Global asymptotic stability of Cohen - Grossberg neural networks of neutral type
Sufficient conditions for existence and global asymptotic stability of a unique equilibrium point of a Cohen – Grossberg neural network of neutral type are obtained. An example is given.
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irk-123456789-1746882021-01-28T01:27:13Z Global asymptotic stability of Cohen - Grossberg neural networks of neutral type Akca, H. Covachev, V. Covacheva, Z. Sufficient conditions for existence and global asymptotic stability of a unique equilibrium point of a Cohen – Grossberg neural network of neutral type are obtained. An example is given. Отримано достатнi умови iснування та глобальної стiйкостi єдиної точки рiвноваги для нейронної мережi Коена – Гроссберга нейтрального типу. 2014 Article Global asymptotic stability of Cohen - Grossberg neural networks of neutral type / H. Akca, V. Covachev, Z. Covacheva // Нелінійні коливання. — 2014. — Т. 17, № 1. — С. 3-15. — Бібліогр.: 18 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/174688 517.9 en Нелінійні коливання Інститут математики НАН України |
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Sufficient conditions for existence and global asymptotic stability of a unique equilibrium point of a Cohen – Grossberg neural network of neutral type are obtained. An example is given. |
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Article |
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Akca, H. Covachev, V. Covacheva, Z. |
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Akca, H. Covachev, V. Covacheva, Z. Global asymptotic stability of Cohen - Grossberg neural networks of neutral type Нелінійні коливання |
| author_facet |
Akca, H. Covachev, V. Covacheva, Z. |
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Akca, H. |
| title |
Global asymptotic stability of Cohen - Grossberg neural networks of neutral type |
| title_short |
Global asymptotic stability of Cohen - Grossberg neural networks of neutral type |
| title_full |
Global asymptotic stability of Cohen - Grossberg neural networks of neutral type |
| title_fullStr |
Global asymptotic stability of Cohen - Grossberg neural networks of neutral type |
| title_full_unstemmed |
Global asymptotic stability of Cohen - Grossberg neural networks of neutral type |
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global asymptotic stability of cohen - grossberg neural networks of neutral type |
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Інститут математики НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/174688 |
| citation_txt |
Global asymptotic stability of Cohen - Grossberg neural networks of neutral type / H. Akca, V. Covachev, Z. Covacheva // Нелінійні коливання. — 2014. — Т. 17, № 1. — С. 3-15. — Бібліогр.: 18 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
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| first_indexed |
2025-07-15T11:44:19Z |
| last_indexed |
2025-07-15T11:44:19Z |
| _version_ |
1837713179489075200 |
| fulltext |
UDC 517.9
GLOBAL ASYMPTOTIC STABILITY OF COHEN – GROSSBERG NEURAL
NETWORKS OF NEUTRAL TYPE*
ГЛОБАЛЬНА АСИМПТОТИЧНА СТIЙКIСТЬ НЕЙРОННИХ МЕРЕЖ
КОЕНА – ГРОССБЕРГА НЕЙТРАЛЬНОГО ТИПУ
H. Akça
College Arts and Sci. Abu Dhabi Univ.
P. O. Box 59911, Adu Dhabi, UAE
e-mail: Haydar.Akca@adu.ac.ae
V. Covachev
Inst. Math. Bulgar. Acad. Sci., Sofia, Bulgaria
e-mail: vcovachev@hotmail.com
Z. Covacheva
Middle East College, Muscat, Oman
and Higher College Telecommunications and Post, Sofia, Bulgaria
e-mail: zkovacheva@hotmail.com
Sufficient conditions for existence and global asymptotic stability of a unique equilibrium point of a
Cohen – Grossberg neural network of neutral type are obtained. An example is given.
Отримано достатнi умови iснування та глобальної стiйкостi єдиної точки рiвноваги для ней-
ронної мережi Коена – Гроссберга нейтрального типу.
