REGIONAL EXPONENTIAL REDUCED OBSERVABILITY IN DISTRIBUTED PARAMETER SYSTEMS

The regional exponential reduced observability concept in the presence for linear dynamical systems is addressed for a class of distributed parameter systems governed by strongly continuous semi group in Hilbert space. Thus, theexistence of necessary and sufficient conditions is established for regional exponential reduced estimator in parabolic infinite dimensional systems. More precisely, the introduced approach is developed by using the decomposed system and reduced system in connection with various new concepts of (stability, detectability, estimator, observability and strategic sensors).Finally, we alsoshow that there exists a dynamical systemfor two-phase exchange system described by the coupled parabolic equations is not exponentially reduced observable in usual sense, but it may be regionally exponentially reduced observable.


NTRODUCTION
One of the most important concepts in infinite dimensional systems analysis is observability concept. Many researches of these concept included the notion of exponential observer( estimator), where Luenberger introduced this notion for finite dimensional systems [22], and has been generalized to infinite dimensional systems described by strongly continuous linear semi-group operators by Gressang and Lamont [20]. The purpose of an exponential estimator is to provide an exponential state estimation for the considered system state [16]. New concept of regional analysis for a class of distributed parameter systems was extended by Al-Saphory and El Jai et al. as in ref.s [1-7, 18, 16, 25-29]. Various asymptotic characterizations have been established and explored in connection with sensors structures [1,6]. In this paper, we introduce and study the notion of exponential regional reduced state observability in a given region of the domain Ω. Thus the developed approach is an extension of previous works to the regional case as in [2]. Moreover the relationship between this notion, regional detectability and strategic sensors are studied and discussed. The main reason behind the study of this notion (reduced observability), there exist some problem in the real world cannot observe the system state in the whole domain, but in a part of this domain. The scenario described by (Figure 1) below, one is interested in estimating the state in the green zone rather than in the entire space [12]. This problem falls into a class of so-called regional observation and estimation problem introduced by Al-Saphory and El-Jai and their workers as in [1][2][3][4][5][6][7][25][26][27][28][29].

Fig. 1: Zone control with fixed and mobile sensors
This paper is organized as follows. Section 2 is devoted to the introduction of regional exponential detectability and considered system with -detectability and -observability. We study the links of this notion with the regional exponential observability and strategic sensors. In Section 3, we study a regional exponential observability through the relations between -estimator reconstruction method and -observability. In section 4 we introduce regional exponential reduced observability notion for a distributed parameter system in terms of regional exponential reduced detectability and reduced strategic sensors. In the last section, we illustrate applications with different domains and circular strategic sensors of two-phase exchange systems.

REGIONAL EXPONENTIAL DETECTABILITY
The detectability is in some sense a dual notion of stabilizability [15]. This notion was considered and studied in the whole domainΩ.

2.1Considered Systems
Let Ω be a bounded and open subset of , with boundary Ω. Let 0, , > 0 a time measurement interval and be a non-empty given subregion ofΩ. We denote = Ω × (0, ∞)and Θ = Ω × 0, ∞ . Let , , and be separable Hilbert spaces, where is the state space, the control space and the observation space. We consider = 2 Ω , = 2 (0, ∞, ) and = 2 0, ∞, where and hold for the number of actuators and sensors [17]. The considered distributed parameter systems are described by the following parabolic equations where is a second-order linear differential operator, which generates a strongly continuous semigroup ( ) ≥0 on the Hilbert space = 2 Ω , and is self-adjoint with compact resolvent. The operators ∈ ℒ( , ) and ∈ ℒ( , ) depend on the structures of actuators and sensors [17] see ( Figure 2). S e p t e m b e r 1 1 , 2 0 1 5 That means, in the case of pointwise (internal or boundary) and boundary zone sensors (actuators), we have ∉ ℒ( , ) and ∉ ℒ( , ) [12,22]. Thus, the system (1) has a unique solution given by The problem is that how to give an approach which enable to estimate the system state in a sub-region . The regional exponential reduced estimator is defined when the output give a part of the state vector in this region.

