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In this paper, the characterization of regional asymptotic gradient observer have been given for a parabolic system. The approach of this characterization derived from Luenberger observer theory which is enable to estimate asymptotically the gradient state of the original system in a subregion í µí¼”í µí¼” of a spatial domain Ω in order that the regional asymptotic gradient observability notion can be achieved. Furthermore, we show that the strategic sensors allows the existence of regional asymptotic gradient observer and a sufficient condition have been given for such regional asymptotic gradient observer in identity case. Thus, the obtained result are applied for different types of measurements, domains and boundary conditions. 344 Raheam A. Al-Saphory et al. Mathematics Subject Classification: 93A30; 93B07; 93B30; 93C20 Keywords: ω Gí µí°¹í µí°¹í µí°¹í µí°¹-strategic sensors, asymptotic ω Gí µí°¹í µí°¹í µí°¹í µí°¹-detectability, asymptotic ω Gí µí°¹í µí°¹í µí°¹í µí°¹-observers
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International Journal of Contemporary Mathematical Sciences
Vol. 11, 2016, no. 7, 343 - 358
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ijcms.2016.6633
Asymptotic Regional Gradient Full-Order
Observer in Distributed Parabolic Systems
Raheam A. Al-Saphory
Department of Mathematics
College of Education for Pure Sciences Tikrit University, Tikrit, Iraq
Naseif J. Al-Jawari
Department of Mathematics College of Science
Al-Mustansriyah University, Baghdad, Iraq
Asmaa N. Al-Janabi
Department of Mathematics College of Science
Al-Mustansriyah University, Baghdad, Iraq
Copyright © 2016 Raheam A. Al-Saphory, Naseif J. Al-Jawari and Asmaa N. Al-Janabi. This
article is distributed under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, the characterization of regional asymptotic gradient
observer have been given for a parabolic system. The approach of this
characterization derived from Luenberger observer theory which is enable to
estimate asymptotically the gradient state of the original system in a subregion
of a spatial domain in order that the regional asymptotic gradient observability
notion can be achieved. Furthermore, we show that the strategic sensors allows
the existence of regional asymptotic gradient observer and a sufficient condition
have been given for such regional asymptotic gradient observer in identity case.
Thus, the obtained result are applied for different types of measurements, domains
and boundary conditions.
344 Raheam A. Al-Saphory et al.
Mathematics Subject Classification: 93A30; 93B07; 93B30; 93C20
Keywords: -strategic sensors, asymptotic -detectability, asymptotic
-observers
1. Introduction
There are many situations in modern technology in which it is necessary
to estimate the state of a dynamic system using only the measured input and
output data of the system [18]. An observer is a dynamic system the purpose of
which is to estimate the state of another dynamic system using only the
measured input and output letter. If the order of is equal to the order of the
observer is called full-order state observer [3, 8, 12]. Thus, asymptotic observer
theory explored by Luenberger in [20] for finite dimensional linear systems and
extended infinite dimensional distributed parameter systems govern by strongly
continuous semi-group in Hibert space by Gressang and Lamont as in [19]. The
study of this approach via another variable like sensors and actuators developed
by El-Jai et al. as in ref.s [3, 8, 12, 14] in order to achieve asymptotic
observability. One of the most important approach in system theory is focused on
reconstruction the state of the system from knowledge of dynamic system and the
output function on a subregion of a spatial domain this problem is called
regional observability problem has been received much attention as in [7, 15-17].
An extension of this notion has been given in [4, 21] to the regional gradient case.
The regional asymptotic notion has been introduced and developed by Al-Saphory
and El- Jai in [2, 11]. Thus, this notion consists in studying the asymptotic
behavior of the system in an internal subregion of a spatial domain Ω. In this
paper, we develop the results of asymptotic regional state reconstruction in [6, 10]
to the asymptotic regional gradient full order observer which allows to estimate
the state gradient of the original system. The main reason for introducing this
notion is that it provides a means to deal with some physical problems concern the
model of single room shown in (Figure 1) below.
Fig. 1: Room observation problem , workspace ω, and input-output vents.
Asymptotic regional gradient full-order observer 345
Now the object to design the room (locate vents, place sensors, etc. …) in order to
observe asymptotically the room vents near workspace (for more details see [13]).
The outline of this paper is organized as follows: Section 2 is devoted to the
problem statement and some basic concept related to the regional gradient
stability, regional asymptotic gradient detectability. Section 3 we focus on
regional asymptotic gradient observer so we introduced and characterization the
existing of identity regional asymptotic gradient observer to provide an identity
regional asymptotic gradient estimator of gradient state for the original system in
terms of sensors structure. In the last section we have been applied these result to
the two dimensional distributed parameter systems for different zone and
pointwise sensors case.
2. Problem Formulation and Preliminaries
Let Ω be a regular bounded open subset of , with smooth boundary 
and be subregion of , [0, ], > 0 be a time measurement interval. We
denoted = ×]0, [, =×]0, [. We considered distributed parabolic
systems is described by the following partial deferential equations