1. Introduction. An artificial neural network is an information processing paradigm that is
inspired by the way biological nervous systems, such as the brain, process information. The
key element of this paradigm is the novel structure of the information processing system. It is
composed of a large number of highly interconnected processing elements (neurons) working
in unison to solve specific problems. Although the initial intent of artificial neural networks
was to explore and reproduce human information processing tasks such as speech, vision, and
knowledge processing, artificial neural networks also demonstrated their superior capability
for classification and function approximation problems. This has great potential for solving
complex problems such as systems control, data compression, optimization problems, pattern
recognition, and system identification.
Cohen – Grossberg neural network [10] and its various generalizations with or without trans-
mission delays and impulsive state displacements have been a subject of intense investigation
recently [3, 6, 7, 13, 16, 17]. In a Cohen – Grossberg neural network model, the feedback terms
consist of amplification and stabilizing functions which are generally nonlinear. These terms
provide a model with a special kind of generalization wherein many neural network models that
are capable for content addressable memory such as additive neural networks, cellular neural
∗ The article is based on a report at the International Mathematical Conference DIF-2013.
c© H. Akça, V. Covachev, Z. Covacheva, 2014
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1 3
4 H. AKÇA, V. COVACHEV, Z. COVACHEVA
networks and bidirectional associative memory networks and also biological models such as
Lotka – Volterra models of population dynamics are included as special cases.
In contrast to retarded systems, in neutral systems time delays appear explicitly in the state
velocity vector. Neutral systems can be applied to describe more complicated nonlinear engi-
neering and bioscience models, including those describing chemical reactors, transmission lines,
partial element equivalent circuits in very large-scale integrated systems, and Lotka – Volterra
systems [18, 14, 4, 1, 2, 9, 15]. Neural networks can be implemented using very large-scale
integrated circuits. Therefore, both retarded-type delays and neutral type delays are inherent in
the dynamics of neural networks.
In the present paper we consider a Cohen – Grossberg neural network of neutral type more
general than in [8]. A discrete-time analogue of this system provided with impulse conditions
was considered in our previous paper [11]. Sufficient conditions for global asymptotic stabi-
lity of the unique equilibrium point of the system are obtained by exploiting an appropriate
Lyapunov functional. The conditions obtained are much more precise than in [8]. An example
is given.
2. Preliminaries. We consider a Cohen – Grossberg neural network of neutral type consisting
of m ≥ 2 elementary processing units (or neurons) whose state variables xi (i = 1,m which
henceforth will stand for i = 1, 2, . . . ,m) are governed by the system
ẋi(t) +
m∑
j=1
eij ẋj(t− τj) = ai(xi(t))
−bi(xi(t)) +
m∑
j=1
cijfj(xj(t)) +
m∑
j=1
dijgj(xj(t− τj)) + Ii
,
(1)
i = 1,m, t > t0 = 0,
with initial values prescribed by continuous functions xi(s) = φi(s) for s ∈ [−τ, 0], τ =
= maxj=1,m{τj}. In (1), ai(xi) denotes an amplification function; bi(xi) denotes an appropri-
ate function which supports the stabilizing (or negative) feedback term −ai(xi)bi(xi) of the
unit i; fj(xj), gj(xj) denote activation functions; the parameters cij , dij are real numbers that
represent the weights (or strengths) of the synaptic connections between the jth unit and the
ith unit, respectively without and with time delays τj ; the real numbers eij show how the state
velocities of the neurons are delay feed-forward connected in the network; the real constant Ii
represents an input signal introduced from outside the network to the ith unit.
Let E be the unit (m ×m)-matrix. Denote by E and |E| the (m ×m)-matrices with entries
eij and |eij |, respectively.
Definition 1 [5]. A real matrix A = (aij)m×m is said to be an M -matrix if aij ≤ 0 for
i, j = 1,m, i 6= j, and all successive principal minors of A are positive.
The assumptions that accompany the neural network (1) are given as follows:
A1. The amplification functions ai : R → R+ are continuous and bounded in the sense that
0 < ai ≤ ai(x) ≤ ai for x ∈ R, i = 1,m,
for some constants ai, ai.
