# Strategic Sensors and Regional Exponential Observability

Authors: Raheam A. Mansor Al-Saphory
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

The aim of this paper is to introduce the concept of regional exponential observability in connection with the strategic sensors. Then, we give characterization of such sensors in order that regional exponential observability can be achieved. The obtained results are applied to two-dimensional systems, and various cases of sensors are considered. We also show that there exists a dynamical system for diffusion system which is not exponentially observable in the usual sense but it may be regionally exponentially observable.


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## Body

## 1. Introduction

In system theory, the observability is related to the possibility of reconstruction of the state from the knowledge of system dynamics and the output [1–4]. The notion of regional analysis was extended by El Jai et al. [5, 6]. The study of this notion is motivated by certain concrete-real problem, in thermic, mechanic environment [7–9]. If a system is defined on a domain Ω and represented by the model as in (Figure 1), then we are interested in the regional state on ω of the domain Ω.Figure 1
The domain ofΩ, the subregion ω, and the sensors locations.The concept of regional asymptotic analysis was introduced recently by Al-Saphory and El Jai in [10–12], consisting in studying the behaviour of the system not in all the domain Ω but only on particular region ω of the domain.The purpose of this paper is to give some results related to the link between regional exponential observability and strategic sensors. We consider a class of distributed system and we explore various results connected with the different types of measurements, domains, and boundary conditions.The paper is organized as follows. Section2 devotes to the introduction of exponential regional observability problem. We give the formulation problem and preliminaries. We need some notions concerning the exponential behaviour (ω-strategic sensor, ω-detectability, and ω-observer). Section 3 is related to the characterization notion of ωE-observable by the use of strategic sensors. In Section 4, we illustrate applications with many situations of sensor locations.


## 2. Regional Exponential Observability

### 2.1. Problem Statement

LetΩ be an open bounded subset of Rn, with boundary ∂Ω and let [0,T], T>0 be a time measurement interval. Suppose that ω be a nonempty given subregion of Ω. We denote Θ=Ω×(0,∞) and ∏=∂Ω×(0,∞). The considered distributed parameter systems is described by the following parabolic systems:
(2.1)∂x∂t(ξ,t)=Ax(ξ,t)+Bu(t)Θx(ξ,0)=x∘(ξ)Ωx(η,t)=0∏
augmented with the output function
(2.2)y(⋅,t)=Cx(⋅,t),
where A is a second-order linear differential operator, which generates a strongly continuous semigroup (SA(t))t≥0 on the Hilbert space X=L2(Ω) and is self-adjoint with compact resolvent. The operatorsB∈L(Rp,X) and C∈L(Rq,X) depend on the structures of actuators and sensors [13, 14]. The spaces X,U, and O are separable Hilbert spaces where X is the state space, U=L2(0,∞,Rp) is the control space, and O=L2(0,∞,Rq) is the observation space, where p and q are the numbers of actuators and sensors. Under the given assumption [15], the system (2.1) has a unique solution:
(2.3)x(ξ,t)=SA(t)x∘(ξ)+∫0tSA(t-τ)Bu(τ)dτ.The problem is that how to observe exponentially the current state in a given subregionω (see Figure 1), using convenient sensors and to give a sufficient condition for the existence of a regional exponential observability.


