STRATEGIC SENSORS AND ASYMPTOTIC REGIONAL BOUNDARY GRADIENT OBSERVATION

In this research paper an extension of asymptotical regional gradient observability concept to the case of boundary region of ∂Ω has been discussed and analyzed together with different types of strategic sensors. Our purpose is to show that, the number and location of sensor may be some interest in the existence of asymptotical regional gradient observation state in the boundary region for a parabolic system type.


Introduction
The controllability and observability are two important major concepts of modern control system theory. The observability measures means the ability of the particular sensor configuration to supply all necessary information to estimate or reconstruct all states of the system [1][2]. Notions of regional observability concepts are provided and developed by El Jai et al. [3][4] and extended to the case where the subregion is a part of the boundary of the system evolution domain, in [5,6]. The concept of regional gradient observability was developed in many real world problem, where one is cares about the knowledge of the gradient state only in a critical subregion of the system domain without the knowledge of the state itself. This concept was introduced for parabolic systems but little has been done for hyperbolic by Zerrik et al., Al-Saphory et al. and Boutoulout et al. [7,8,9,10].
The concept of asymptotic regional analysis was submitted recently by Al-Saphory , El Jai and et al. in [11,12,13,14,15,16], consists in studying the behavior of systems not in all the domain ߗ but only on particular regions ߱ or Г of the domain. Our interest in this study, is to establish some results on asymptotically regional boundary gradient observability in connection with sensors structure, their numbers, and locations on a subregion Г of the boundary ߲ߗ. The established results are also applied to the case of diffusion systems in two dimensions. The important reasons behind the study is to bring the light on the link between the asymptotic regional boundary gradient observability and sensor structure (location and number (Figure 1)). This paper is organized as follows: Section 2 focuses on the system considerations and the problem statement of asymptotic regional boundary gradient observability. In the third section , we will introduce and characterize regional boundary gradient strategic, sensors. In section four, an ,approach introduced which allows the determination of regional boundary gradient state on ߁ ீ . In section five, the relation between asymptotic regional boundary gradient observability and strategic sensor and also, we show that a gradient state which is not asymptotically observable in the usual sense may be asymptotically observable on Γ ୋ . Finally, the last section illustrates applications to a two-dimensional diffusion process, in addition to various situations which will be examined.

Asymptotic Regional Boundary Gradient Observability
In this subsection we give firstly, the statement of the problem with the hypothesis of considered system, and then the concept of asymptotic regional gradient observability is explained, and we provide a theorem, which gives the approach observed the current gradient state ‫,ߦ(ݔ∇‬ ‫)ݐ‬ of the original system (1) asymptotically.
2.1 Problem Statement ,The considered system is represented by the parabolic equations: Where Ω is the domain when the above system is defined as bounded open subset of ܴ with boundary ∂Ω, [0, ܶ] is the time interval for ܶ > 0, ‫ܣ‬ is a self adjoint linear differential operator of order two with compact resolvent, and which generates a strongly continuous semi-group (ܵ ‫))ݐ(‬ ௧ஹ on the state space ܺ = ‫ܪ‬ ଵ (Ω) which is Sobolev space of order one. The operators ‫ܤ‬ ∈ ‫ܴ(ܮ‬ , ܺ) and ‫ܥ‬ ∈ ‫,ܺ(ܮ‬ ܴ ), where ‫‬ given number of actuators and ‫ݍ‬ given the number of sensors. The initial gradient state ‫ݔ∇‬ (ߦ) is supposed to be unknown and located in ‫ܪ‬ ଵ (Ω ഥ ). The measurements of system (1) are obtained through internal or boundary zone or pointwise sensors which characterize the output function Under these hypotheses, the above system has a unique solution [1,17] The operator ∇ denotes the gradient is given by And the adjoint of ∇ denotes by ∇ * is given by where ‫ݒ‬ is a solution of the Dirichlet problem The operator defined as ߛ : ‫ܪ‬ ଵ (Ω) → ‫ܪ‬ ଵ ଶ ⁄ (߲Ω) is trace operator with order zero which is a linear, subjective and continuous. And, the extension of ߛ denoted by ߛ defined as ߛ: ‫ܪ(‬ ଵ (Ω)) → ‫ܪ(‬ ଵ ଶ ⁄ (߲Ω)) and the adjoints are respectively given by ߛ * , ߛ * .
For a sub-boundary region of ߲Ω the function ߯ defined by ‫ݔ‬ | means the restriction of the gradient state ‫ݔ∇‬ in a boundary subregion, and Where the adjoints are respectively given by ߯ * , ߯ * .
Definition 2.1: The autonomous system associated with system (1) and the output function (2) is called exactly regional boundary gradient observable (or exactly Γ ீ -observable) if Definition 2.2:The autonomous system associated with system (1) and the output function (2) is called weakly regional boundary gradient observable (or weakly Γ ீ -observable) if
Hence, we get the following result. Proof: The steps of the proof is depend on the rank condition in [19,20,5], the main difference step is that the rank condition for the proposition 2.5 need only to hold for ‫݇݊ܽݎ‬ ‫ܩ‬ = ‫ݎ‬ , ∀݉, ݉ = 1, … , ‫∎.ܬ‬

ࢣ -Observability and Strategic Sensor
The problem of ߁ ீ -observability is consists of estimation of the current gradient state asymptotically in a given boundary gradient sub-region of the boundary ߲ߗ. This approach is introduced by the following main theorem.

Remark 5.3:
We can get that: (1)Asymptotically observable system (on Ω) is ߁ ீ -observable, (2) An exactly ߁ ீ -observable system is ߁ ீ -observable, (3)A ߁ ீ -observable system is Γ ீ ଵ -observable, for every subset Γ ீ ଵ subset of ߁ ீ . Remark 5.4: The usefulness of this section is that a gradient state which is not asymptotically observable in the usual sense may be asymptotically observable on Γ ீ , this is illustrated by the following counter example.

.(Internal Filament Sensor)
Here the observation on the curve ߪ = ‫)ߛ(ܫ‬ with ߛ ∈ ‫ܥ‬ ଵ (0,1) (see Figure 6), hence, we have the following. Remark 6.4: We can extended this case into different type of sensors (internal or boundary, zonal or pointwise) as in [24].

Conclusion
The goal of this paper is related to the asymptotic regional boundary gradient observability provides with strategic sensors. In the sense, It permits us to avoid some "bad" sensor locations in order to guarantee asymptotic observability achievement for the system. Many benefit results related with choice of sensors structure are given and illustrated in specific situations. For future research, an interesting direction would be the extension of these results to the Neumann condition problem. Also, many questions are still opened and need to study the possibility to develop to the boundary gradient case, for example, the problems of gradient detectabitlity [25] and stabilizability [26] may be in quasi Banach space [27].