Regional Boundary Strategic Sensors Characterizations

This paper, deals with the linear infinite dimensional distributed parameter systems in a Hilbert space where the dynamics of the system is governed by strongly continuous semi-groups. More precisely, for parabolic distributed systems the characterizations of regional boundary strategic sensors have been discussed and analyzed in different cases of regional boundary observability in infinite time interval. Furthermore, the results so obtained are applied in two-dimensional systems and the sensors are studied under which conditions guarantee regional boundary observability in a sub-region of the system domain boundary. Also, the authors show that, the existence of a given sensor for the diffusion system is not strategic in the usual sense, but it may be regional boundary strategic of this system.


INTRODUCTION
The important problem of strategic sensor in distributed parameter systems has much attention in literatures ( [1][2] and references therein), in order that to estimate current state of the considered system [3][4]. This problem may be called the observability notion in control systems theory [5]. Thus, the observation problem is depended on the possibility of the state reconstruction from the knowledge of system dynamics and output function by using an approach to choose the best sensor may be strategic [6][7]. Recently, regional strategic sensors characterizations is developed by El-Jai, Zerrik and Al-Saphory et al. for different cases in finite [8][9][10][11] or infinite time interval, may be represented as regional asymptotic systems analysis [12][13][14][15][16] and focused on state estimation in a sub-region of the domain Ω [17][18]. The purpose of this paper is to extend the previous results as in ref. [12] to the regional boundary case where the interested region Γ is a part of the domain boundary ∂Ω. The main reason behind the study of this notion is that, there exists some problem in the real world cannot observe the system state in the whole domain, but it is possible in a part of the considered domain [15][16][18][19][20][21]. The scenario described by energy exchange problem, where the aim is to determine the energy exchanged in a casting plasma on a plane target which is perpendicular to the direction of the flow from measurements (internal pointwise sensors) carried out by thermocouples (Figure 1),

Fig. 1: Model of energy exchanged problem on Γ
where (1) is the torch of plasma, (2) is the probe of (steal), (3) is the insulator, Γ is the face of exchange and 1 , 2 sensor locations. This paper is organized as follows: The second section is focused on the considered system and the problem of regional boundary observability. The third section is devoted to the mathematical concepts of regional boundary observability and the characterization of regional boundary strategic sensors in various situations are studied. In the last section, we illustrate applications with many situations of sensor locations.

REGIONAL BOUNDARY STRATEGIC SENSORS
In this section, we are interested to study and characterize the notion of strategic sensors on a sub -region of the domain boundary of the considered systems and present some original results related to this notion.

Problem Statement
Let Ω be an open regular and bounded subset of , with smooth boundary Ω. Suppose that Γ be a nonempty given sub-region of with positive measurement. For ˃ 0 let us set Θ = Ω × (0, ∞) and Π = ∂Ω × (0, ∞). The considered systems is described by the following state space equations where Ω ̅ holds for closure of and 0 ( ) is unknown initial state in 1 (Ω ̅ ). The system (1) is defined with a Neumann boundary conditions, ⁄ holds for the outward normal derivative. The measurements maybe given by the use of zone, pointwise or lines sensors which is located insides of Ω or on the boundary [1]. Thus, the augmented output function to (1) is defied by where is a second order linear differential operator, which is generated a strongly continuous semi -group ( ( )) ≥0 on the Hilbert space = 1 (Ω) and, it is self-adjoint with compact resolvent. The operator ∈ ( , 1 (Ω) ) and ∈ ( , 1 (Ω ̅ ) ) , depend on the structures of actuators and sensors [1][2]. The spaces , and be are separable Hilbert spaces where is the state space, = 2 (0, ∞, ) is the control space and = 2 (0, ∞ , ) is the observation space, where and are the numbers of actuators and sensors. Under the given assumption, the system (1) has a unique solution [20]: The problem is that, how to present sufficient conditions for regional boundary strategic sensors which enables to observe the current state in a given sub region Γ (see below Figure 2 ), using convenient sensors. Mathematical model in (Figure 2) is more general spatial case in (Figure 1).

