Int. J. Dynam. Control
DOI 10.1007/s40435-017-0315-9
On the output controllability of a class of discrete nonlinear
distributed systems: a fixed point theorem approach
Mustapha Lhous
1
· Mostafa Rachik
2
· Jamal Bouyaghroumni
2
·
Abdessamad Tridane
3
Received: 30 April 2016 / Revised: 8 February 2017 / Accepted: 8 March 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract Given a desired signal y
d
= ( y
d
i
)
i ∈{0,..., N }
, we
investigate the optimal control, which applied to nonlinear
discrete distributed system x
i +1
= Ax
i
+ Ex
i
+ Bu
i
, to
give a desired output y
d
. Techniques based on the fixed point
theorems for solving this problem are presented. An example
and numerical simulation is also given.
Keywords Output controllability · Optimal control ·
Nonlinear system · Fixed point theorem
1 Introduction
The research devoted the controllability was started in the
1960s by Kalman and refers to linear dynamical systems.
Because the most of practical dynamical systems are non-
linear, that’s why, in recent years various controllability
problems for different types of nonlinear or semilinear
dynamical systems have been considered [1–9].
There are large type of controllability such as completely
controllability, small controllability, local controllability,
regional controllability, near controllabilitry, null controlla-
bility and output controllability [4–6, 8–14].
B Mustapha Lhous
mlhous17@gmail.com
1
Laboratory of Modeling, Analysis, Control and Statistics,
Department of Mathematics and Computer Science, Faculty
of Sciences Ain Chock, Hassan II University of Casablanca,
5366 Maarif, Casablanca, Morocco
2
Laboratory of Analysis Modelling and Simulation,
Department of Mathematics and Computer Science, Faculty
of Sciences Ben M’sik, Hassan II University of Casablanca,
7955 Sidi Othman, Casablanca, Morocco
3
Department of Mathematical Sciences, United Arab Emirates
University, P.O. Box 15551, Al-Ain, United Arab Emirates
In the present paper we investigate the output controlla-
bility of a class of nonlinear infinite-dimensional discrete
systems. More precisely, we consider the nonlinear system
whose state is described by the following difference equa-
tion
( S)
x
i +1
= Ax
i
+ Ex
i
+ Bu
i
, i ∈{0,..., N − 1},
x
0
,
the corresponding output signal is
y
i
= Cx
i
, ∀ i ∈{0,..., N }.
The operator A : X −→ X is supposed to be bounded on
the Hilbert space X (the state space), E : X −→ X is a
nonlinear operator, B ∈ L(U, X ) and C ∈ L( X , Y ) where
the Hilbert space U is the input space and the Hilbert space
Y is the output one.
Given a desired output y
d
= ( y
d
i
)
i ∈{1,..., N }
, we investigate
the optimal control u = (u
i
)
i ∈{0,1,..., N −1}
which minimizes
the functional cost
J (u ) = u
2
over all controls satisfying
Cx
i
= y
d
i
, ∀ i ∈{1,..., N }.
To solve this problem and inspired by what was done in
[15, 16] we use, in the first part, a state space technique to
show that the problem of input retrieval can be seen as a
problem of optimal control with constraints on the final state
[17]. In the second part, we use a technique based on the
fixed point theorem (see [2, 3, 7, 18–21]). We establish that
the set of admissible controls is completely characterized by
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