Regional Boundary Gradient Detectability in Distributed Parameter Systems

The aim of this paper is study and explore the notion of the regional boundary gradient detectability in connection with the choice of strategic gradient sensors on sub-region of the considered system domain boundary. More precisely, the principal reason behind introducing this notion is that the possibility to design a dynamic system (may be called regional boundary gradient observer) which enable to estimate the unknown system state gradient. Then for linear infinite dimensional systems in a Hilbert space, we give various new results related with different measurements. In addition, we provided a description of the regional boundary gradient strategic sensors for completion the regional boundary gradient observability and regional boundary gradient detectability. Finally, we present and illustrate some applications of sensors structures which relate by regional boundary gradient detectability in diffusion distributed parameter systems.


Introduction
Many real problems in the sensors and detectability of distributed parameter systems can be reformulated as problems of infinite dimensional systems in a domain Ω [1][2][3]. The analysis of distributed parameter systems consists a set of notions as observability, detectability, stability, and observer is represented by infinite dimensional systems of partial differential equations [4][5][6][7][8]. Recently, the concept of regional and regional boundary analysis has been introduced by El Jai, Zerrik, Al-Saphory et al., for finite time horizon [9][10][11][12][13] and for infinite time horizon [14][15][16]. Thus, this concept which is given important tool to solve many problems in real world not in total domain of the considered system state but in sub -region of the domain or of the boundary [17][18][19]. Later, another direction of regional analysis has been extended is the system state gradient for observability, detectability and strategic sensors [20][21][22][23][24][25][26].
The purpose of this paper is to study and investigate the concept of regional boundary gradient detectability by using the choice of sensors. The principle reason for considering this case is that, firstly exist systems which are detectable on some boundary sub-region but are not detectable in any neighborhood of Γ ⊂ . Secondly, it is closer to a real situation, the treatment of water by using a bioreactor where the objective is to detect the concentration of substrate at the boundary output of the bioreactor in order the water regulation is achieved (for example see figure 1) [27]. The outline of this paper is organized as follow.
Section 2 concerns the class of considered system, definition, characterizations in connection with sensors and preliminaries of regional boundary gradient observability and detectability. Section 3, devotes to the problem of crossing method from internal region to boundary case by using trace operator estime the gradient of state in region Γ. Section 4, gives an applications to various situations of sensors locations on the regional boundary gradient detectability in diffusion parabolic distributed systems. Finally, section tackles the relation between regional boundary detectability of state gradient and regional boundary observer.

Regional Boundary Gradient Detectability
In this section, we extend the results in [15,21] to the boundary regional case by considering Γ ⊂ Ω. Thus, we will give some definitions which will be used to explain the notion of regional boundary gradient observability and detectability in ( 1 2 ⁄ ( Γ ) ) (state space).
with ∈ ( ̅ ) (domain of ̅ ) is a second order linear differential operator, which generates a strongly continuous semi-group ( ( ) ) ≥0 on the state space and is self-adjoint with compact resolvent.
• The operators ∈ ( , ) and ∈ ( , ) are depend on the structure of actuators and sensors as in [8] see (figure 2) which is a mathematical model is more general spatial case in (figure 1).
• Under the given assumption above, the system (1) has a unique solution given by the following form [1][2].
• The problem is how to detect the current state in a given sub-boundary Γ, and to give a sufficient conditio n for the existence of a regional boundary gradient detectability.
• The measurements can be obtained by the use of zone or pointwise sensors, which may be located in Ω or Ω [3].
• We first recall a sensors are defined by any couple ( , ) 1≤ ≤ where be a non-empty closed subsets of Ω ̅ , which is spatial supports of sensors and ∈ 2 ( ) represent the distributions of the sensing measurements on . Then, according to the choice of the parameters and , we have different types of sensors: • It may be zone, if ⊂ Ω ̅ and ∈ 2 ( ) . In this case, the operator is bounded and the output function (2) may be given by the form • It may be pointwise, if = { } with ∈ Ω ̅ and = (. − ), where is the Dirac mass concentrated in . In this case, the operator is un bounded and the output function (2) may be given by the form • It may be boundary zone, if Γ ⊂ Ω and ∈ 2 ( Γ ) , the output function (2) may be given by the form • The initial state 0 and its gradient ∇ 0 are supposed to be unknown, the problem concerns the reconstructio n of the initial gradient ∇ 0 on the sub-region Γ of the system domain Ω.
• Now, we consider the operator given by the form where is bounded linear operator as in [7][8]. Thus, the adjoint operator * of is defined by * : → , and represntedby the form * * = ∫ * ( ) * * ( ) 0 • The operator ∇ denotes the gradient is given by with the adjoint of ∇ denotes by ∇ * is given by where is a solution of the Dirichlet problem in Ω • The trace operator of order zero is described by [28] 0 : 1 ( Ω ) → 1 2 ⁄ ( Ω ) which is linear, subjective and continuous [2]. Thus, the extension of the trace operator of order zero which is denoted by defined as and the adjoints are respectively given by 0 * , * .
• For a sub-boundary Γ of Ω and let ̃Γ be the function defined by With | Γ is the restriction of the state to Γ, and where the adjoints are respectively given by ̃Γ * , Γ * .