### 2.2.ω-Strategic Sensor

The purpose of this subsection is to give the characterization for sensors in order that the system (2.1) is regionally exponentially observable in ω. (i)
Sensors are any couple(Di,fi)1≤i≤q where Di denote closed subsets of Ω¯, which is spatial supports of sensors and fi∈L2(Di) define the spatial distributions of measurements on Di.According to the choice of the parametersDi and fi, we have various types of sensors. These sensors may be types of zones when Di⊂Ω. The output function (2.2) can be written in the form
(2.4)y(⋅,t)=Cx(⋅,t)=∫Dix(ξ,t)fi(ξ)dξ.
Sensors may also be pointwise when Di={bi} and fi=δbi(x-bi) where δbi is Dirac mass concentrated in bi. Then, the output function (2.2) can be given by the form
(2.5)y(⋅,t)=Cx(⋅,t)=∫Ωx(ξ,t)δbi(ξ-bi)dξ.
In the case of internal pointwise sensors, the operator C is unbounded and some precaution must be taken in [13, 14]. In the case when (2.1) is autonomous system, (2.3) allows to give the following equation:
(2.6)x(ξ,t)=SA(t)x∘(ξ).(ii)
Define the operatorK:X→O,
(2.7)x⟶CSA(⋅)x
which is in the case of internal zone sensors is linear and bounded [16]. The adjoint operator K* of K  is defined by
(2.8)K*y=∫0tSA*(s)C*y(s)ds.(iii)
For the regionω of the domain Ω, the operator χω is defined by
(2.9)χω:L2(Ω)⟶L2(ω)x⟶χωx=x|ω,
where x|ω is the restriction of x to ω.(iv)
An autonomous system associated to (2.1)-(2.2) is exactly (resp., weakly) ω-observable if
(2.10)Im⁡χωK*=L2(ω)(resp.Im⁡χωK*(⋅)̅=L2(ω)).(v)
A sequence of sensors(Di,fi)1≤i≤q is ω-strategic if the system (2.1)-(2.2) is weakly ω-observable [5].The concept ofω-strategic has been extended to the regional boundary case as in [17]. Assume that the set (φnj) of eigenfunctions of  L2(Ω) orthonormal in L2(ω) is associated with eigenvalues λn of multiplicity rn and suppose that the system (2.1) has J unstable modes. Then, we have the following result.Proposition 2.1.
The sequence of sensors(Di,fi)1≤i≤q is ω-strategic if and only if(1)
q≥r,(2)
rankGn=rn,for alln,n=1,…,J with
(2.11)Gn=(Gn)ij={〈φnj,fi(⋅)〉L2(DI),inthezonecase,φnj(bi),inthepointwisecase,
where suprn=r and J=1,…,rn.Proof.
The proof of this proposition is similar to the rank condition in [16]; the main difference is that the rank condition is as follows(2.12)rank⁡Gn=rn,∀n.
For Proposition 2.1., we need only to hold for rank Gn=rn,foralln,n=1,…,J.


### 2.3.ωE-Observability

Regional exponential observability characterization needs some notions which are related to the exponential behaviour (stability, detectability, and observer). The concept of exponential behaviour has been extended recently by Al-Saphory and El Jai as in [12].Definition 2.2.
A semigroup is exponentially regionally stable inL2(ω) (or ωE-stable) if, for every initial state x∘(·)∈L2(Ω), the solution of the autonomous system associated with (2.1) converges exponentially to zero when  t→∞.Definition 2.3.
The system (2.1) is said to be exponentially stable on ω (or ωE-stable) if the operator A generates a semigroup which is exponentially stable in L2(ω). It is easy to see that the system (2.1) is ωE-stable if and only if, for some positive constants Mω and αω,(2.13)‖χωSA(⋅)‖L2(ω)≤Mωe-αωtt≥0.
If (SA(t))t≥0 is ωE-stable, then, for all x∘(·)∈L2(Ω), the solution of autonomous system associated with (2.1) satisfies
(2.14)‖x∘(t)‖L2(ω)=‖χωSA(⋅)x∘‖L2(ω)≤Mωe-αωt‖x∘‖L2(ω)
and then
(2.15)lim⁡t→∞‖x(t)‖L2(ω)=0.Definition 2.4.
The system (2.1) together with output (2.2) is said to be exponentially detectable on ω (or ωE-detectable) if there exists an operator Hω:Rq→L2(ω) such that (A-HωC) generates a strongly continuous semigroup  (SHω(t))t≥0 which is ωE-stable.Definition 2.5.
Consider the system (2.1)-(2.2) together with the dynamical system
(2.16)∂z∂t(ξ,t)=Fωx(ξ,t)+Gωu(t)+Hωy(t)Θz(ξ,0)=z∘(ξ)Ωz(η,t)=0∏,
where Fω generates a strongly continuous semigroup  (SFω(t))t≥0 which is stable on Hilbert space Z,Gω∈L(Rp,Z) and Hω∈L(Rq,Z). The system (2.16) defines an ωE-estimator for χωTx(ξ,t) if(1)
limt→∞∥z(·,t)-χωTx(·,t)∥L2(ω)=0,(2)
χωT maps D(A) in D(Fω) where z(ξ,t) is the solution of the system (2.16).Definition 2.6.
The system (2.16) specifies an ωE-observer for the system (2.1)-(2.2) if the following conditions hold:(1)
there existMω∈L(Rq,L2(ω)) and Nω∈L(L2(ω)) such that
(2.17)MωC+NωχωT=Iω,(2)
χωTA+FωχωT=GωC and Hω=χωTB,(3)
the system (2.16) defines an ωE-observer.Definition 2.7.
The system (2.16) is said to be ωE-observer for the system (2.1)-(2.2) if X=Z and χωT=Iω. In this case, we have  Fω=A-GωC  and Hω=B. Then, the dynamical system (2.16) becomes
(2.18)∂z∂t(ξ,t)=Az(ξ,t)+Bu(t)-Gω(Cz(ξ,t)-y(⋅,t))Θz(ξ,0)=0Ωz(η,t)=0∏.Definition 2.8.
The system (2.1)-(2.2) is ωE-observable if there exists a dynamical system which is exponential ωE-observer, for the original system.  Now, the approach which is observed is that the current state x(ξ,t) exponentially is given by the following result.


