Asymptotic Regional Boundary Observer in Distributed Parameter Systems via Sensors Structures

The purpose of this paper is to study the concept of asymptotic regional observer in connection with the characterizations of sensors. We give various results related to different types of measurements, of domains and boundary conditions. Furthermore, we show that the measurements structures allow the existence of regional observer and we give a sufficient condition for such observer. We also show that, there exists a dynamical system for the considered system is not observer in the usual sense, but it may be regional boundary observer.


Introduction
The observer theory was introduced by Luenberger in [1] and is generalized to infinite dimensional control systems described by semi-group operators in [2].The analysis of distributed parameter systems has received much attention in the literatures [3,4,5].The study of this notion, through the structure of the sensors and the actuators, was developed by El Jai and Pritchard [6,7].
The notion of regional analysis was extended by El Jai and Zerrik et al. [8,9].This study motivated by certain concrete-real problems, in thermic, mechanic, environment [10,11].If a system defined on a domain Ω is represented as in the (Fig. 1), then we are interested in the regional state on ω or Г of the domain Ω.
The concept of asymptotic regional analysis was introduced recently by Al-Saphory and El Jai in [12,13,14], consists in studying the behavior of systems not in all the domain Ω but only on particular regions ω or Г of the domain.In this paper, we are only interested in the asymptotic regional boundary state on a region Г of the boundary ∂Ω.The purpose of this paper is to give some results related to the link between the Г-observer and the number of sensors, their locations, the geometrical domains and boundary conditions.The paper is organized as follows.Section 2 concerns the formulation problem and preliminaries.Section 3, devotes to the introduction of Г-detectability problem.In section 4, we give some definitions and characterizations concerns Г-observer and strategic sensors.We show that there exists a counter-example of the case Г-observer which is not observer, in the whole domaine.In the last section, we illustrate applications with various situations of sensor locations and we characterize such a regional boundary observer to a diffusion system.

Problem Statement
Let Ω be a regular, bounded and open set of R n , with smooth boundary Ω ∂ and Γ be a nonempty given subregion of Ω ∂ , with positive measurement.We denote Q = Ω × ]0, ∞ [ and Θ = Ω ∂ × ]0, ∞ [.We consider the system described by the following parabolic equation and we assume that Z, U, O be separable Hilbert spaces where Z is the state space, U the control space and O the observation space.Usually we consider Z = H 1 ( Ω ), U = L 2 (0, ∞ , R p ) and O = L 2 (0, ∞ , R q ) where p and q hold for the number of actuators and sensors.A is a second order linear differential operator which generates a strongly continuous semi-group (S A (t)) 0 ≥ t on Z and is selfadjoint with compact resolvent.The operators B∈ L (R p , Z) and C ∈ L (Z, R q ) depend on the structure of actuators and sensors [7].Under the given assumptions, the system (2.1) has an unique solution given by ( ) 3) The measurements can be obtained by the use of zone, pointwise or lines sensors which may be located in Ω (or Ω ∂ ) [7]. • Let us recall that a sensor by any couple (D, f ) where D denote closed subsets of Ω , which is spatial support of sensor and ∈ f L 2 (D) defines the spatial distribution of measurement on D.
According to the choice of the parameters D and f, we have various types of sensors.A sensors may be a zone types when D Ω ⊂ .The output function (2.2) can be written in the form A sensor may also be a point when D = {b} and f = δ (. -b) where δ is the Dirac mass concentrated in b.Then, the output function (2.2) may be given by the form In the case of boundary zone sensor, we consider 2) can then be written in the form The operator C is unbounded and some precautions must be taken in [7,15].
• The operator K defined by and in the case of internal zone sensors is, linear and bounded with an adjoint • The trace operator of order zero ( ) ( ) is linear, surjective and continuous with adjoint denoted by γ 0 * • Consider a subdomain Γ of Ω ∂ and let χ Γ be the function defined by where Γ z is the restriction of the state z to Γ .We denote by * Γ χ the adjoint of Γ χ .
• Let ω χ be the function defined by where Γ z is the restriction of the state z to .ω • The operator ) ( ) ( : Recalling some definitions concern the notion of Γ -observability as in [9,16] : • The autonomous system associated to (2.1)-(2.2) is said to be exactly (respectively weakly) Γ -observable if : D f ≤ ≤ of sensors is said to be Γ-strategic if the system (2.1) together with the output function (2.2) is weakly Γ-observable.For the dual results concerning the actuators structures [17] .

Regional boundary detectability
The main reason for introducing Γ -detectability is, the possibility to construct an Γ -estimator for the current state of the original system.This concept is extended by Al-Saphory and El Jai in [13].For this objective we recall some definitions concern this concept : • If a system is Ω ∂ -detectable, then it is possible to construct an asymptotic Ω ∂ -observer for the original system.If we consider the system Remark 3.1.In this paper, we only need the relation (3.1) to be true on a given subregion Γ of the region Ω ∂ we may refer to this as Γ -stability.
• The system (2.1) is said to be Γ -stable, if the operator A generates a semi-group which is stable on H 2 / 1 ( Γ ).
• The system (2.1)-(2.2) is said to be Γ -detectable if there exists an operator generates a strongly continuous semi-group 0 )) ( ( For more detail [18].However, one can easily have the following results : Corollary 3.2.If the system (2.1) together with output function (2.2) is exactly Γ -observable, then it is Γ -detectable.
This result leads to: Thus, the notion of Γ -detectability is a weaker property than the exact Γ -observability as in (Ref.[16].