A2.The stabilizing functions bi : R → R are Lipschitz continuous and monotone increasing,
namely,
0 < bi ≤
bi(x)− bi(y)
x− y
≤ bi for x 6= y, x, y ∈ R, i = 1,m,
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
GLOBAL ASYMPTOTIC STABILITY OF COHEN – GROSSBERG NEURAL NETWORKS . . . 5
for some constants bi, bi.
A3. The activation functions fj , gj : R → R are Lipschitz continuous in the sense of
sup
x 6=y
∣∣∣∣fj(x)− fj(y)
x− y
∣∣∣∣ = Fj , sup
x 6=y
∣∣∣∣gj(x)− gj(y)
x− y
∣∣∣∣ = Gj
for x, y ∈ R, j = 1,m, where Fj , Gj denote positive constants.
A4. ‖E‖ < 1, where ‖ · ‖ is the spectral matrix norm, and E − |E| is an M -matrix.
The “stability condition” ‖E‖ < 1 guarantees the existence and uniqueness of the solution
of the Cauchy problem. Since E − |E| is an M -matrix, it is nonsingular and its inverse has
nonnegative entries only.
Under these assumptions and the given initial conditions, there is a unique solution of
system (1). The solution is a vector x(t) = (x1(t), x2(t), . . . , xm(t))T in which xi(t) are conti-
nuously differentiable for t ∈ (0, β), where β is some positive number, possibly∞, An equili-
brium point of system (1) is denoted by x∗ = (x∗1, x
∗
2, . . . , x
∗
m)T where the components x∗i are
governed by the algebraic system
bi(x
∗
i ) =
m∑
j=1
cijfj(x
∗
j ) +
m∑
j=1
dijgj(x
∗
j ) + Ii, i = 1,m. (2)
Definition 2. The equilibrium point x∗ = (x∗1, x
∗
2, . . . , x
∗
m)T of system (1) is said to be globally
asymptotically stable if any other solution x(t) = (x1(t), x2(t), . . . , xm(t))T of system (1) is defi-
ned for all t > 0 and satisfies
lim
t→∞
x(t) = x∗.
3. Existence and global asymptotic stability of an equilibrium point. Sufficient conditions
for existence and uniqueness of the solution x∗ of the algebraic system (2) are given by the
following theorem.
Theorem 1 ([11], Theorem 4.2). Let the assumptions A2, A3 hold. Suppose, further, that the
following inequalities are valid:
bi −
1
2
m∑
j=1
(|cij |Fj + |cji|Fi)−
1
2
m∑
j=1
(|dij |Gj + |dji|Gi) > 0, i = 1,m. (3)
Then system (1) has a unique equilibrium point x∗ = (x∗1, x
∗
2, . . . , x
∗
m)T .
Further on we give sufficient conditions for the global asymptotic stability of the equilibrium
point x∗ of system (1).
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
6 H. AKÇA, V. COVACHEV, Z. COVACHEVA
Theorem 2. Let the assumptions A1 –A4 hold. Suppose, further, that the inequalities
aibi −
1
2
m∑
j=1
(ai|cij |Fj + aj |cji|Fi)−
1
2
m∑
j=1
(ai|dij |Gj + aj |dji|Gi)−
− 1
2
m∑
j=1
(
aibi|eij |+ ajbj |eji|
)
− 1
2
m∑
j=1
aj
m∑
k=1
(|cji| |ejk|Fi + |cjk| |eji|Fk)−
− 1
2
m∑
j=1
aj
m∑
k=1
(|dji| |ejk|Gi + |djk| |eji|Gk) > 0, i = 1,m, (4)
are valid and system (1) has an equilibrium point x∗ = (x∗1, x
∗
2, . . . , x
∗
m)T whose components
satisfy (2). Then the equilibrium point x∗ is globally asymptotically stable.
Remark 1. Inequalities (3) can be deduced from (4) for ai = ai = 1, eij = 0 for i, j = 1,m.
However, in general inequalities (4) do not imply (3).
Remark 2. Inequalities (4) were given in [11] (Theorem 4.3) as a part of the sufficient condi-
tions for the global asymptotic stability of the equilibrium point of the discrete-time counterpart
of system (1) provided with impulsive conditions, for small values of the discretization step h.