## 2.1. Problem Statement

LetΩ be an open bounded subset of Rn, with boundary ∂Ω and let [0,T], T>0 be a time measurement interval. Suppose that ω be a nonempty given subregion of Ω. We denote Θ=Ω×(0,∞) and ∏=∂Ω×(0,∞). The considered distributed parameter systems is described by the following parabolic systems:
(2.1)∂x∂t(ξ,t)=Ax(ξ,t)+Bu(t)Θx(ξ,0)=x∘(ξ)Ωx(η,t)=0∏
augmented with the output function
(2.2)y(⋅,t)=Cx(⋅,t),
where A is a second-order linear differential operator, which generates a strongly continuous semigroup (SA(t))t≥0 on the Hilbert space X=L2(Ω) and is self-adjoint with compact resolvent. The operatorsB∈L(Rp,X) and C∈L(Rq,X) depend on the structures of actuators and sensors [13, 14]. The spaces X,U, and O are separable Hilbert spaces where X is the state space, U=L2(0,∞,Rp) is the control space, and O=L2(0,∞,Rq) is the observation space, where p and q are the numbers of actuators and sensors. Under the given assumption [15], the system (2.1) has a unique solution:
(2.3)x(ξ,t)=SA(t)x∘(ξ)+∫0tSA(t-τ)Bu(τ)dτ.The problem is that how to observe exponentially the current state in a given subregionω (see Figure 1), using convenient sensors and to give a sufficient condition for the existence of a regional exponential observability.