Definitions and Characterizations
We extend some definitions and characterizations in the Hilbert space 2 Ω as ref.s [15,19].
Definition 2.1:The semi-group ( ( )) ≥0 is said to be exponential stable on Ω or (Ω -stable) if there exist two positive constants and such that If ( ( )) ≥0 is an Ω -stable semi-group, then for all 0 . ∈ , the solution of the associated autonomous system satisfies we shall consider the following usual definition of stability.

Definition 2.2:
The system (1) is said to be Ω -stable if the operator generates a semi-group which is Ω -stable.

Definition 2.3:
The system (1) together with the output (2) is said to be detectable on Ω if there exists an operator ∶ → 2 Ω such that ( − ) generates a strongly continuous semi-group ( ( )) ≥0 which is Ω -stable.
Thus, if a system is (Ω -detectable), then it is possible to construct an exponential Ω-estimator for the system state [9].

Remark 2.4:
In this paper, we only need the relation (4) to be true on a given subdomain ⊂ Ω, i.e., if we consider a subdomain of the domain Ω and let be the function defined by where | is the restriction of to . Thus and then lim →∞ (. , ) 2 = 0.
We may refer to this as regional exponential stability (or -stability), which is the equivalent for the considered class of systems to the exponential stability.

Definition 2.5:
The system (1) is said to be regionally -stable if the operator generates a semi-group which is regional exponential stable (or -stable).
In this section, we shall extend the definition of detectability by using equation (5) to the regional case by considering as subregion of Ω. S e p t e m b e r 1 1 , 2 0 1 5 Definition 2.6: The system (1)-(2) is said to be -detectable if there exists an operator ∶ → 2 such that − generates a strongly continuous semi-group( ( )) ≥0 , which is -stable.
The main reason for introducing the concept of -detectability is the possibility of constructing an -estimator for the state of system (1).

-Detectability and -Observability
It has been shown that a system which is exactly observable is detectable [16]. For linear systems, we recall the observability [2]. Now consider the autonomous system of (1) by the following form , = , where . ,0 is supposed to be unknown. The knowledge of . ,0 allows one to observe the state , 0 at any time . Measurements are obtained by the output function (2). The solution of the system (6) is given by: . , = . ,0 .
As in El Jai and Pritchard [17], we will develop a characterization result that links the -detectability in terms of sensors structures. So, we recall some definitions related to sensors.


A sensor is defined by any couple , where a non-empty closed subset of Ω, is the spatial support of the sensor, and ∈ 2 ( ) defines the spatial distribution of the sensing measurements on . S e p t e m b e r 1 1 , 2 0 1 5 In the case of a pointwise sensor, is reduced to a point { } and = . − , where is the Dirac mass concentrated in . Depending on the choice of the parameters and we have various types of sensors, the output function (2) may be written in the form In the case of boundary measurements (pointwise or zone) the support of sensors is subset of Ω. Then, the output function (2) given by = Ω , ( − ) (Boundary pointwise case) (12) Now in the case where the zone measurements, with = ⊂ Ω and ∈ 2 . Then, the output function (2) given by  The sensors (zone or pointwise) ( , ) 1≤ ≤ are said to be -strategic sensors if the system (1)-(2) is weaklyobservable.
Proof:by the result on observability considering * [14], we can proof this theorem. We see that if the system is satisfy the condition (2) above. Since = , therefore, the sensor of the system (1)-(2) is strategic sensor, and this system (1)-(2) is weakly -observable, then it's exactly -observable, finally we have the system (1)-(2) is -detectable.

REGIONAL EXPONENTIAL OBSERVABILITY
In this section, we give an approach which allows constructing an -estimator of , . This method avoids the calculation of the inverse operators, and the consideration of the initial state [18]. It enables to observe the current state in without needing the effect of the initial state of the original system.

-Estimator Reconstruction Method
We consider the system and the output specified by the following form: Let ⊂ Ω be a given subdomain (region) of Ω and assume that for ∈ ℒ( 2 Ω ), and = (where is defined in (5)) there exists a system with state (. , ) such that Thus, if we can build a system which is an exponential estimator for , , then it will also be an exponential estimator for . , , that is to say an exponential estimator to the restriction of . , to the region . The equations (2)-(15) give If we assume that there exist two linear bounded operators and , where : ℝ → 2 and : 2 → 2 , such that + = , then by deriving , we have , = , = , + = , + . , + .