Remark 3. In [8] it is assumed that gj = fj , the functions bi(xi) and b−1
i (xi) are continuosly
differentiable, and b′i(xi) are bounded both below and above by positive constants. Instead of
the m inequalities (4) a single inequality is presented, which in our notation can be written as
min
i=1,m
(aibi)− max
i=1,m
(aibi)‖E‖ − max
i=1,m
ai
(
max
i=1,m
Fi‖C‖+ max
i=1,m
Gi‖D‖
)
(1 + ‖E‖) > 0, (5)
where C and D are (m×m)-matrices with entries cij and dij , respectively.
Though condition (5) seems much simpler than (4), in our opinion it is much less precise si-
nce the individual lower and upper bounds, Lipschitz constants, and matrix entries are replaced
by their minima or maxima, and matrix norms. Below we shall give an example of a system sati-
sfying conditions (4) but not (5).
Proof. Upon introducing the translations
ui(t) = xi(t)− x∗i , ϕi(s) = φi(s)− x∗i
we derive the system
u̇i(t) +
m∑
j=1
eij u̇j(t− τj) = ãi(ui(t))
−b̃i(ui(t)) +
m∑
j=1
cij f̃j(uj(t))+
+
m∑
j=1
dij g̃j(uj(t− τj))
, t > t0 = 0, (6)
ui(s) = ϕi(s), s ∈ [−τ, 0], i = 1,m,
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
GLOBAL ASYMPTOTIC STABILITY OF COHEN – GROSSBERG NEURAL NETWORKS . . . 7
where
ãi(ui) = ai(ui + x∗i ), b̃i(ui) = bi(ui + x∗i )− bi(x∗i ),
f̃j(uj) = fj(uj + x∗j )− fj(x∗j ), g̃j(uj) = gj(uj + x∗j )− gj(x∗j ).
This system inherits the assumptions A1 –A4 given before. It suffices to examine the stability
characteristics of the trivial equilibrium point u∗ = 0 of system (6).
We define a Lyapunov functional V (t) by
V (t) =
1
2
m∑
i=1
ui(t) +
m∑
j=1
eijuj(t− τj)
2
+ ωi
t∫
t−τi
u2
i (s) ds
,
where the positive constants ωi, i = 1,m,will be determined later. First we notice that the value
V (0) =
1
2
m∑
i=1
ϕi(0) +
m∑
j=1
eijϕj(−τj)
2
+ ωi
0∫
−τi
ϕ2
i (s) ds
is completely determined from the initial values of the system. Then, calculating the rate of
change of V (t) along the solutions of (6), we successively find
V̇ (t) =
m∑
i=1
ui(t) +
m∑
j=1
eijuj(t− τj)
u̇i(t) +
m∑
j=1
eij u̇j(t− τj)
+
ωi
2
(u2
i (t)− u2
i (t− τi))
=
=
m∑
i=1
ui(t) +
m∑
j=1
eijuj(t− τj)
ãi(ui(t))
−b̃i(ui(t)) +
m∑
j=1
cij f̃j(uj(t)) +
+
m∑
j=1
dij g̃j(uj(t− τj))
+
ωi
2
(
u2
i (t)− u2
i (t− τi)
)=
=
m∑
i=1
−ãi(ui(t))b̃i(ui(t))ui(t) + ãi(ui(t))ui(t)
m∑
j=1
cij f̃j(uj(t)) +
m∑
j=1
dij g̃j(uj(t− τj))
+
+ ãi(ui(t))