## 2.2.ω-Strategic Sensor

The purpose of this subsection is to give the characterization for sensors in order that the system (2.1) is regionally exponentially observable in ω. (i)
Sensors are any couple(Di,fi)1≤i≤q where Di denote closed subsets of Ω¯, which is spatial supports of sensors and fi∈L2(Di) define the spatial distributions of measurements on Di.According to the choice of the parametersDi and fi, we have various types of sensors. These sensors may be types of zones when Di⊂Ω. The output function (2.2) can be written in the form
(2.4)y(⋅,t)=Cx(⋅,t)=∫Dix(ξ,t)fi(ξ)dξ.
Sensors may also be pointwise when Di={bi} and fi=δbi(x-bi) where δbi is Dirac mass concentrated in bi. Then, the output function (2.2) can be given by the form
(2.5)y(⋅,t)=Cx(⋅,t)=∫Ωx(ξ,t)δbi(ξ-bi)dξ.
In the case of internal pointwise sensors, the operator C is unbounded and some precaution must be taken in [13, 14]. In the case when (2.1) is autonomous system, (2.3) allows to give the following equation:
(2.6)x(ξ,t)=SA(t)x∘(ξ).(ii)
Define the operatorK:X→O,
(2.7)x⟶CSA(⋅)x
which is in the case of internal zone sensors is linear and bounded [16]. The adjoint operator K* of K  is defined by
(2.8)K*y=∫0tSA*(s)C*y(s)ds.(iii)
For the regionω of the domain Ω, the operator χω is defined by
(2.9)χω:L2(Ω)⟶L2(ω)x⟶χωx=x|ω,
where x|ω is the restriction of x to ω.(iv)
An autonomous system associated to (2.1)-(2.2) is exactly (resp., weakly) ω-observable if
(2.10)Im⁡χωK*=L2(ω)(resp.Im⁡χωK*(⋅)̅=L2(ω)).(v)
A sequence of sensors(Di,fi)1≤i≤q is ω-strategic if the system (2.1)-(2.2) is weakly ω-observable [5].The concept ofω-strategic has been extended to the regional boundary case as in [17]. Assume that the set (φnj) of eigenfunctions of  L2(Ω) orthonormal in L2(ω) is associated with eigenvalues λn of multiplicity rn and suppose that the system (2.1) has J unstable modes. Then, we have the following result.Proposition 2.1.
The sequence of sensors(Di,fi)1≤i≤q is ω-strategic if and only if(1)
q≥r,(2)
rankGn=rn,for alln,n=1,…,J with
(2.11)Gn=(Gn)ij={〈φnj,fi(⋅)〉L2(DI),inthezonecase,φnj(bi),inthepointwisecase,
where suprn=r and J=1,…,rn.Proof.
The proof of this proposition is similar to the rank condition in [16]; the main difference is that the rank condition is as follows(2.12)rank⁡Gn=rn,∀n.
For Proposition 2.1., we need only to hold for rank Gn=rn,foralln,n=1,…,J.


## 2.3.ωE-Observability

Regional exponential observability characterization needs some notions which are related to the exponential behaviour (stability, detectability, and observer). The concept of exponential behaviour has been extended recently by Al-Saphory and El Jai as in [12].Definition 2.2.
A semigroup is exponentially regionally stable inL2(ω) (or ωE-stable) if, for every initial state x∘(·)∈L2(Ω), the solution of the autonomous system associated with (2.1) converges exponentially to zero when  t→∞.Definition 2.3.
The system (2.1) is said to be exponentially stable on ω (or ωE-stable) if the operator A generates a semigroup which is exponentially stable in L2(ω). It is easy to see that the system (2.1) is ωE-stable if and only if, for some positive constants Mω and αω,(2.13)‖χωSA(⋅)‖L2(ω)≤Mωe-αωtt≥0.
If (SA(t))t≥0 is ωE-stable, then, for all x∘(·)∈L2(Ω), the solution of autonomous system associated with (2.1) satisfies
(2.14)‖x∘(t)‖L2(ω)=‖χωSA(⋅)x∘‖L2(ω)≤Mωe-αωt‖x∘‖L2(ω)
and then
(2.15)lim⁡t→∞‖x(t)‖L2(ω)=0.Definition 2.4.
The system (2.1) together with output (2.2) is said to be exponentially detectable on ω (or ωE-detectable) if there exists an operator Hω:Rq→L2(ω) such that (A-HωC) generates a strongly continuous semigroup  (SHω(t))t≥0 which is ωE-stable.Definition 2.5.
Consider the system (2.1)-(2.2) together with the dynamical system
(2.16)∂z∂t(ξ,t)=Fωx(ξ,t)+Gωu(t)+Hωy(t)Θz(ξ,0)=z∘(ξ)Ωz(η,t)=0∏,
where Fω generates a strongly continuous semigroup  (SFω(t))t≥0 which is stable on Hilbert space Z,Gω∈L(Rp,Z) and Hω∈L(Rq,Z). The system (2.16) defines an ωE-estimator for χωTx(ξ,t) if(1)
limt→∞∥z(·,t)-χωTx(·,t)∥L2(ω)=0,(2)
χωT maps D(A) in D(Fω) where z(ξ,t) is the solution of the system (2.16).Definition 2.6.
The system (2.16) specifies an ωE-observer for the system (2.1)-(2.2) if the following conditions hold:(1)
there existMω∈L(Rq,L2(ω)) and Nω∈L(L2(ω)) such that
(2.17)MωC+NωχωT=Iω,(2)
χωTA+FωχωT=GωC and Hω=χωTB,(3)
the system (2.16) defines an ωE-observer.Definition 2.7.
The system (2.16) is said to be ωE-observer for the system (2.1)-(2.2) if X=Z and χωT=Iω. In this case, we have  Fω=A-GωC  and Hω=B. Then, the dynamical system (2.16) becomes
(2.18)∂z∂t(ξ,t)=Az(ξ,t)+Bu(t)-Gω(Cz(ξ,t)-y(⋅,t))Θz(ξ,0)=0Ωz(η,t)=0∏.Definition 2.8.
The system (2.1)-(2.2) is ωE-observable if there exists a dynamical system which is exponential ωE-observer, for the original system.  Now, the approach which is observed is that the current state x(ξ,t) exponentially is given by the following result.


