2B −H − εAT A) > 3 +0.25ε

, (4.10)

where H = diag.(ν1, ν2). It is sufficient to have

λmin(2B − εAT A)− ν1 ∨ ν2 > 3 +0.25ε

. (4.11)

Numerically we find the optimal ε ≈ 0.155 and compute the corresponding

eigenvalue λmin(2B − εAT A) ≈ 4.87733. Hence, (4.11) yields the stability

condition

ν1 ∨ ν2 < 0.2644. (4.12)

We now demonstrate how to apply Corollary 3.3 to improve this result.

According to this corollary, we need to seek four positive numbers εij , 1 ≤

i, j ≤ 2, such that

min

10− ν1 − 0.5(ε11 + 4ε12), 6− ν2 − 0.5(2ε21 + ε22)

> max

3 + 0.5( 1

ε11+

4ε12

), 1 + 0.5

( 2ε21

+1

ε22

). (4.13)

By choosing ε11 = ε22 = 1, (4.13) reduces to

min

9.5− ν1 − 2ε12, 5.5− ν2 − ε21

> max

3.5 +

2ε12

, 1.5 +1

ε21

. (4.14)

Now look for ε12 and ε21 such that

3.5 +2

ε12= 1.5 +

1ε21

, i.e. ε21 =ε12

2(ε12 + 1).

By setting ε12 = ε, (4.14) becomes

min

9.5− ν1 − 2ε, 5.5− ν2 −ε

2(ε + 1)

> 3.5 +

2ε. (4.15)

If we do not know which of ν1 and ν2 is larger, it would be better to select

ε such that

9.5− 2ε = 5.5− ε

2(ε + 1)

which yields ε = 2.17116. Substituting this into (4.15) gives the stability

condition

ν1 ∨ ν2 < 0.7364, (4.16)

which improves condition (4.12) by 178%. Alternatively, if we know that ν1

would be larger than ν2, then we can choose, for example, ε = 2 to obtain

the stability condition

ν1 < 1 and ν2 <23. (4.17)

which is better than (4.12) again. Clearly, we may select different εij to give

the other alternative conditions on stability.

Of course, one may argue that (4.11) is stronger than (4.10) so condition

(4.12) is conservative. However, if letting ν1 = ν2 = 0.7364, or ν1 = 1 and

n2 = 2/3, we can show numerically that

maxε>0

[λmin(2B −H − εAT A)− 3− 0.25

ε

]< 0, (4.18)

which means that (4.10) will never hold for any ε > 0. In other words, Corol-

lary 3.6 will not give any better result than Corollary 3.3 in this particular

example. However, as demonstrated, it is not easy to select εij even in this

example of dimension 2, and it could be very difficult indeed in the case of

higher dimensions.

To conclude this section let us stress that the examples above show the

advantage and disadvantage of different corollaries. In theory, they comple-

ment each other. Therefore, in application, one should use one or the other

based on the structure of the given network in order to obtain better stability

conditions.

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