Biomedical Engineering Reference
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evolution of the state, it is necessary to specify
the initial state variables
x
(
t
) in a time interval
of length t, i.e.,
with pi
i
≥ 0, li
i
≥ 0, r
ij
≥ 0, δ
ij
≥ 0 some free pa-
rameters.
Differentiating it with respect to
t
along the
solutions of (25) we may find the so-called deriva-
tive functional
,
x
t
=
t
x
(
+
)
=
(
)
−
≤
≤
0
0
0
(Gu
et al.
, 2003).
W
:
as below
Global Asymptotic Stability for Time
Delay Hopfield Networks via
Lyapunov-Like Results
m
[
(
)
∑
2
W
(
z
(
⋅
))
=
−
a
z
(
0
−
a
g
z
(
0
z
(
0
−
i
i
i
i
i
i
i
i
i
=
1
m
(
)
[
(
)
]
∑
−
z
(
0
+
g
z
(
0
c
g
z
(
−
)
+
i
i
i
i
i
ij
j
j
ij
If we consider the standard equations for the
Hopfield network (5) and the delay at the intercon-
nection level as being associated with each neural
cell, the following model is obtained (Gopalsamy
& He, 1994)
j
=
1
m
m
[
]
.
( )
(
)
∑∑
2
2
2
2
+
z
(
0
+
g
z
(
0
−
z
(
−
)
−
g
z
(
−
)
ij
j
ij
j
j
ij
j
ij
ij
j
j
ij
1
1
(27)
The problem of the sign for
W
gives the fol-
lowing choice of the free parameters in (26) (see,
Danciu & Răsvan, 2007)
m
(
)
∑
=
x
(
t
)
=
−
a
x
−
c
f
x
(
t
−
)
+
I
,
i
=
1
m
i
i
i
ij
j
j
ij
i
j
1
(24)
c
2
m
m
∑
ij
∑
2
>
0
and
=
a
−
(
+
)
>
0
where the time delays t
ij
are positive real num-
bers.
Let
x
be an equilibrium for (24). In order to
study the equilibrium at the origin one uses the
system in deviations
i
i
i
ji
ji
j
=
1
ji
j
=
1
−
1
−
1
c
2
c
2
m
m
∑
ij
∑
ij
2
(
a
−
)
<
<
2
(
a
+
)
i
i
i
i
i
j
=
1
ji
j
=
1
ji
(28)
m
(
)
The application of the standard stability theo-
rems for time delay systems (Hale & Verduyn
Lunel, 1993) will give the asymptotic stability of
the equilibrium
z =
0 (
∑
=
z
(
t
)
=
−
a
z
−
c
g
z
(
t
−
)
,
i
=
1
m
i
i
i
ij
j
j
ij
j
1
(25)
x
=
). We thus obtain the
following result (Danciu & Răsvan, 2007)
x
j
f
satisfy
the usual sigmoid conditions then
g
j
defined by
(11) are such. Denoting
:
As previously shown, if
=
max
Theorem 3.
Consider system (24) with a
i
> 0
and c
ij
such that it is possible to choose
r
ij
> 0
and
δ
ij
> 0
in order to satisfy
σ
i
> 0
with
σ
i
defined in (28). Then the equilibrium is globally
asymptotically stable.
, one chooses
ij
i
,
j
−
.
For time delay systems the Lyapunov-Kraso-
vskii functional is the analogue of the Lyapunov
function of the case without delays. One considers
the Lyapunov-Krasovskii functional suggested
by some early papers (Kitamura
et al.
, 1967;
Nishimura
et al.
, 1969),
(
,
0
m
)
as state space
Remark that the choice of the neural networks'
parameters is not influenced here by the values
of the delays, i.e. one obtains
delay-independent
sufficient
conditions
that characterize the local be-
havior of an equilibrium, expressed as previously
in the language of the weights of the RNN.
V
:
as
+
z
(
0
)
m
1
i
( )
∑
2
∫
V
z
(
⋅
)
=
z
(
0
+
g
(
)
d
+
i
i
i
i
2
i
=
1
0
0
m
(
)
∑
∫
2
2
+
z
(
)
+
g
(
z
(
))
d
ij
j
ij
j
j
j
=
−
1
ij
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