## 3. Strategic Sensors andωE-Observability

In this section, we give an approach which allows to construct anωE-estimator of x(ξ,t). This method avoids the consideration of initial state [6]; it enables to observe exponentially the current state in ω without needing the effect of the initial state of the considered system.Theorem 3.1.
Suppose that the sequence of sensors(Di,fi)1≤i≤q is ω-strategic and the spectrum of A contain J eigenvalues with nonnegative real parts. Then, the system (2.1)-(2.2) is ωE-observable by the following dynamical system:
(3.1)∂z∂t(ξ,t)=Az(ξ,t)+Bu(t)-GωC(z(ξ,t)-y(⋅,t))Θz(ξ,0)=z∘(ξ)Ωz(η,t)=0∏.Proof.
The proof is limited to the case of zone sensors in the following steps.
Step 1.
Under the assumptions of Section2.1, 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 x(ξ,t)=[x1(ξ,t)+x2(ξ,t)]tr where x1(ξ,t) is the state component of the unstable part of the system (2.1) and may be written in the form
(3.2)∂x1∂t(ξ,t)=A1x1(ξ,t)+PBu(t)Θx1(ξ,0)=x∘1(ξ)Ωx1(η,t)=0∏
and x2(ξ,t) is the component state of the part of the system (2.1) given by
(3.3)∂x2∂t(ξ,t)=A2x2(ξ,t)+(I-P)Bu(t)Θx2(ξ,0)=x∘2(ξ)Ωx2(η,t)=0∏.
The operator A1 is represented by matrix of order (∑n=1Jrn,∑n=1Jrn) given by
(3.4)A1=diag⁡⌊λ1,…,λ1,λ2,…,λ2,…,λj,…,λj⌋,PB=[G1tr⁡,G2tr⁡,…,GJtr⁡].
Step 2.
Since the sequence suite of sensors(Di,fi)1≤i≤q is ω-strategic for the unstable part of the system (2.1). The subsystem (3.2) is weakly ω-observable [5], and since it is of finite dimensional, it is exactly ω-observable [2]. Therefore, it is ωE-detectable and hence there exists an operator Hω1 such that  A1-Hω1C which satisfies the following: ∃Mω1,αω1>0 such that ∥e(A1-Hω1C)t∥≤Mω1e-αω1t and, then, we have
(3.5)‖x1(⋅,t)‖L2(ω)≤Mω1e-αω1t‖Px∘‖L2(ω).
Since the semigroup generated by the operator A2 is ωE-stable, there exists Mω2,αω2>0 such that
(3.6)‖x2(⋅,t)‖L2(ω)≤Mω1e-αω1‖(I-P)x∘2(⋅)‖L2(ω)+∫0tMω2e-αω2(t-τ)‖(I-P)x∘2(⋅)‖L2(ω)‖u(τ)‖dτ
and therefore  ∥x(ξ,t)∥L2(ω)→0 when t→∞. Finally, the system (2.1)-(2.2) is ωE-detectable.
Step 3.
Lete(ξ,t)=x(ξ,t)-z(ξ,t) where z(ξ,t) is solution of the system (3.1). Driving the above equation and using (2.1) and (3.1), we obtain
(3.7)∂e∂t(ξ,t)=∂x∂t(ξ,t)-∂z∂t(ξ,t)=Ax(ξ,t)+Bu(t)-Az(ξ,t)-Bu(t)+HωC(z(ξ,t)-x(⋅,t))=(A-HωC)e(ξ,t).
Since the system (2.1)-(2.2) is ωE-detectable, there exists an operator Hω∈L(Rq,L2(ω)), such that the operator (A-HωC) generates exponentially regionally stable, strongly continuous semigroup (SHω(t))t≥0 on L2(ω) which satisfies the following relations:
(3.8)∃Mω,αω>0suchthat‖χωSHω(t)‖L2(ω)≤Mωe-αωt.
Finally, we have