(,)=(,)+()
(, 0)=()
(,)= 0
(1)
Augmented with the output function
(. , )=( , ) (2)
where is a second order linear differential operator, which generator a strongly
continuous semi-group ()on the Hilbert space and is self-adjoin with
compact resolvent. The operator (,) and (,), depend on the
structure of actuators and sensors [16]. The space , and be separable Hilbert
spaces where is the state space, =(0, ,) is the control space and =
(0, ,) is the observation space where and are the numbers of actuators
and sensors (see Figure 2) which is mathematical model more general spatial case
in (Figure 1).
Fig. 2: The domain of , the sub-region , various sensors locations.
346 Raheam A. Al-Saphory et al.
Under the given assumption, the system (1) has a unique solution [18]:
( , )=()()+()()
(3)
The measurements are obtained through the output function by using of zone,
point wise which may located in Ω (or ). [16]
(. , )=( , ) (4)
We first recall a sensors is defined by any couple (D, f ), where D is its
spatial support represented by a nonempty part of
and f represents the
distribution of the sensing measurements on D.
Depending on the nature of D and f , we could have various type of
sensors. A sensor may be pointwise if D={b} with
and =(. ),
where is the Dirac mass concentrated at b. In this case the operator C is
unbounded [10] and the output function (2) can be written in the form
()=(,)=(,)()
It may be zonal when
and (). The output function (2) can be
written in the form
()=(,)()
In the case of boundary zone sensor, we consider = and
(), the output function (2) can be written as
(. , )=(. , )=(,)
()
We define the operator by the form
:=(.)
With : is the adjoint operator of K defined by
=()()
Consider the gradient operator :()(())
Asymptotic regional gradient full-order observer 347
:()( )
=
where is the restriction of to and it’s adjoint is denoted by
.
Finally, we introduced the operator = from into (( )) where
its adjoint given by .
The problem is how to build an approach which observe (estimates) regional
gradient state in subregion in of asymptotically by using a dynamic system
(an observer) in identity case only may be called full-order observer in region .
The important of an observer is that to estimates all the gradient state variables,
regardless of whether some are available for direct measurements or not [18].
The systems (1)-(2) are said to be exactly regionally gradient observable on
(exactly observable) if
 = = (()).
The systems (1)(2) are said to be weakly regionally gradient observable on
(weakly observable) if

= 
= (()).
It is equivalent to say that the systems (1)-(2) are weakly –observable if
 == {0}.
If The systems (1)(2) are is weakly –observable, then ( , 0) is given by
= ()=,
where is the pseudo-inverse of the operator (see ref. [15, 17]).
A sensor (,) is gradient strategic on (-strategic) if the observed system
is weakly -observable.
As well known the observability [8, 18] and asymptotic observability [14, 16, 18-
20] are important concepts to estimate the unknown state of the considered
dynamic system from the input and output functions. Thus, These notions are
studied and introduced to the regional distributed parameter systems analysis with
different characterizations by El-Jai , Zerrik and Al-Saphory et al. in many paper
for example [2, 4-7, 9-11, 13, 15, 17, 21] in connection with strategic sensors.
Definition 2.1: The system (1) is said to be asymptotically regionally gradient
stable (asymptotically -stable) if the operator generates a semi-group which
is asymptotically gradient stable on the (()). It is easy to see that the system
(1) is asymptotically -stable, if and only if for some positive constants
,, we have
(. )(), () ,0
348 Raheam A. Al-Saphory et al.
If (()) is -stable semi-group in (()), then for all (),
the solution of associated system satisfies