m∑
j=1
eijuj(t− τj)
−b̃i(ui(t)) +
m∑
j=1
cij f̃j(uj(t)) +
+
m∑
j=1
dij g̃j(uj(t− τj))
+
ωi
2
(
u2
i (t)− u2
i (t− τi)
) ≤
≤
m∑
i=1
−aibiu2
i (t) + ai|ui(t)|
m∑
j=1
|cij |Fj |uj(t)|+
m∑
j=1
|dij |Gj |uj(t− τj)|
+
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
8 H. AKÇA, V. COVACHEV, Z. COVACHEVA
+ ai
m∑
j=1
|eij | |uj(t− τj)|
bi |ui(t)|+ m∑
j=1
|cij |Fj |uj(t)|+
m∑
j=1
|dij |Gj |uj(t− τj)|
+
+
ωi
2
(
u2
i (t)− u2
i (t− τi)
) ≤
≤
m∑
i=1
−aibiu2
i (t) +
ai
2
m∑
j=1
|cij |Fj
(
u2
i (t) + u2
j (t)
)
+
ai
2
m∑
j=1
|dij |Gj
(
u2
i (t) + u2
j (t− τj)
)
+
+
aibi
2
m∑
j=1
|eij |
(
u2
i (t) + u2
j (t− τj)
)
+
ai
2
m∑
j=1
m∑
k=1
|eij | |cik|Fk
(
u2
k(t) + u2
j (t− τj)
)
+
+
ai
2
m∑
j=1
m∑
k=1
|eij | |dik|Gk
(
u2
j (t− τj) + u2
k(t− τk)
)
+
ωi
2
(
u2
i (t)− u2
i (t− τi)
) =
=
m∑
i=1
−
aibi − 1
2
ai m∑
j=1
|cij |Fj + Fi
m∑
j=1
|cji|aj
− ai
2
m∑
j=1
|dij |Gj −
− aibi
2
m∑
j=1
|eij | −
Fi
2
m∑
j=1
m∑
k=1
|cki| |ekj |ak −
ωi
2
u2
i (t)+
+
1
2
Gi m∑
j=1
|dji|aj +
m∑
j=1
|eji|ajbj +
m∑
j=1
m∑
k=1
|eji| |cjk|ajFk +
+
m∑
j=1
m∑
k=1
(|eji| |djk|ajGk + |ekj | |dki|akGi)− ωi
u2
i (t− τi)
.
Choose
ωi = Gi
m∑
j=1
|dji|aj +
m∑
j=1
|eji|ajbj +
m∑
j=1
m∑
k=1
|eji| |cjk|ajFk+
+
m∑
j=1
m∑
k=1
(|eji| |djk|ajGk + |ekj | |dki|akGi) > 0,
then after some simplifications we obtain
V̇ (t) ≤ −
m∑
i=1
aibi − 1
2
m∑
j=1
(ai|cij |Fj + aj |cji|Fi) −
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
GLOBAL ASYMPTOTIC STABILITY OF COHEN – GROSSBERG NEURAL NETWORKS . . . 9
− 1
2
m∑
j=1
(ai|dij |Gj + aj |dji|Gi)−
1
2
m∑
j=1
(
aibi|eij |+ ajbj |eji|
)
−
− 1
2
m∑
j=1
aj
m∑
k=1
(|cji| |ejk|Fi + |cjk| |eji|Fk)−
1
2
m∑
j=1
aj
m∑
k=1
(|dji| |ejk|Gi + |djk| |eji|Gk)
u2
i (t).
According to inequalities (4) there exists µ > 0 such that
µ = min
i=1,m
aibi − 1
2
m∑
j=1
(ai|cij |Fj + aj |cji|Fi)−
1
2
m∑
j=1
(ai|dij |Gj + aj |dji|Gi) −
− 1
2
m∑
j=1
(
aibi|eij |+ ajbj |eji|
)
− 1
2
m∑
j=1
aj
m∑
k=1
(|cji| |ejk|Fi + |cjk| |eji|Fk)−
− 1
2
m∑
j=1
aj
m∑
k=1
(|dji| |ejk|Gi + |djk| |eji|Gk)
,
then
V̇ (t) ≤ −µ‖u(t)‖2, t > 0, (7)
where ‖v‖ =
(∑m
i=1 v
2
i
)1/2 is the Euclidean norm of the vector v = (v1, v2, . . . , vm)T ∈ Rm.
Inequality (7) shows that for any solution u(t) of system (6) the function V (t) is monotone
decreasing and it is bounded below by 0. Thus there exists the limit L = limt→∞ V (t) ≥ 0.