(3.9)‖e(⋅,t)‖L2(ω)≤‖χωSHω(t)‖L2(ω)‖e∘(⋅)‖≤Mωe-αωt‖e∘(⋅)‖
with e∘(·)=x∘(·)-z∘(·) and therefore e(ξ,t) converges exponentially to zero as t→∞. Thus, the dynamical system (3.1) observes exponentially the regional state x(ξ,t) of the system original system and (2.1)-(2.2) is ωE-observable.Remark 3.2.
We can deduce that(1)
a system which is exactlyω-observable is exponentially ω-observable,(2)
a system which is exponentially observable is exponentiallyω-observable,(3)
a system which is exponentiallyω-observable is exponentially ω1-observable, in every subset ω1 of ω, but the converse is not true. This may be proven in the following example.Example 3.3.
Consider the system(3.10)∂x∂t(ξ,t)=Δx(ξ,t)+x(ξ,t)Θx(ξ,0)=x∘(ξ)Ωz(η,t)=0∏
augmented with the output function
(3.11)y(t)=∫Ωx(ξ,t)δ(ξ-bi)dξ,
where Ω=(0,1) and bi∈Ω are the location of sensors (bi,δbi) as in (Figure 2). The operator A=(Δ+1) generates a strongly continuous semigroup (SA(t))t≥0 on the Hilbert space L2(ω) [15]. Consider the dynamical system
(3.12)∂z∂t(ξ,t)=Δz(ξ,t)+z(ξ,t)-HC(z(ξ,t)-x(ξ,t))(0,1),t>0,z(ξ,0)=z∘(ξ)(0,1),z(0,t)=z(1,t)=0t>0,
where H∈L(Rq,Z),Z is the Hilbert space, and C:Z→Rq is linear  operator. If bi∈Q, then the sensors (bi,δbi) are not strategic for the unstable subsystem (3.10) [1] and therefore the system (3.10)-(3.11) is not exponentially detectable in Ω [14]. Then, the dynamical system (3.12) is not observer and then (3.10)-(3.11) is not exponentially observable [16].Figure 2
The domainΩ, the subregion ω, and  locations bi of internal pointwise sensors.Now, we consider the regionω=[0,β]⊂(0,1) and the dynamical system
(3.13)∂z∂t(ξ,t)=Δz(ξ,t)+z(ξ,t)-HωC(z(ξ,t)-x(ξ,t))(0,1),t>0,z(ξ,0)=z∘(ξ)(0,1),z(0,t)=z(1,t)=0t>0,
where Hω∈L(Rq,L2(ω)). If bi/β∉Q, then the sensors (bi,δbi) are ω-strategic for the unstable subsystem of (3.10) [7] and then the system (3.10)-(3.11) is ωE-detectable. Therefore, the system (3.10)-(3.11) is ωE-observable by ωE-observer [12].


## 4. Application to Sensor Location

In this section, we present an application of the above results to a two-dimensional system defined onΩ=(0,1)×(0,1) by the form
(4.1)∂x∂t(ξ1,ξ2,t)=Δx(ξ1,ξ2t)+Bu(t)Θx(ξ1,ξ2,0)=x∘(ξ1,ξ2)Ωx(η1,η2,t)=0∏
together with output function by (2.4), (2.5). Let ω=(α1,β1)×(α2,β2) be the considered region which is subset of (0,1) × (0,1). In this case, the eigenfunctions of system (4.1) are given by
(4.2)φij(ξ1,ξ2)=2(β1-α1)(β2-α2)siniπ(ξ1-α1β1-α1)sinjπ(ξ2-α2β2-α2)
associated with eigenvalues
(4.3)λij=-(i2(β1-α1)2+j2(β2-α2)2).The following results give information on the location of internal zone or pointwiseω-strategic sensors.