-Observability
In this case, we consider = , and = , so the operator equation − = of the -observable reduces to = − , where and are known. Thus, the operator must be determined such that the operator is stable. For the system (14), consider the dynamic system Thus, a sufficient condition for existence of -estimator is formulated in the following proposition.

REGIONAL REDUCED EXPONENTIAL OBSERVABILITY
In this section we need some of additional assumptions, concerning the semigroup, its infinitesimal generator, and the observation space, under which condition can be given a regional reduced estimator for the state system (1)-(2).

General Decomposed System
Now, under the assumption of strongly continuous semigroup we have the system (1)-(2) is reduced as in the additional assumptions allow a decomposition of (1) to a form of the stabilizing operator . These assumptions are as follows.
(1) has a pure point spectrum, denoted by .
(4) The subspace associated with each finite dimensional point of in the half plane : ≥ − .
These five assumptions are strong. The Hille-Yosida theorem implies that the set of spectral point of lying in the half plane : ≥ − forms a bounded spectral set. Denote this spectral set by 1 . Using the spectral set 1 , a reduced form of (1) can be derived. Denote − 1 by 2 . As is a closed operator with nonempty resolvent set, operational calculus can be used to completely reduce the operator in terms of the spectral sets 1 and 2 [20]. 1 and 2 determine subspaces 1 and 2 , and projections 1 : → 1 , 2 : → 2 , such that Where = 1 + 2 , ∈ , 1 ∈ 1 , 2 ∈ 2 , 1 ∈ ℒ( , 1 )and 2 ∈ ℒ( , 2 ) as is dense in , 1 is dense in 1 , and 2 is dense in 2 .
1 and 2 are closed operators as is closed operator. If is the infinitesimal generator of a strongly continuous semigroup, then the Hille-Yosida theorem shows that both 1 and 2 are infinitesimal generators. Using the decomposition of and given by (21) Augmented with the output function Equations (24)- (25) are called the reduced form of (1)-(2).
Since 1 is the restriction of to 1 , and 1 = ∩ 1 , the spectrum of 1 is 1 [21]. As the points of 1 are isolated, each point by itself is a spectral set, and the spectral sets so formed are pairwise disjoint. Thus a projection and subspace can be associated with each point ∈ 1 , and the subspace 1 completely reduced to 1 = 11 ⨁ 12 ⨁ … ⨁ where is the number of points in 1 . Each is finite dimensional by assumption, hence 1 is finite dimensional, and 1 is a bounded operator. Then choosing bases for 1 and , (24) can be represented as a linear constant coefficient ordinary differential equation, and restricted to 1 can be expressed as a matrix.
In terms of the finite dimensional bases for 1 and , the homogeneous equations corresponding to (24) . , = 1 , Where is the coordinate space associated with the basis for 1 , and : 1 → in terms of the bases of 1 and .
An estimate will now be made of the solutions of (28). A having a pure point spectrum implies that 2 has a pure point spectrum, while 2 ( ) being a compact operator for some > 0 implies that 2 ( ) is a compact operator. As 2 ( ) is a compact operator, its spectrum consists of only point spectrum, denoted by ( 2 ( )) is given by Then the spectral radius of 2 ( ) satisfies using a lemma of Hale [18]. For any > 0 there exists an ( ) ≥ 1 such that 2 ( ) 2 0 ≤ ( 1 ) (− + 1 ) 2 0 for all ≥ 0 and 2 0 ∈ 2 . Thus, (27) is exponentially stable.

General Reduced System
In the case where the output function (2) gives information about a part of the state vector , , it is necessary to define an exponential estimator enables to construct the unknown part of the state. Consider now = 1 ⨁ 2 where 1 and 2 are subspaces of . Under the hypothesis of subsection 4.1, the system (1) can be decomposed [16,20] by S e p t e m b e r 1 1 , 2 0 1 5 where 1 ∈ 1 , 2 ∈ 2 , 1 ∈ ℒ( 1 , )and 2 ∈ ℒ( 2 , ). Using the decomposition above, the system (1) can be written by the form where , = 1 , ⨁ 2 ( , ). The problem consists in constructing a regional exponential estimator that enables one to estimate the unknown part 2 ( , ) equivalently; the problem is reduced to define the dynamical system for (31). Thus, equations (30)-(31) allow the following system: with the output function . , = 12 (. , ) where the state in system (32) plays the role of the state 2 in system (30).