Let us integrate inequality (7) from 0 to t,
V (t)− V (0) ≤ −µ
t∫
0
‖u(s)‖2 ds
for all t > 0, that is,
t∫
0
‖u(s)‖2 ds ≤ (V (0)− V (t))/µ.
The last inequality and L = limt→∞ V (t) ≥ 0 show that
∞∫
0
‖u(t)‖2 dt < ∞. (8)
Below we show that the zero solution of system (6) is stable and (8) implies limt→∞ ‖u(t)‖ = 0,
that is, limt→∞ ‖x(t) − x∗‖ = 0. This means that the equilibrium point x∗ of system (1) is
globally asymptotically stable.
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
10 H. AKÇA, V. COVACHEV, Z. COVACHEVA
We complete the proof by arguments using fragments from the proofs of Theorems 1.1, 1.3
and 1.4 in [12] (Chapter 8). In the sequel for a vector v = (v1, v2, . . . , vm)T ∈ Rm we shall also
use the norm
|v| = max
i=1,m
|vi|.
First we shall prove that for any ε > 0 there exists δ1 > 0 such that if∣∣∣∣∣ui(t) +
m∑
i=1
eijuj(t− τij)
∣∣∣∣∣ ≤ δ1 for t ≥ 0, i = 1,m, and sup
s∈[−τ,0]
|ϕ(s)| ≤ δ1,
then |u(t)| ≤ ε for t ≥ 0.
Let T be an arbitrary positive number. For 0 ≤ t ≤ T we have
|ui(t)| ≤
∣∣∣∣∣ui(t) +
m∑
i=1
eijuj(t− τij)
∣∣∣∣∣+
∣∣∣∣∣
m∑
i=1
eijuj(t− τij)
∣∣∣∣∣ ≤
≤ δ1 +
m∑
i=1
|eij | |uj(t− τij)| ≤
≤ δ1 +
m∑
i=1
|eij | sup
−τ≤t≤T
|uj(t)| ≤
≤ δ1 +
m∑
i=1
|eij |
(
sup
0≤t≤T
|uj(t)|+ sup
−τ≤s≤0
|ϕj(s)|
)
≤
≤
m∑
i=1
|eij | sup
0≤t≤T
|uj(t)|+ δ1
(
1 +
m∑
i=1
|eij |
)
,
thus
sup
0≤t≤T
|ui(t)| ≤
m∑
i=1
|eij | sup
0≤t≤T
|uj(t)|+ δ1
(
1 +
m∑
i=1
|eij |
)
or
sup
0≤t≤T
|ui(t)| −
m∑
i=1
|eij | sup
0≤t≤T
|uj(t)| ≤ δ1
(
1 +
m∑
i=1
|eij |
)
for i = 1,m.
If we introduce the vectors
U(T ) =
(
sup
0≤t≤T
|u1(t)|, sup
0≤t≤T
|u2(t)|, . . . , sup
0≤t≤T
|um(t)|
)T
and e = (1, 1, . . . , 1)T ,
we can write the last inequalities in a matrix form as
(E − |E|)U(T ) ≤ δ1(E + |E|)e,
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GLOBAL ASYMPTOTIC STABILITY OF COHEN – GROSSBERG NEURAL NETWORKS . . . 11
meaning inequalities between the respective components of the vectors. Since, by condition A4,
E − |E| is an M -matrix, we obtain
U(T ) ≤ δ1(E − |E|)−1(E + |E|)e.
We have
sup
0≤t≤T
|u(t)| = sup
0≤t≤T
max
i=1,m
|ui(t)| = max
i=1,m
sup
0≤t≤T
|ui(t)| = |U(T )| ≤ δ1|(E − |E|)−1(E + |E|)e|.
If we choose δ1 > 0 so small that δ1|(E − |E|)−1(E + |E|)e| < ε, then |u(t)| ≤ ε for 0 ≤ t ≤ T,
where T was an arbitrary positive number. Thus, |u(t)| ≤ ε for t ≥ 0.