(.,)(())=
(. )(())= 0 (5)
Definition 2.2: The systems (1)-(2) are said to be asymptotically regionally
gradient detectable (asymptotically -detectable) if there exists an operator
:(()) such that  generates a strongly continuous
semi-group (()) which is asymptotically -stable on (()).
Remark 2.3: In this paper, we only need the relation (5) to be true on a sub-
region of the region
lim
(.,)(())= 0.
Thus, from the previous results we can deduce the following results:
Corollary 2.4: If The systems (1)(2) are is exactly -observable, then it is
asymptotically detectable.
Corollary 2.5: For every (()) there exists > 0, such that
(.,)(()) = 0
Remark 2.6: For parabolic systems, the notion of asymptotic –detectability is
far less restrictive than the exact -observability.
3. Sensors and asymptotic Observer
In this section we present the sufficient conditions which are guarantee the
existence of an asymptotic regional gradient full-order observer (asymptotic
Observer ) which allows to construct an estimator of the state
( , ).
3.2 Definitions and characterizations
Definition 3.1: Suppose there exists a dynamical system with state (.,)
given by

(,)=(,)+()+((,)(,))
(, 0)=()
(,)= 0
(6)
Asymptotic regional gradient full-order observer 349
In this case the operator in general case [17] is given by =
where = the identity operator. Thus the operator generator a
strongly continuous semi-group ( ()) on separable Hilbert space
which is asymptotically stable.
Thus, ,> 0 such that
 (. )  ,0.
and let (,),( , ) such that the solution of (6) similar to (3)
( , )=()()+()[()+()]
Definition 3.2: The system (6) defines asymptotic identity (full-order) 
estimator for (,)=( , )=( , )(()) where (,) is the
solution of the systems (1)-(2) if lim
(.,)( , )(())= 0, and 
maps () into () where (,) is the solution of system (6).
Remark 3.3: The dynamic system (6) specifies an asymptotic  –observer of
the systems given by (1) and (2) if the following holds:
1- There exists (, (())) and ((())) such that
+=.
2- =C and = B.
3- The system (6) defines an asymptotic estimator for ( , ).
The object of an asymptotic - observer is to provide an approximation to the
original system state gradient. This approximation is given by
()=()+().
Definition 3.4:The systems (1)-(2) are asymptotically -observable, if there
exists a dynamic system which is asymptotic -observer for the original
system.
3.2 Asymptotic -Observer reconstruction
In this case, we need to consider = and =, then the operator
observer equation becomes as = where and are known. Thus,
the operator must be determined such that the operator is asymptotically
-stable. This observer is an extension of asymptotic observer as in [2, 10-11,
16]. Now Consider again system (1) together with output function (2) described
by the following form
350 Raheam A. Al-Saphory et al.

(,)=(,)+()
(, 0)=()
(,)= 0
()=(. , ) (7)
Let be a given subdomain of Ω and suppose that ((())), and
(,)=(,) there exists a system with state (,) such that
(,)=(,)=(,) with =, where is the identity
operator with respect to regional gradient state estimator. Then
(,)=(,) (8)
From equation (7) and (8) we have
=
If we assume that there exist two bounded linear operators :((())
and
: ((())((()), such that += then by
deriving (,) in (8) we have