Definitions and characterizations
In this sub-section, we introduce some definitions and descriptions of regional boundary gradient observability, detectability and strategic sensors, which is derived of [20][21][22][23][24][25][26]. Consider the autonomous system of (1) define by The solution of (7) is given by the following form Definition 2.1: (a) The system (7) augmented with the output function (2) (or the systems (7)- (2)) are said to be an exactly regionally boundary gradient observable on Γ, if The systems (7)-(2) are said to be an approximately regionally boundary gradient observable on Γ, if Now, we give a notion of the regional boundary gradient strategic sensors.

Definition 2.2:
A sensor ( , ) is said to be regionally boundary gradient strategic on Γ (or Γ -strategic), if the observed systems are an approximately Γ -observable.

Proposition 2.7:
If the systems (1)-(2) are an exactly Γ -observable, then it is Γ -detectable. This results gives the following inequality: ∃ > 0, such that Proof‫׃‬ We conclude the proof of this proposition is conclude from the results on observability considering Γ ∇ * . We have the following forms [2] 1-⊂ .

-Strategic sensors and -detectability
In the sub-section, we shall develop the characterization result that links an Γ -detectable and sensors structures.
For that purpose, we assume that the operator has a complete set of eigenfunctions 1 ( Ω ̅) (which is Sobolev space of order one) [1] denoted orthonormal in ( 1 2 ⁄ ( Γ ) ) and the associated eigenvalues are of multiplicity and suppose that the system (1) has unstable modes.
Thus, the sufficient condition of an Γ -detectability is given by the following theorem.
2-A sensor is Г -strategic if the system is exactly ̅ -observable.

Corollary 3.2:
From the previous results, we have.
2-If the system is approximately ̅ -observable, then the system is approximately Г -observable.
Definition ‫׃3. 3‬ The system (1) is -stable, if the solution of autonomous system associated with (1) together with (2) converges exponentially to zero when → ∞.
Proof: For the proof see ref. [15], with miner changment. Now, the method of crossing from internal -detectability into Γ -detectability will be given in the following theorem.

Application to Sensors Locations
In this section, we will explore different results related to different types of measurements and, we give the results on the locations of internal and boundary (pointwise and zone). Consider the two dimensional of diffusion system on a rectangular domain will may be described by the following form with measurements obtained by output function given as in (2) If we suppose that 1 2 2

Case of Zone Sensor
We study and discuss the following cases.

• Boundary Zone Sensor
Now, the measurements are given by the output with Г 0 ⊂ Ω is the boundary support of the sensor and ∈ 2 ( Γ 0 ) . In the case, where the support of the sensor ( , ) is one of side as in (figure 5), then we have the following proposition.

Case of Pointwise Sensor
We investigate different pointwise cases.

• Internal Pointwise Sensor
In this case, we can give the output function by the following form where = ( 1 , 2 ) is the location of pointwise sensor in Ω as defined in ( figure 6). Then we obtain the following result.

5.
-Detectability and -Observer In this section, we show that the regional boundary gradient detectability is the possibility to defined a regional boundary gradient observer which is enable to estimate the state gradient of considered system in part Γ of boundary Ω. This approach is derived from the previuos researchs as in [7,[13][14][15][16]21].
Remark 5.7: If the system is a regionally boundary gradient detectable, then it is possible to construct an regionally boundary gradient observer for the original system.

Conclusion
We have explored the original results devoted to the concept of regional boundary gradient detectability to the state gradient for parabolic distributed system in Hilbert sapce. Then, we have shown that, the possibility to design a dynamic system which is enable to estime the state gradient in sub -region Γ of the boundary Ω bu using detectability and strategic sensors in different situations. Moreover, many proble m still opend like the development of these results to case of haperbolic distributed parameter systems as in [25].