Next we shall show that the zero solution of system (6) is stable, that is, for any ε > 0 there
exists δ > 0 such that if sups∈[−τ,0] |ϕ(s)| ≤ δ, then |u(t)| ≤ ε for t ≥ 0. For any t ≥ 0 we get
1
2
m∑
i=1
ui(t) +
m∑
j=1
eijuj(t− τj)
2
≤ V (t) ≤ V (0) =
=
1
2
m∑
i=1
ϕi(0) +
m∑
j=1
eijϕj(−τj)
2
+ ωi
0∫
−τi
ϕ2
i (s) ds
≤
≤ δ2
2
m∑
i=1
1 +
m∑
j=1
|eij |
2
+ ωiτi
.
If we choose δ ∈ (0, δ1) so small that
δ2
m∑
i=1
1 +
m∑
j=1
|eij |
2
+ ωiτi
≤ δ2
1 ,
then
m∑
i=1
ui(t) +
m∑
j=1
eijuj(t− τj)
2
≤ δ2
1 ,
which implies that ∣∣∣∣∣ui(t) +
m∑
i=1
eijuj(t− τij)
∣∣∣∣∣ ≤ δ1 for t ≥ 0, i = 1,m,
and, consequently, |u(t)| ≤ ε for t ≥ 0.
Because of stability of the zero solution of system (6) we can assume that |u(t)| ≤ h for
some positive constant h when sups∈[−τ,0] |ϕ(s)| ≤ δ. Suppose that limt→∞ u(t) = 0 is not true.
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12 H. AKÇA, V. COVACHEV, Z. COVACHEVA
In this case there exists a number ν > 0 and an increasing sequence {tk} such that tk → ∞ and
|u(tk)| ≥ ν for k ∈ N. For the sake of brevity we write system (6) in the form
d
dt
ui(t) +
m∑
j=1
eijuj(t− τj)
= Fi(u(t), u(t− τ)), i = 1,m, t > 0, (9)
where
Fi(u, ū) := ãi(ui)
−b̃i(ui) +
m∑
j=1
cij f̃j(uj) +
m∑
j=1
dij g̃j(ūj)
, i = 1,m.
We denote
Ci = sup
|u|,|ū|≤h
|Fi|, i = 1,m.
For t ≥ 0 and ∆ > 0 integrate equation (9) from t to t+ ∆ to obtain
ui(t+ ∆)− ui(t) =
= −
m∑
j=1
eij(uj(t+ ∆− τj)− uj(t− τj))+
+
t+∆∫
t
Fi(u(s), u(s− τ)) ds,
hence
|ui(t+ ∆)− ui(t)| ≤
m∑
j=1
|eij ||uj(t+ ∆− τj)− uj(t− τj)|+
+
t+∆∫
t
|Fi(u(s), u(s− τ))| ds ≤
≤
m∑
j=1
|eij | sup
t≥−τ
|uj(t+ ∆)− uj(t)|+ Ci∆ ≤
≤
m∑
j=1
|eij |
(
sup
t≥0
|uj(t+ ∆)− uj(t)|+ sup
s∈[−τ,0]
|uj(s+ ∆)− uj(s)|
)
+ Ci∆,
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GLOBAL ASYMPTOTIC STABILITY OF COHEN – GROSSBERG NEURAL NETWORKS . . . 13
thus
sup
t≥0
|ui(t+ ∆)− ui(t)| ≤
≤
m∑
j=1
|eij |
(
sup
t≥0
|uj(t+ ∆)− uj(t)|+ sup
s∈[−τ,0]
|uj(s+ ∆)− uj(s)|
)
+ Ci∆
or
sup
t≥0
|ui(t+ ∆)− ui(t)| −
m∑
j=1
|eij | sup
t≥0
|uj(t+ ∆)− uj(t)| ≤
≤
m∑
j=1
|eij | sup
s∈[−τ,0]
|uj(s+ ∆)− uj(s)|+ Ci∆, i = 1,m.