(,)=
(,)=(,)+()
= (,)+(,) + ()
Since the operator = , then we have

(,)= (,)+()+(,)
Therefore

(,)=(,)+()+(.,)
and since =C and = B then we have

(,)=(,)+()+((,)(,))
(, 0)=()
(,)= 0
(9)
Let us consider a complete sets of eigenfunctions  in (()) orthonormal to
(()) associated with the eigenvalue of multiplicity and suppose the
system (1) has unstable mode. Then, the sufficient condition of an -observer
is formulated in the following main result.
Theorem 3.5: Suppose that there are zone sensors (,) and the
spectrum of contains eigenvalues with non-negative real parts. Then the
dynamic system (9) is -observer system for the system (7), that is
lim
[(,)(,)] = 0, if :
Asymptotic regional gradient full-order observer 351
1-There exists (, (())) and ((())) such that
+ = .
2- =C , = B.
3-
4- rank = , ,= 1, … , with
= ()=
(),(.)>() for zone sensors
() for pointwise sensors
<
 ,(.)>() for boundary zone sensors
where sup =< and j = 1, … , .
Proof:
First step: The proof is limited to the case of pointwise sensors. Under the
assumptions of section 2, the system (1) can be decomposed by the projections
and on two parts, unstable and stable. The state vector may be given by
(,)= [(,),(,)] where (,) is the state component of the
unstable part of the system (1) may be written in the form

 (,)=(,)+()
(, 0)=()
(,)= 0
(10)
and (,) is the component state of the stable part of the system (1) given by

 (,)=(,)+ ()()
(, 0)=()
(,)= 0
(11)
The operator is represented by a matrix of order ( ,
 )
 given
by =diag[,…,,, … , , … , ,…,] and = [,, … , ]. The
condition (4) of this theorem, allows that the suit (,) of sensors is -
strategic for the unstable part of the system (1), the subsystem (10) is weakly -
observable [4] and since it is finite dimensional, then it is exactly -observable
[5]. Therefor it is asymptotically -detectable, and hence there exists an operator
such that (
) which is satisfied the following:
,
,
> 0 such that (
)(())

,
and then we have
(. , )(())
,
(.)(()) .
352 Raheam A. Al-Saphory et al.
Since the semi-group generated by the operator is stable on (()) , then
there exist 
,
> 0 [20] such that
(. , )(()) 
,
()()(.)(())
+
, 
()(
)(.)(()) ()
and therefor (,)0 when . Finally, the system (9) are asymptotically
-detectable.
Second step: From equation (8), we have (,)=(,) with the observer error
is given by the following form
(,)=(,) (,)
Where (,) is a solution of the dynamic system (9). Derive the above equation,
and by using equation (8) and condition 2, we can get the following forms

(,)=
(,)
(,)= 
(,)
(,)
=(,)+() (,)()(,)
=(,)()(,)(,)
=((,)(,))
= (,)
Thus, from the first part of this proof we obtain (,)=(0, ) is
asymptotically -stable with (0, )=()().
(,)(()))()()(()))
therefore lim
(,)= 0. Now, let the approximate solution to the gradient state of
the original system is
(,)=(. , )+(,) with = and =, then we
have
(,)=(,)
Now, we can calculate the error of gradient state estimator
(,)=(,)(,)=(,)(,)+(,)(,)
= (,)(,)=(,)=(0, )
is asymptotically -stable with (0, )=()(). Consequently we get
lim
(. , )( , )(())= lim
(.,)( , )(())= 0.
Then, the dynamical system (9) is -0bserver to the system (7).
Asymptotic regional gradient full-order observer 353
Corollary 4.4 From the previous results, we can deduce that:
1. Theorem 3.5 gives the sufficient conditions which guarantee the dynamic
system (9) is a -observer for the system (7).
2. If a system which is an -observer, then it is -observer for system (7).
3. If a system is -observer, then it is 
-observer for every subset of
, but the converse is not true [6].
4. Application to asymptotic -observer in diffusion system
In this section we consider the distributed diffusion systems defined on
;
k
where
21 k
. Various results related to different types of sensor have
been extended. In the case of two-dimensional, we take
] [ ] [
21
2
,0 ,0 aa ×=
and
] [
×= ,0
22
Q
,
] [
×=Σ ,0
2
2
, with boundaries
] [
×
=Θ ,
0
22
.
4.1. Two-dimensional system with rectangular domain
4.1.1. Case of zone sensors
Consider a two dimensional system defined in =]0, [×]0, [ by parabolic
equation

(,,)=
(,,)+
(,,)
(,, 0)=(,)
(,,)= 0
(12)
Augmented with output function measured by internal or boundary zone sensors
(. , )=(
,,)(,) or ( (. , )=
(
,,)(,)) (13)
Where and , , see (Figure 3).
Fig. 3: shows the domain
2
region
2
ω
and locations of internal (boundary) zone
sensors
354 Raheam A. Al-Saphory et al.
Let =],[×],[ be a subregion of . The eigenfunctions of the operator
(
+
) are defined by
(,)=
()()