Definition 4.9:
The system (32) The system is said to be regional exponential reduced stability (or -stable) if the operator A 22 generates a semi-group which is -stable.
In this section, we shall extend the definition of Ω -detectable (35) to the regional case by considering as subregion of Ω.
The importance of reduced ω -detectability is possible to define a reduced -estimator for system state may be given by the following important result: given by 22 1 = diag 2 1 , … , 2 1 , … , 2 , … , 2 and 2 = 2 1 , 2 2 , … , 2 From condition (2) of this theorem, then the suite of sensors ( , ) 1≤ ≤ is -strategic for the unstable part of the system (32), the subsystem (37) is weakly regionally reduced-observable in (or weakly -observable) and since it is finite dimensional, then it is exactly regionally reduced-observable in (or exactly -observable).
Here, we construct the -estimator for parabolic distributed parameter system (1), we need to present the following remakes Remark 4.13: the dynamical system (39) observes the regional reduced state of the system (1) if the following conditions satisfy: 1. ∃ ∈ , 2 and ∈ 2 such that: Remark 4.14: the system (1) is -observable if there exists an -estimators (39) which estimate the regional exponential reduced state of this system. Now, we present the sufficient condition of the regional exponential reduced observability notion as in the following main result.

APPLICATIONS TO EXCHANGE SYSTEMS
Consider the case of two-phase exchange system described by the following coupled parabolic equations: and consider Ω = 0,1 × (0,1) with subregion = 1 , 1 × 2 , 2 ⊂ Ω. Suppose that it is possible to measure the states 1 . , , by using zone sensors( , ) 1≤ ≤ . The output function (2) is given by S e p t e m b e r 1 1 , 2 0 1 5 In this section, we give the specific results related to some examples of sensors locations and we apply these results to different situations of the domain, which usually follow from symmetry considerations.
We consider the two-dimensional system defined on Ω = 0,1 × (0,1) with the case of system described by the following equations: Let = 1 , 1 × 2 , 2 , In this case the eigenfunctions and eigenvalues for the dynamic system (52) for Neumann conditions are given by We examine the tow cases illustrated in fig. (2)-(3).

Internal Rectangular Sensor
For discussing this case, suppose the system (52)-(53) where the sensor supports is the located inΩ. The output function can be written by the form where ⊂ Ω, is the location of zone sensor as in (Figure 3).

Fig. 3: Rectangular domain, region and location with rectangular support sensor
Then, the sensor ( , ) 1≤ ≤ may be sufficient for -observability, and there exists ℋ ∈ ℒ( , 2 ) such that the operator ( 22 − ℋ 12 ) generates a strongly continuous stable semi-group on the space 2 . Thus we have If 1 is symmetric about 1 = 2 the integral in the square bracket is zero and hence = 1 , 0 0 = 0.

5.2
Internal circular sensor S e p t e m b e r 1 1 , 2 0 1 5 Consider the system (52) augmented with the output function = 1 . , where the sensor supports is located inside the domainΩ. The output = 1 . , can be written by thefollowing form where = ( , ) ⊂ Ω, is the location of zone sensor as in (Figure 4).

Remark 5.3:
These results can be extended to the following: (1) Case of Neumann or mixed boundary conditions.

CONCLUSION
The concept developed in this paper is related to the regional exponential reduced observability in connection with the strategic sensors. Various interesting results concerning the choice of circular sensors are given and illustrated in specific situations. Many questions still opened. This is the case of, for example, the problem of finding the optimal sensor location S e p t e m b e r 1 1 , 2 0 1 5 ensuring such an objective. The result of regional exponential reduced observability concept of hyperbolic linear or semi linear or nonlinear systems is under consideration.