If we introduce the vectors
ρ(∆) =
(
sup
t≥0
|u1(t+ ∆)− u1(t)|, sup
t≥0
|u2(t+ ∆)− u2(t)|, . . . , sup
t≥0
|um(t+ ∆)− um(t)|
)T
,
σ(∆) =
=
(
sup
s∈[−τ,0]
|u1(s+ ∆)− u1(s)|, sup
s∈[−τ,0]
|u2(s+ ∆)− u2(s)|, . . . , sup
s∈[−τ,0]
|um(s+ ∆)− um(s)|
)T
and C = (C1, C2, . . . , Cm)T ,we can write the last inequalities in a matrix form as (E−|E|)ρ(∆) ≤
≤ |E|σ(∆) + ∆C. From here as above we obtain ρ(∆) ≤ (E − |E|)−1(|E|σ(∆) + ∆C) and
sup
t≥0
|u(t+ ∆)− u(t)| ≤ |(E − |E|)−1(|E|σ(∆) + ∆C)|. (10)
Let η > 0 and ∆ ≤ η. Since u(t) is uniformly continuous on the interval [−τ, η], the right-hand
side of (10) can be made arbitrarily small for sufficiently small values of ∆. Thus we can choose
η > 0 so that |u(t+ ∆)− u(t)| ≤ ν/2 for all t ≥ 0 and ∆ ∈ [0, η]. In particular,
|u(tk + ∆)| ≥ |u(tk)| − |u(tk + ∆)− u(tk)| ≥ ν − ν
2
=
ν
2
or
|u(t)| ≥ ν
2
and ‖u(t)‖2 ≥ ν2
4
for t ∈ [tk, tk + η], k ∈ N.
Without loss of generality we can assume that the intervals [tk, tk + η] are disjoint (otherwise
we choose a subsequence). Then
∞∫
0
‖u(t)‖2 dt ≥
∞∑
k=1
tk+η∫
tk
‖u(t)‖2 dt ≥
∞∑
k=1
η
ν2
4
= ∞,
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14 H. AKÇA, V. COVACHEV, Z. COVACHEVA
which contradicts (8). Thus limt→∞ u(t) = 0 is true and the proof is complete.
4. Example. Consider the system
ẋ1(t) + 0.1ẋ1(t− τ1) + 0.15ẋ2(t− τ2) = (2 + 0.01 sinx1(t)) [−2x1(t) + 0.1 arctanx1(t)+
+ 0.15 arctanx2(t) + 0.1 arctanx1(t− τ1) + 0.15 arctanx2(t− τ2) + 1] ,
(11)
ẋ2(t)− 0.2ẋ1(t− τ1) + 0.2ẋ2(t− τ2) = (3− 0.02 sinx2(t)) [−3x2(t) + 0.15 arctanx1(t)−
− 0.2 arctanx2(t) + 0.1 arctanx1(t− τ1)− −0.2 arctanx2(t− τ2) + 1] , t > 0,
with arbitrary delays τ1, τ2 and initial conditions xi(s) = φ(s), i = 1, 2, s ∈ [−max{τ1, τ2}, 0].
System (11) has the form (1). It satisfies assumptions A1 –A4 with a1 = 1.99, a1 = 2.01,
a2 = 2.98, a2 = 3.02, b1 = b1 = 2, b2 = b2 = 3, F1 = F2 = G1 = G2 = 1, ‖E‖ =
= 0.2863903109 and
E − |E| =
[
0.9 −0.15
−0.2 0.8
]
is an M -matrix.
It is easy to see that system (11) satisfies inequalities (3). In fact, the left-hand sides of these
inequalities are equal respectively to 1.525 and 2.325 for i = 1 and 2. Thus system (11) has a
unique equilibrium point x∗. We can find that x∗ = (0.6027869379, 0.3353919007)T .
Furthermore, system (11) satisfies the assumptions of Theorem 2. In fact, the left-hand si-
des of inequalities (4) are equal respectively to 0.5415 and 4.449 for i = 1 and 2. Thus the
equilibrium point x∗ of system (11) is globally asymptotically stable.
On the other side, system (11) does not satisfy condition (5) since the left-hand side of the
inequality equals −0.608775797.
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Received 12.09.13,
after revision — 12.12.13
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