REGIONAL EXPONENTIAL GENERAL OBSERVABILITY IN DISTRIBUTED PARAMETER SYSTEMS

ABSTRACT The purpose of this paper is to explore the concept of regional exponential general observability for a class of distributed parameter systems, in connection with the regional strategic sensors. Then, we give characterization of the strategic sensors in order that exponential regional general observability can be achieved. Furthermore, we apply these results to one and two infinite dimensional systems, and various cases of sensors are considered and discussed. We also show that there exists a dynamical system for diffusion system is not exponentially general observable in the usual sense, but it may be regionally exponentially general observable.


INTODUCTION
The theory of observer was introduced by Luenberger (1966) [26] and has been generalized to system described by semigroup operators governed by partial differential infinite dimensional equations in Hilbert state space (Gressang and Lamont 1975) [25]. In system theory, the exponential observability is related to the possibility to estimate the state from knowledge of the state estimator system and output function [15]. The concept of actuators and sensors are introduced and applied to the controllability, observability and exponential estimator of systems described by partial differential equations as in [16][17].
The notion of regional analysis was extended by El Jai et al. [4,6,12,[21][22][23].The study of this notion motivated by certain concrete-real problem, in thermic, mechanic, environment or in isentropic superdense star as in [9,22,[23][24]. The concept of regional asymptotic and exponential analysis was introduced recently by Al-Saphory and El Jai et al. in [1][2][3][4][5][6][7][8][9][10][11], consists in studying the behaviour of the system not in the entire domain Ω but only on particular region inside the domain or on it is boundary. The aim of this paper is to extend some results related to regional exponential observability and strategic sensor to the general case as in ref.s [3,7]. We consider a class of distributed system and we explore various results connected with the different types of measurement, domains, and boundary conditions.
The purpose behind to develop this paper is that when we have fundamental aspects of a system knowledge, which developed by theoreticians from a mathematical point of view, and there is very often a wide gap between this and a concrete comprehension of the system [15][16]. Thus, sensors and actuators can play a fundamental role in the understanding of any real system [17][18][19][20].It is now time to link some of this theoretical works with concrete considerations of input-output problems. The main reason for discovering this concept is that it provides a means to deal with some physical problems concern the model of single room shown in (Figure 1) below. Here the goal to design the room (locate vents, place sensors, etc. …)in order to observe exponentially the room temperature near workspace (for more details see [13]). The outline of this paper is organized as follows: Section 2 is related to define the problem statement, regional strategic sensors, definitions and characterizations. Section 3 defines the regional exponential estimator (general case) for a distributed parameter system in terms of regional detectability and strategic sensors. We discuss under which condition such a regional observability exists and give a counter-example of an -estimator which is not an estimator in the whole domain Ω. Section 4 is related to the characterization notion of -observable by the use of strategic sensors. Section 5, we illustrate applications with many situations of sensors locations in one or two dimensional systems.

Problem Statement
Let Ω be an open regular bounded set of , with smooth boundary ∂Ω and let [0,T] , T> 0 be a time measurement interval. Suppose that ω be a nonempty given sub-region of Ω. We denote Θ = Ω × 0, ∞ and Π = ∂Ω × (0, ∞). The considered distributed parameter system is described by the following parabolic equations: augmented with the output function , = ( , ) (2) S e p t e m b e r 1 1 , 2 0 1 5 whereA is a second-order linear differential operator, which generates a strongly continuous semi-group ( ( )) ≥0 on the Hilbert space = 2 (Ω) and is self-adjoint with compact resolvent. The spaces , U, and are separable Hilbert space where is the state space, = 2 (0, ∞, ) is the control space, and = 2 (0, ∞, ) is the observation space, where pand q are the numbers actuators and sensors [11], see ( Figure 2) which is mathematical model more general spatial case in (Figure 1).

Fig. 2:
The domain of Ω , the subregion ω, and the sensors location The operators ∈ ℒ , and ∈ ℒ( , ) depend on the structures of actuators and sensors [9,11]. Under the given assumption [19], the system (1) has a unique solution: The problem is that how to construct an approach which observe exponential general current state in sub-region ω.

Regional Strategic Sensors
This subsection consists of the concept of the sensors, which was coined by A. El Jai as in [16][17]. Thus, sensors can play a fundamental role in understanding of any real system. Moreover, sensors form an important link between a system and its environment [19][20]. The purpose of this subsection is to give the characterization for sensors in order that the system (1) is regionally exponentially general observable in ω.
• Sensors are any couple ( , ) 1≤ ≤ where denote closed subsets of Ω , which is spatial supports of sensors and ∈ 2 ( ) define the spatial distributions of measurement on( ).
• Sensors may also be pointwise when = { } and = − where is Dirac mass concentrated in . Then, the output function (2) can be given by the form • In the case of boundary zone sensor, we consider Ω = Γ with Γ ⊂ Ωand ∈ 2 (Γ). Then, the output function (2) can be written in the form , = , = ∫ ( , ) Γ (6) •In the case, of internal pointwise sensors the operator C is unbounded and some precaution must be taken in [14].
•For the following equation •Consider a subdomain of Ω and let be the function defined by where | is the restriction of the state to ω.
• Here, we recall the concept of regional observability as in [12,21]. The concept of ωstrategic has been extended to the regional boundary case for ∈ [0, ] as in [26,27] and to the case ∈ (0, ∞) as in [2,10]. It is said that a suite of ( , ) 1≤ ≤ is -strategic if there exist at least one sensor ( 1 , 1 ) which is -strategic.
Let us consider the orthonormal set ( ) of eigenfunctions of 2 (Ω) orthonormal in 2 (Ω) is associated with eigenvalues of multiplicity and suppose that the system (1) has J unstable modes. Then, we have the following result.
Proof: The proof of this proposition is similar to the rank condition in [12]; the main difference is that the rank condition as follows For proposition 2.4, we need only to hold for Rank = , for all , = 1, … , . ∎

Definitions and Characterizations
This section presents some definitions and characterizations related to the regional exponential behaviour.