Fig. 2:
The domain of Ω, the sub-regions ω and Γ, various sensors locations.

Definitions and Characterizations
The regional boundary observability concept has been developed recently by El Jai et al. as in [18][19][20][21][22] and extended to the regional boundary asymptotic state by Al-Saphory and El Jai in ref.s [1][2][3][4][5]. To recall regional boundary observability, consider the associated autonomous system to (1) given by Thus, the knowledge of ( , 0 ) permits to observe regional boundary state ( , ) at any time . Consider now the following points: ▪ The solution of system (4) is given by the following form, ▪ The operator is defined by following then, we obtain where is bounded linear operator (this is valuable on some output function) [23].
Proof: The proof of this property is deduced from the usual results on observability by considering Γ 0 * as in [4]. Let , and be Banach reflexive space and ∈ ( , ), ∈ ( , ), then we have (1) The notion of approximate Γ-observability is far less restrictive than the exact Γ-observability.
(2) From the equation (14) there exists a reconstruction error operator that gives an estimation ̃0 of the initial state 0 in [22]. Then, we have Proposition 2.8: The regional boundary observability concept is more convenient for the analysis of real systems [20]. Then, we can deduce that: (1) The definitions 2.3 and 2.4 are more general and can be applied to the case where Γ = Ω.
(2) The equation (17) shows that the regional boundary state reconstruction will be more precise than if we estimate the state in the boundary of the domain Ω.
(3) If a system is exactly Ω-observable, then, it is exactly Γ-observable, but the converse is not true in general. Now, we prove that property (3) of remark 2.6.

Proof:
We see that if the system is exactly observable on Ω, then it is exactly Γ-observable and this is a consequence of (17) and then by the same way with miner tanique as in regional case [12], we can show that, if Γ ⊂ Ω, then and hence From equations (16), (17), (18), (19), and (20), we have Then from proposition 2.6 and remark 2.5, we can deduce that the system (4)-(2) is exactly Γ-observable.

SUFFICIENT CONDITIONS FOR -STRATEGIC SENSORS
The purpose of this section is to give the sufficient condition for the characterization of sensors in order that the system (1) is regionally boundary approximately observable in a region Γ.

Concept of Sensors
This subsection recalls and studies the concept of the sensors, which was introduced by A. El Jai [6][7]. Thus, we know that the sensors form an important link between a system and its environment [17][18]. In any case of sensors is considered via a space variable, mathematically speaking, the space variable is present in all systems described by partial differential equations [12].

Definition 3.1:
A sensor may be defined by any couple ( , ), where , a non-empty closed subset of Ω, ̅ is the spatial support of sensor and ∈ ( ) defines the spatial distribution of the sensing measurements on .

Remark 3.2:
According to the choice of the parameters and we have various types of sensors. Sensor may be a zone types denoted by ( , ), where ⊂ Ω, then, the output function (2) can be written in the form Also, sensors maybe pointwises when is the Dirac mass concentrated in . Thus, the output function (2) can be given by the form In the case of boundary zone sensors ( , ) where = with ⊂ Ω and ∈ 2 ( ).Therefore, the output function (2) can then be written in the form The operator is unbounded and some precautions must be taken in [3,11].

Definition 3.4:
A suit of ( , ) 1≤ ≤ is said to be Ω-strategic if there exists at least one sensor ( 1 , 1 ) which is approximately Ω-strategic.

Definition 3.6:
A suit of ( , ) 1≤ ≤ is said to be Γ-strategic if there exists at least one sensor ( 1 , 1 ) which is approximately Γ-strategic.
Thus, we can deduce that the following result:
(3) One can find various sensors which are not Ω-strategic for the systems, but may be Γ-strategic and achieve the observability in Γ. This is illustrated in the following counter-example.
In this section, we are interested to develop the results which are related to the strategic sensors and give the sufficient conditions for each sensor. For this purpose, we assume that there exists a complete se t of eigenfunctions of in 1 (Ω), associated to the eigenvalues with a multiplicity and suppose that the functions defined by = Γ 0 is a complete set in 1/2 (Γ) defined by, is a complete set in 1 (Ω). If the system (2.1) has unstable modes, then we have the following result.