Asymptotic Γ -observer and strategic sensors
In this section, we propose an approach which allows to determinate a regional asymptotic estimator of z(ξ, t) on Γ, based on the internal asymptotic Γ-observer.This method need some definitions concern regional observer as in [11].
• The system (4.1) is said to be a reduced-order ω -observer for the system (2.2) • The boundary regional observer in Γ may be seen as internal regional observer in ω if we consider the following transformations.Let ℜ be the continuous linear extension operator [19] ℜ : • Let r>0 is an arbitrary and sufficiently small real and let the sets where B(z, r) is the ball of radius r centered in ) , ( t z ξ and where Γ is a part of r ω (Fig. 2).
From this result, we can deduce the following proposition : (4.4) Since the system (4.1) is regional observer on r ω so we can deduce that : The system (4.1) is regional observer on r ω , there exists a dynamical system with The equations (4.5) and (2.2) allow and there exist two linear bounded operators R and S satisfy the relation * 0 There exists an operator r F ω is regionally stable on , r ω then it is regionally stable on Γ [18].
Finally the system (4.1) is a regional boundary observer on .Γ ■

Sufficient condition for Γ -observer
As in (Refs.[7,18]), we develop a characterized result that links the Γ -observer and strategic sensors and we give a sufficient condition for Γ -observer.For that purpose, we assume that there exists a complete set of eigenfunctions n ϕ of A in H 1 ( Ω ), associated to the eigenvalues n λ with a multiplicity n s and suppose that the functions n ψ defined by ) ( ) (Γ .If the system (2.1) has J unstable modes, then we have the following theorem.
Theorem 4.3.Suppose that there are q zone sensors and the spectrum of A contains J eigenvalues with non-negative real parts.If the following conditions are satisfied : Proof : The proof is limited to the case of zone sensors in the following stapes : 1.
Step Under the assumptions of section 2, the system (2.1) can be decomposed by the projections P and I -P 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 (2.1), may be written in the form is the component state of the stable part of the system (2.1) given by Step The condition (2) of this theorem, allows that the suite of sensors is Γstrategic, the subsystem (4.7) is weakly Γ -observable [16] and since it is finite dimensional, then it is exactly Γ -observable.Therefore it is Γ -detectable, and hence there exists an operator 1 which is satisfied the following : and then we have Γ of Γ , but the converse is not true.This may be proven in the following example: Example 4.5.Consider a two-dimensional system described by the diffusion equation . The operator  1  is a linear operator.Consider the boundary sensor . Thus, the output function can be written by , then the system (4.11)-(4.9) is not weakly observable in Ω , i.e. the sensor ) , ( 0 f Γ is not strategic and therefore the system is not detectable in Ω .Thus, the dynamical system (4.10) is not observer for the system (4.11)-(4.9) (see [6]).Here, we consider the region (Fig. 3) and the dynamical system In this case, the system (4.11)-(4.9) is weakly observable in Γ and the sensor ) , ( 0 f Γ is Γ -strategic [9].Thus, the system (4.11)-(4.9) is Γ -detectable [13].Finally the dynamical system (4.9) is Γ -observer for the system (4.11)-(4.9) [18].

Application to sensors structures
In this section, we consider the distributed diffusion systems defined on . We explore various results related to different types of measurements, domains and boundary conditions.

Case of a zone sensor
We study the following cases :

Rectangular domain
In this case, the system is given by is the location of zone sensor and above system represent the heatconduction problem (see Ref. [20] The eigenfunctions and the eigenvalues are given by and one sensor can be guaranteed Γ -strategic sensor (see [21]).The dynamical system (4.6) may be given by Let the measurement support is rectangular with and 2 f is symmetric with respect to 2 0 2 ξ ξ = , then we have the following result : Corollary 5.1.The dynamical system (5.4) is Γ -observer for the system (5.1) if a } then we have : f is symmetric with respect to , then the dynamical system (5.4) is Γ -observer for the system (5.1) if

Disk domain
The system (5.1) may be given by the following form are defined as in (Fig. 5).Let the eigenfunctions and eigenvalues concerning the region for all 0 = nm .In this case, the Γ -strategic sensor is required at least two zone sensors (see [6]).The dynamical system (5.4) can be written by When the sensors are located on Ω ∂ and the function as in (Fig. 5).So, we have.

Case a pointwise sensor
In this subsection, we consider the following cases : , then the dynamical system (5.4) is Γ -observer for the system (5.9). 2. Filament case : Suppose that the observation is given by the filament sensor where ) ( Im γ σ = is symmetric with respect to the line ) , ( , the dynamical system (5.4) is Γ -observer for the system (5.9)., then the dynamical system (5.4) is Γobserver for the system (5.9).

5.2.2
The domain ) D(0, Ω a = The system (5.5) may be given by the following form then the dynamical system (5.8) is Γ -observer for the system (5.10).

Conclusion
The concept studied in this paper is related to the case of identity Γ -observer in connection with the sensors structure.For this class of parabolic distributed parameter systems, many interesting results concerning the choice of the sensor are explored and illustrated in specific situations of the domain.In similar way, we can characterize the general case (reduced-order) Γ -observer by sensors structure.An important extension of these results, related to the problem of regional boundary gradient observer, in connection with the structure of sensors; is under consideration.

Figure 2 :
Figure 2: The considered domain Ω and the subregion ω r.