Definition 2.5:
The semi-group ( ( )) ≥0 is said to be exponentially stable if there exist two positive constants and such that If ( ( )) ≥0 is an exponentially stable semi-group, then for all 0 (. ) ∈ , the solution of the associated autonomous system satisfies we shall think about the next usual definition of stability.
Definition 2.6: The system (1) is said to be stable if the operator generates a semi-group which is exponentially stable.

Definition 2.7:
A semi-group is exponentially regionally stable in 2 ( ) (or -stable) if, for every initial state 0 (. ) ∈ 2 Ω , the solution of the autonomous system associated with (1) converges exponentially to zero when → ∞.

Definition 2.8:
The system (1) is said to be exponentially stable onω (or -stable) if the operator A generates a semigroup which is ( -stable).

REGIONAL EXPONENTIAL GENERAL OBSERVABILITY
In this section, we shall extend these results as is refer [3,7] to the general regional case by considering ω as sub-region of Ω. Thus, a new exponential general approach is introduced depending on general estimator in the region may be called regional exponential general observability ( -observability). The characterization of this concept needs some assumptions which are related to the exponential behaviour.

-Estimator Reconstruction Method
Consider now the system and the output specified by Let ⊂ Ω be a given subdomain (region) of Ω and assume that for ∈ ℒ( 2 Ω ), and = (where is defined in (10)) 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)-(18) give = If we assume that there exist two linear bounded operators R and S, where : → 2 ( ), and ∶ 2 ( ) → 2 ( ), such that + = , then by deriving ( , ) in (3.18) we have S e p t e m b e r 1 1 , 2 0 1 5 , = , = , + ( ) = , + , + ( ) Consider now the system (which is destined to be the regional exponential general estimator) where generates a strongly continuous semi-group ( ( )) ≥0 , which is regionally exponentially stable on = 2 ( ), i.e., and ∈ ℒ( , 2 ) and ℋ ∈ ℒ( , 2 ). The solution of (20) is given by Now, we want to show that under convenient hypothesis, the state of the system (20) is a regional exponential general estimator of , (Figure 3). For this purpose we need to decompose the system (1) as in next subsection.

General Decomposed System
Now, under the assumption of strongly continuous semi-group 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 ℋ. These assumptions are as follows.
(1) has a pure point spectrum, denoted by .
the spectrum of contained in the closed half plane : ≥ − . ≥ − forms a bounded spectral set [25]. 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 [15].
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 (23) Augmented with the output function Equations (26)- (27) 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 [25]. 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 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 , (26) 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 (26) . , = 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 (29). 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 [16]. For any > 0 there exists an ( ) ≥ 1 such that S e p t e m b e r 1 1 , 2 0 1 5 2 ( ) 2 0 ≤ ( 1 ) (− + 1 ) 2 0 for all ≥ 0 and 2 0 ∈ 2 . Thus, (29) is exponentially stable.

-Observability and -Detectability
This subsection, explores the relation between regional exponential general observability and regional exponential general detectability in order to characterize the concept of regional exponential general observability.
Here, we introduce some definitions and characterization concern to the concept of regional exponential general observability.
• A system which isΩ -observable, it is -observable • If a system is -observable in , then it is 1 -observable every subset 1 ⊂ , but the converse is not true. This is illustrated in the following counter example.

APPLICATION TO TWO DIMENSTIONAL DIFFUSION SYSTEM
In this section, we consider two-dimensional system defined on Ω = (0,1) × (0, 1) by the form together with output function by (4), (5). Let = ( 1 , 1 ) × ( 2 , 2 ) be the considered region which is subset of (0,1) × (0, 1). In this case, the eigenfunctions of system (46) are given by The following results give information on the location of internal zone or pointwise regional strategic sensors.

Boundary Zone Sensor
We consider the system (46) with the Dirichlet boundary conditions and output function (6). We study this case with different geometrical domains.

Boundary Pointwise Sensor
Let us consider the system (46) with Dirichlet boundary condition, so, we can study the following.
In this case the sensor ( , ) is located on Ω (Figure 9). The output function is given by    (1) Case of Neumann or mixed boundary conditions [19][20].

CONCLUSION
The original concept developed in this paper is related to sufficient condition of regional exponential general observability in connection with the strategic sensors and locations. Various interesting results concerning the choice of sensor S e p t e m b e r 1 1 , 2 0 1 5 structures are given and illustrated in specific situations. Then, we have explained the regional exponential detection problem for a class of distributed parameter systems with regional observability and regional estimator. We have given an extension of exponential regional state reconstruction in the considered sub-region ω based on the structures of sensors. Thus, we have characterized the existence of such -observable (general case).The state to be exponentially estimated on a part of boundary of the domain is under consideration.