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s
d
(
c
i
)=1
,∀
11
≤i≤
20. With
i
0
= 15 the problem is solved using classical
genetic algorithms and the obtained solution has an explicit form given by :
u
t
=
a
t
⊗b
t
⊗c
t
(15)
where
a
t
=
s
t
(
c
14
),
b
t
=
s
t
(
c
15
) + 2 and
c
t
=
s
t
(
c
16
).
By successive applications of the control (15), we obtain in Fig. 3, the evo-
lution from a given initial configuration to the desired configuration which is
reached after
T
= 23 steps
Time
Time
w
t=23
t=23
t=21
t=21
t=18
t=18
t=15
t=15
w
t=12
t=12
t=9
t=9
t=6
t=6
t=3
t=3
t=0
t=0
11
11
w
w
20
20
Sites
Sites
Fig.3.
Thesuccessiveconfigurationsfromtime0,obtainedbyapplicationofcontrol.
TwoDimensionalCase.
Consider a square lattice composed with 10
×
10
cells indexed as
c
i,j
,
i,j
=1
,···,
10. The state set is given by
S
=
{
0
,
1
,
2
}
.If
s
t
(
c
i
,j
) denotes the state of
c
i,j
at time
t
, the CA evolution obeys the following
rule
f
(
s
t
(
N
(
c
i,j
))) =
f
1
(
s
t
(
N
(
c
i,j
))) +
u
t
χ
L
1
(16)
where
L
1
is a sublattice of
L
which denotes the support of the control
u
t
and
N
designates the von Neumann neighbourhood. We consider the particular case
when
f
1
is of totalistic type which calculates
s
t
+1
(
c
i,j
) as a sum modulo 3 of its
neighbouring cell states
s
t
(
c
i,j−
1
)
⊕s
t
(
c
i,j
+1
)
⊕s
t
(
c
i−
1
,j
)
⊕s
t
(
c
i
+1
,j
).
Given the same desired configuration
s
d
(
c
i,j
)=1
∀c
i,j
∈ω
, the aim is to find a
control
u
t
which is active on
L
1
and allows the system to reach the state
s
d
at
time
T
on the subregion
ω
=
{c
i,j
,
4
≤i,j≤
6
}
.For
L
1
=
{c
3
,
3
,c
3
,
4
,c
4
,
3
,c
4
,
4
}
,
the used genetic algorithm gives a result in the form :
u
t
=
s
t
(
c
i
+1
,j
)
⊕
1
which makes the system regionally controllable on
ω
(see Fig. 4).