### 4.1. Internal Zone Sensor

Consider the system (4.1) together with output function (2.2) where the sensor supports D are located in Ω. The output (2.2) can be written by the form
(4.4)y(t)=∫Dx(ξ1,ξ2,t)f(ξ1,ξ2)dξ1dξ2,
where D⊂Ω is location of zone sensor and f∈L2(D). In this case of Figure 3, the eigenfunctions and the eigenvalues are given by (4.2) and (4.3). However,  if we suppose that
(4.5)(β1-α1)2(β2-α2)2∉Q,
then r=1 and one sensor may be sufficient to achieve ωE-observability [18]. In this case, the dynamical system (3.1) is given by
(4.6)∂z∂t(ξ1,ξ2,t)=Δz(ξ1,ξ2,t)+z(ξ1,ξ2,t)+Bu(t)-Hω<x(⋅,t),fi(⋅)>-Cz(ξ,t)Θz(ξ1,ξ2,0)=z∘(ξ1,ξ2)Ωz(η1,η2,t)=0∏.
Let the measurement support be rectangular with
(4.7)D=[ξ1-l1,ξ1+l2]×[ξ2-l2,ξ2+l2]∈Ω,
then we have the following result.Figure 3
DomainΩ, subregion ω, and location D of internal zone sensor.Corollary 4.1.
Iff1 is symmetric about ξ1=ξ∘1 and f2 is symmetric about ξ2=ξ∘2, then the system (4.1)–(4.4) is ωE-observable by the dynamical system (4.6) if
(4.8)i(ξ∘1-α1)(β1-α1),i(ξ∘2-α2)(β2-α2)∉Nforsomei=1,2,…,J.


### 4.2. Internal Pointwise Sensor

Let us consider the case of pointwise sensor located inside ofΩ. The system (4.1) is augmented with the following output function:
(4.9)y(t)=∫x(ξ1,ξ2,t)δ(ξ1-b1,ξ2-b2)dξ1dξ2,
where b=(b1,b2) is the location of pointwise sensor as defined  in Figure 4.Figure 4
Rectangular domainΩ, region ω, and location b of internal pointwise sensor.If(β1-α1)/(β2-α2)∉Q, then m=1 and one sensor (b,δb) may be sufficient for ωE-observability. Then, the dynamical system is given by
(4.10)∂z∂t(ξ1,ξ2,t)=Δz(ξ1,ξ2,t)+z(ξ1,ξ2,t)+Bu(t)+Hω(x(b1,b2,t)-y(t))Θz(ξ1,ξ2,0)=z∘(ξ1,ξ2)Ωz(η1,η2,t)=0∏.Thus, we obtain the following.Corollary 4.2.
The system (4.1)–(4.9) is not ωE-observable by the dynamical system (4.10) if i(b1-α1)/(β1-α1) and i(b2-α2)/(β2-α2)∈N, for every i,1≤i≤J.


### 4.3. Internal Filament Sensor

Consider the case of the observation on the curveσ=Im(γ) with γ∈C1(0,1) (see Figure 5), then we have the following.Figure 5
Rectangular domain and locationσ of internal filament sensors.Corollary 4.3.
If  the observation recovered by filament sensor(σ,δσ) such that it is symmetric with respect to the line ξ=ξ∘, then the system (4.1)–(4.9) is not ωE-observable by (4.10)  if i(ξ∘1-α1)/(β1-α1) and i(ξ∘2-α2)/(β2-α2)∈N for all i=1,...,q.Remark 4.4.
These results can be extended to the following:(1)
case of Neumann or mixed boundary conditions [1, 2],(2)
case of disc domainΩ=(D,1) and ω=(0,rω) where ω⊂Ω and 0<rω<1 [10],(3)
case of boundary sensors whereC∉L(X,Rq); we refer to see [13, 14].