Associated with the eigenvalues
=(
()+
())
Now, consider the dynamical system

(,,)=
(,,)+
(,,) (,,)()
(,, 0)=(,)
(,,)= 0
(14)
And suppose that the sensors is -strategic for unstable subsystem part of the
system (8), then we have the following results:
Proposition 4.1:
1- Internal zone case: Suppose that is symmetric about = and is
symmetric about =, then the dynamic system (14) is -observer systems
(12)-(13) if ()
 and ()
N for every ,= 1, … , .
2- One side boundary zone case: Suppose that
Γ
and f is symmetric with
respect to
1
01
ηη
=
, then the dynamic system (14) is -observer for the
systems (12)-(13) if
( ) ( )
Ni
1111
/
αβαη
for every, = 1, , . . . , .
3- Two side boundary zone case:
Let
( )
{ } { }
[ ]
+××=Γ×Γ=Γ
20110
21
21
,00, laal
ηη
and
1
Γ
f
is symmetric
with respect to
1
01
ηη
=
, and the function
2
Γ
f
is symmetric with respect to
2
02
η
η
=
, then the dynamic system (14) is --observer for the system (12)-
(13), if
( )
( )
( )
( )
Nji 22201
110 /and
/21
αβαηαβαη
for every i, j =1 ,...,
J.
4.1.2. Case of pointwise sensors
Consider again the systems (12)-(13) augmented with output function
measured by internal or boundary pointwise sensors (Figure 4).
Asymptotic regional gradient full-order observer 355
Fig. 4: Rectangular domain and locations b, σ of pointwise sensors
(. , )=(
,,)(,)((.,)=

(
 ,,)(,) (15)
Proposition 4.2:
1- The internal pointwise case:
If
( ) ( ) ( ) ( )
Nbjbi
22221111
/
and/
αβαα
βα
, for every , = 1, , , then
the dynamic system (14) is -observer for the systems (12)-(15).
2- Filament pointwise case: Suppose that the observation is given by the filament
sensor
( )
σ
δσ
,
, where
)
Im(
γσ
=
is symmetric with respect to the line
( )
21
,bbb =
.
The dynamic system (14) is -observer for the system (12)-(15), if
( ) ( )
1111
/
α
βα
bi
and
( ) ( )
Nbj 2222 /
αβα
, for every
.1,. . . , Ji, j=
3- The boundary pointwise case:
If
( ) ( )
and/111
1
αβα
bi
( ) ( )
Nbj
2222
/
αβα
, for  , =
1, .. . ,, then the dynamic system (14) is -observer for the system (12)-(15).
4.2. Two-dimensional systems with circular domain
Remark 4.3: We can extend these results to the case of two dimensional systems
with circular domain in different sensor structures as in [2, 10].
4.3. One-dimensional systems domain case
Remark 4.4: We can extend the above results of the two dimensional systems
(12)-(13) to case of one dimensional systems case if we take
] [
a,0
1
=
. We
denote
] [
×= ,0
11
Q
,
] [
×=Σ ,0
1
1
with boundaries
] [
×=Θ ,0
11
as in ref.s [2, 4-7, 9-11, 13, 15, 17, 21].
Remark 4.5: We know that the previous results have been developed with
Dirichlet boundary conditions, then we can extend with Neumann or mixed
boundary conditions as in [1, 17].
356 Raheam A. Al-Saphory et al.
5. Conclusion
The concept studied in this paper is related to the -observer in
connection with sensors structure for a class of distributed parameter systems.
More precisely, we have been given a sufficient condition for existing an -
observer which allows to estimate the gradient state in a subregion . For future
work, one can be extension these result to the problem of regional boundary
gradient observer in connection with the sensors structures as in [5].
Acknowledgments. Our thanks in advance to the editors and experts for
considering this paper to publish in this estimated journal and for their efforts.
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Received: July 2, 2016; Published: August 10, 2016
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