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Remark1.
It should benoted that convergenceof genetic algorithms is not
guaranteed.We havethentospecifyaterminationconditionasforinstancethe
attainmentofanappropriatefitnesslevel.
4ProblemStatement
4.1Preliminaries
Let us first recall some definitions related to controllability and regional con-
trollability of distributed parameter systems. Consider a continuous system de-
scribed by a partial differential equation defined on an open bounded set
Ω∈IR
n
!
˙
z
(
t
)=
Az
(
t
)+
Bu
(
t
)
t∈
]0
,T
[
z
(0) =
z
0
(7)
Under some specific assumptions satisfied by the operators
A
and
B
and given
smooth boundary conditions, we can ensure the existence and uniqueness of the
system solution denoted by
z
(
x,t
). Let us now consider a given desired state
z
d
defined on
Ω
which is assumed to be enough smooth. The controllability problem
consists in finding a control
u
in a given regular space which steers the system
to
z
d
at a given time
T
. A large variety of works dealing with the controllability
problem have been developed in a big literature see e.g [5, 13, 17].
Inspired by real applications, the controllability concept was recently relaxed
to regional controllability which consist in steering a system from a given initial
state to a prescribed state defined only on a subregion
ω⊂Ω
. This notion was
firstly introduced in [6, 24] and various works related to regional analysis which
concern essentially linear systems have been developed.
In order to define the above mentioned notions in terms of CA's models, we
need to introduce control as an external input affecting the evolution even if
CA's in the classical sense are autonomous systems. Let
A
=(
L,S,N,f
)bea
CA and
L
1
beasubsetofthelattice
L
.
Definition2.
AcontroluisanyrealvaluewhichisassignedtoL
1
maybe
reducedtoonecellcandvaryingintimeandspace).Aisthenanon-autonomous
CAdenotedby
(
L,S,N,f,u
)
withamodifiedformoftransitionfunctionwhich
canbedenotedbuf
u
(L
1
andexpressedasfollows
s
t
+1
(
c
)=
f
u
(
s
t
(
N
(
c
))) =
f
(
s
t
(
N
(
c
))) +
χ
L
1
u
t
(
c
)
(8)
wheres
t
+1
(
c
)
isthestateofthecellcattimet
+1
ands
t
(
N
(
c
))
denotesthe
stateofitsneighbourhoodattimet.TheglobalcontrolledCAfunctionisthen
denotedbyF
u
.
