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Regional￿Controllability￿with￿Cellular￿Automata
Models
Samira￿El￿Yacoubi 1 , Abdelhaq El Jai 1 ,￿and￿Nezha￿Ammor 2
1￿
Laboratory￿of￿Systems￿Theory￿,￿University￿of￿Perpignan,￿52,￿Avenue￿de￿Villeneuve,
66860￿Perpignan,￿France
{ yacoubi,￿eljai } @univ-perp.fr
http://www.univ-perp.fr/see/rch/lts/index.htm
2￿
Department￿of￿mathematics,￿University￿Mohammed￿V,￿Agdal
Rabat,￿Morocco
nammor@yahoo.fr
Abstract. In relation with spatio-temporal systems theory some re-
gional analysis aspects were recently developed [6, 24] and well studied
in continuous systems described by partial differential equations. The
purpose of this paper is to give a comparative study of the regional
controllability by means of cellular automata models. We show through
various examples how the main features of regional controllability may
be simply described and implemented by cellular automata approach.
To solve this problem, we propose one of the most e " cient evolutionary
techniques based on genetic algorithms.
1
Introduction
The basic concepts of systems theory including controllability, observability,
identification, stability analysis, have been introduced to analyse systems be-
haviour taking into account input-outputs [2, 5, 13, 17]. They have been widely
studied using essentially partial differential equations models. However, these
equations remain very hard to implement for predictive descriptions and evolu-
tions, particularly in the case of controlled systems. The main results have been
found only for linear systems. For non-linear systems, many problems are still
open. Cellular automata (CA's) models can offer in this case some additional
results in so far as they can describe many nonlinear phenomena by means of
simple local rules.
The concept of control is recently introduced in CA's models as a forcing
function called u ( t ) which takes values at each time t in a discrete set of all
possible inputs U [7, 12]. The regional controllability is an extension of control-
lability concept which studies whether a given system may be steered from any
initial state at time t 0 to any final state within a finite time T − t 0 . The no-
tion of regional controllability is motivated by various practical application in
industry, environment, etc. It consists of finding an appropriate control which
allow the system to achieve some objectif not on the whole domain but only
on a subregion. We consider in this paper a numerical approach of the regional
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