Information Technology Reference
In-Depth Information
The propulsion F
i
is equal to the derivative of the total potential of each particle
(drift term), i.e.
2
P
i
D
1
P
j
D
1;j
¤
i
ŒV
a
.jjx
i
F
i
Dr
x
i
fV
i
.x
i
/ C
1
x
j
jj C V
r
.jjx
i
x
j
jj/g)
Dr
x
i
fV
i
.x
i
/gC
P
j
D
1;j
¤
i
Œr
x
i
V
a
.jjx
i
F
i
x
j
jj/ r
x
i
V
r
.jjx
i
x
j
jj/ )
Dr
x
i
fV
i
.x
i
/gC
P
j
D
1;j
¤
i
Œ.x
i
F
i
x
j
/g
a
.jjx
i
x
j
jj/ .x
i
x
j
/g
r
.jjx
i
x
j
jj/ )
Dr
x
i
fV
i
.x
i
/g
P
j
D
1;j
¤
i
g.x
i
F
i
x
j
/
Taking also into account that the force F
i
which excites the particles' motion
(weights variation in time) may be subjected to stochastic perturbations and noise,
that is
Dr
x
i
fr.x
i
/
P
jD1;j¤i
g.x
i
F
i
x
j
/ C
i
g
(8.18)
one has finally that the particles' motion is a stochastic process. Next, substituting
in Eq. (
8.17
) one gets Eq. (
8.31
), i.e.
x
i
.t C 1/ D x
i
.t/ C
i
.t/Œr
x
i
V
i
.x
i
/ C e
i
.t C 1/
P
jD1;j¤i
g.x
i
x
j
/
i D 1;2; ;M; with
i
.t/ D 1;
(8.19)
which verifies that the kinematic model of a multi-particle system is equivalent to a
distributed gradient search algorithm.
8.2.2
Stability Analysis for a Neural Model with Brownian
Weights
The stability of the system consisting of Brownian particles is determined by the
behavior of its center (mean position of the particles x
i
) and of the position of each
particle with respect to this center. The center of the multi-particle system is given
by
M
P
iD1
x
i
M
P
iD1
x
i
) x D
) x
x D E.x
i
/ D
1
1
M
P
iD1
h
x
j
//
i
(8.20)
r
x
i
V
i
.x
i
/
P
jD1;j¤i
.g.x
i
1
D