Remark 3.9:
The previous result can be extended to the case of internal zone, filament and internal o r boundary sensors as in ref.s [12][13][14][15][16].

Remark 3.10:
The important to introduce this notion is that the using to charaterize the regional boundary exponential reduced observability in distributed parameter system as in [24] and this notion is extended to mutipule situations for finite time interval [25][26] or infinite as in [27][28][29].

APPLICATION TO SENSOR LOCATIONS
In this section, we present an application of the above results in two -dimensional systems defined on Ω = The following results give information on the locations of internal, boundary zone or pointwise Γ-strategic sensors.

Zone Sensor Cases
This subsection study various types of domains with different systems.

Rectangular domain
We discuss and examine different type of zone sensors.

Internal rectangular zone case:
Consider the system (30)- (2) where the sensor supports are located inside Ω. Then the output (2) can be written by the form where ⊂ Ω is location of zone sensor and ∈ 2 ( ) . In this case of (Figure 4), the eigenfunctions and the eigenvalues where ℕ is the natural numbers. If = 1 then one sensor ( , ) maybe suffices to achieve Γ-strategic sensor of the corresponding systems (30)-(33) [9][10][11]. Let the measurement support is rectangular with then, we have the following result.

Disc domain
We explore some results concern different type of zone sensors in disc domain.

Internal circular zone case:
In this case, systems (30) may be given by the following form with multiplicity = 2 for all ≠ 0 and = 1 for all = 0. In this case, the Γ-strategic sensor is required at least two zone sensors ( , ) 2≤ ≤ where = ( , ) , = 1,2 (see [14]). If we consider the case of Dirichlet or mixed boundary conditions, we can get various functions [2]. Thus, we develop some practical examples by using the symmetry conditions. If and are symmetric with respect to = , for all 2 ≤ ≤ , then we have . ∉ ℕ for every , = 1, … , ⁄ , then ( 1 , 1 ) and ( 2 , 2 ) are is Γ-strategic to the systems (37)-(38).

Boundary circular zone case:
In this case the system (30) is augmented with output function described by When the sensors (Γ , ) 2≤ ≤ are located on Ω and the function Γ is symmetric with respect to = , for all 2 ≤ ≤ , as in (Figure 8). So, we have.

Pointwise Sensor Cases
This subsection study again various types of domains in different systems.

Internal pointwise case:
Let us consider the case of pointwise sensor located inside of Ω. Thus, the system (30) is augmented with the following output function.
where = ( 1 , 2 ) is the location of pointwise sensor as defined in (Figure 9) In this case may be one pointwise sensor ( , ) is sufficient for strategic sensor on Γ of systems (30)-(42). Thus, we obtain the following result.

Internal filament case:
Consider the case where the information is given on the curve = ( ) with 1 (0, 1) ( Figure 10). If the measurements recovered by filament sensor ( , ) such that is symmetric with respect to the line = , then we have.

Boundary pointwise case:
Now, the system (30) is augmented with the following output function.
where = (0, 2 ) is the location of pointwise sensor ( , ) as defined in (Figure 11). Thus, we obtain the following result.

The domain = [ , ] × [ , ]
We discuss and examine different type of zone sensors.

Internal pointwise case:
Here, the system (37) is augmented with the following output function.

CONCLUSION
The notion of regional boundary strategic sensors have been developed and examined. A various regional boundary observability have been discussed and analyzed which per mit us to avoid some bad sensor locations. In addition, many interesting results concerning the choice of such sensors are given and illustrated in specific situations with diffusion systems. Thus, several questions still opened, for example, the simulati ons of this model are under consideration and the problem of finding an optimal sensor location ensuring such an objective.