## 4.1. Internal Zone Sensor

Consider the system (4.1) together with output function (2.2) where the sensor supports D are located in Ω. The output (2.2) can be written by the form
(4.4)y(t)=∫Dx(ξ1,ξ2,t)f(ξ1,ξ2)dξ1dξ2,
where D⊂Ω is location of zone sensor and f∈L2(D). In this case of Figure 3, the eigenfunctions and the eigenvalues are given by (4.2) and (4.3). However,  if we suppose that
(4.5)(β1-α1)2(β2-α2)2∉Q,
then r=1 and one sensor may be sufficient to achieve ωE-observability [18]. In this case, the dynamical system (3.1) is given by
(4.6)∂z∂t(ξ1,ξ2,t)=Δz(ξ1,ξ2,t)+z(ξ1,ξ2,t)+Bu(t)-Hω<x(⋅,t),fi(⋅)>-Cz(ξ,t)Θz(ξ1,ξ2,0)=z∘(ξ1,ξ2)Ωz(η1,η2,t)=0∏.
Let the measurement support be rectangular with
(4.7)D=[ξ1-l1,ξ1+l2]×[ξ2-l2,ξ2+l2]∈Ω,
then we have the following result.Figure 3
DomainΩ, subregion ω, and location D of internal zone sensor.Corollary 4.1.
Iff1 is symmetric about ξ1=ξ∘1 and f2 is symmetric about ξ2=ξ∘2, then the system (4.1)–(4.4) is ωE-observable by the dynamical system (4.6) if
(4.8)i(ξ∘1-α1)(β1-α1),i(ξ∘2-α2)(β2-α2)∉Nforsomei=1,2,…,J.


## 4.2. Internal Pointwise Sensor

Let us consider the case of pointwise sensor located inside ofΩ. The system (4.1) is augmented with the following output function:
(4.9)y(t)=∫x(ξ1,ξ2,t)δ(ξ1-b1,ξ2-b2)dξ1dξ2,
where b=(b1,b2) is the location of pointwise sensor as defined  in Figure 4.Figure 4
Rectangular domainΩ, region ω, and location b of internal pointwise sensor.If(β1-α1)/(β2-α2)∉Q, then m=1 and one sensor (b,δb) may be sufficient for ωE-observability. Then, the dynamical system is given by
(4.10)∂z∂t(ξ1,ξ2,t)=Δz(ξ1,ξ2,t)+z(ξ1,ξ2,t)+Bu(t)+Hω(x(b1,b2,t)-y(t))Θz(ξ1,ξ2,0)=z∘(ξ1,ξ2)Ωz(η1,η2,t)=0∏.Thus, we obtain the following.Corollary 4.2.
The system (4.1)–(4.9) is not ωE-observable by the dynamical system (4.10) if i(b1-α1)/(β1-α1) and i(b2-α2)/(β2-α2)∈N, for every i,1≤i≤J.


## 4.3. Internal Filament Sensor

Consider the case of the observation on the curveσ=Im(γ) with γ∈C1(0,1) (see Figure 5), then we have the following.Figure 5
Rectangular domain and locationσ of internal filament sensors.Corollary 4.3.
If  the observation recovered by filament sensor(σ,δσ) such that it is symmetric with respect to the line ξ=ξ∘, then the system (4.1)–(4.9) is not ωE-observable by (4.10)  if i(ξ∘1-α1)/(β1-α1) and i(ξ∘2-α2)/(β2-α2)∈N for all i=1,...,q.Remark 4.4.
These results can be extended to the following:(1)
case of Neumann or mixed boundary conditions [1, 2],(2)
case of disc domainΩ=(D,1) and ω=(0,rω) where ω⊂Ω and 0<rω<1 [10],(3)
case of boundary sensors whereC∉L(X,Rq); we refer to see [13, 14].


## 5. Conclusion

The concept developed in this paper is related to the regional exponential observability in connection with the strategic sensors. It permits us to avoid some “bad” sensor locations. Various interesting results concerning the choice of sensors structure are given and illustrated in specific situations. Many questions still opened. This is the case of, for example, the problem of finding the optimal sensor location ensuring such an objective. The dual result of regional controllability concept is under consideration.



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*Source: 673052-2011-08-16.xml*