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.

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 of Ω , 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. Section 2 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

Let Ω 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: 𝜕𝑥𝜕𝑡(𝜉,𝑡)=𝐴𝑥(𝜉,𝑡)+Bu(𝑡)Θ𝑥(𝜉,0)=𝑥∘(𝜉)Ω𝑥(𝜂,𝑡)=0(2.1) augmented with the output function 𝑦(⋅,𝑡)=𝐶𝑥(⋅,𝑡),(2.2) 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: 𝑥(𝜉,𝑡)=𝑆𝐴(𝑡)𝑥∘(𝜉)+𝑡0𝑆𝐴(𝑡−𝜏)Bu(𝜏)𝑑𝜏.(2.3)

The problem is that how to observe exponentially the current state in a given subregion 𝜔 (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 parameters 𝐷𝑖 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 𝑦(⋅,𝑡)=𝐶𝑥(⋅,𝑡)=𝐷𝑖𝑥(𝜉,𝑡)𝑓𝑖(𝜉)𝑑𝜉.(2.4) 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 𝑦(⋅,𝑡)=𝐶𝑥(⋅,𝑡)=Ω𝑥(𝜉,𝑡)𝛿𝑏𝑖𝜉−𝑏𝑖𝑑𝜉.(2.5) 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: 𝑥(𝜉,𝑡)=𝑆𝐴(𝑡)𝑥∘(𝜉).(2.6)(ii)Define the operator 𝐾∶𝑋→𝑂, 𝑥⟶𝐶𝑆𝐴(⋅)𝑥(2.7) which is in the case of internal zone sensors is linear and bounded [16]. The adjoint operator 𝐾∗ of 𝐾 is defined by 𝐾∗𝑦=𝑡0𝑆∗𝐴(𝑠)𝐶∗𝑦(𝑠)𝑑𝑠.(2.8)(iii)For the region 𝜔 of the domain Ω, the operator 𝜒𝜔 is defined by 𝜒𝜔∶𝐿2(Ω)⟶𝐿2(𝜔)𝑥⟶𝜒𝜔𝑥=𝑥|𝜔,(2.9) 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 Im𝜒𝜔𝐾∗=𝐿2(𝜔)resp.Im𝜒𝜔𝐾∗(⋅)=𝐿2(𝜔).(2.10)(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 of 𝜔-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,forall𝑛,𝑛=1,…,𝐽 with 𝐺𝑛=𝐺𝑛𝑖𝑗=⎧⎪⎨⎪⎩𝜑𝑛𝑗,𝑓𝑖(⋅)𝐿2(𝐷𝐼)𝜑,inthezonecase,𝑛𝑗𝑏𝑖,inthepointwisecase,(2.11) 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 followsrank𝐺𝑛=𝑟𝑛,∀𝑛.(2.12) 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 in 𝐿2(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,‖‖𝜒𝜔𝑆𝐴‖‖(⋅)𝐿2(𝜔)≤𝑀𝜔𝑒−𝛼𝜔𝑡𝑡≥0.(2.13) If (𝑆𝐴(𝑡))𝑡≥0 is 𝜔𝐸-stable, then, for all 𝑥∘(⋅)∈𝐿2(Ω), the solution of autonomous system associated with (2.1) satisfies ‖‖𝑥∘‖‖(𝑡)𝐿2(𝜔)=‖‖𝜒𝜔𝑆𝐴(⋅)𝑥∘‖‖𝐿2(𝜔)≤𝑀𝜔𝑒−𝛼𝜔𝑡‖‖𝑥∘‖‖𝐿2(𝜔)(2.14) and then lim𝑡→∞‖𝑥(𝑡)‖𝐿2(𝜔)=0.(2.15)

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 𝜕𝑧𝜕𝑡(𝜉,𝑡)=𝐹𝜔𝑥(𝜉,𝑡)+𝐺𝜔𝑢(𝑡)+𝐻𝜔𝑦(𝑡)Θ𝑧(𝜉,0)=𝑧∘,(𝜉)Ω𝑧(𝜂,𝑡)=0(2.16) 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 exist 𝑀𝜔∈𝐿(𝑅𝑞,𝐿2(𝜔)) and 𝑁𝜔∈𝐿(𝐿2(𝜔)) such that 𝑀𝜔𝐶+𝑁𝜔𝜒𝜔𝑇=𝐼𝜔,(2.17)(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 𝜕𝑧𝜕𝑡(𝜉,𝑡)=𝐴𝑧(𝜉,𝑡)+Bu(𝑡)−𝐺𝜔.(Cz(𝜉,𝑡)−𝑦(⋅,𝑡))Θ𝑧(𝜉,0)=0Ω𝑧(𝜂,𝑡)=0(2.18)

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 and 𝜔𝐸-Observability

In this section, we give an approach which allows to construct an 𝜔𝐸-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: 𝜕𝑧𝜕𝑡(𝜉,𝑡)=𝐴𝑧(𝜉,𝑡)+Bu(𝑡)−𝐺𝜔𝐶(𝑧(𝜉,𝑡)−𝑦(⋅,𝑡))Θ𝑧(𝜉,0)=𝑧∘.(𝜉)Ω𝑧(𝜂,𝑡)=0(3.1)

Proof. The proof is limited to the case of zone sensors in the following steps.
Step 1. Under the assumptions of Section 2.1, 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 𝜕𝑥1𝜕𝑡(𝜉,𝑡)=𝐴1𝑥1𝑥(𝜉,𝑡)+𝑃Bu(𝑡)Θ1(𝜉,0)=𝑥∘1(𝑥𝜉)Ω1(𝜂,𝑡)=0(3.2) and 𝑥2(𝜉,𝑡) is the component state of the part of the system (2.1) given by 𝜕𝑥2𝜕𝑡(𝜉,𝑡)=𝐴2𝑥2𝑥(𝜉,𝑡)+(𝐼−𝑃)Bu(𝑡)Θ2(𝜉,0)=𝑥∘2(𝑥𝜉)Ω2.(𝜂,𝑡)=0(3.3) The operator 𝐴1 is represented by matrix of order (∑𝐽𝑛=1𝑟𝑛,∑𝐽𝑛=1𝑟𝑛) given by 𝐴1𝜆=diag1,…,𝜆1,𝜆2,…,𝜆2,…,𝜆j,…,𝜆j,𝐺PB=1tr,𝐺2tr,…,𝐺𝐽tr.(3.4)
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 𝐻1𝜔 such that 𝐴1−𝐻1𝜔𝐶 which satisfies the following: ∃𝑀1𝜔,𝛼1𝜔>0 such that ‖𝑒(𝐴1−𝐻1𝜔𝐶)𝑡‖≤𝑀1𝜔𝑒−𝛼1𝜔𝑡 and, then, we have ‖‖𝑥1‖‖(⋅,𝑡)𝐿2(𝜔)≤𝑀1𝜔𝑒−𝛼1𝜔𝑡‖‖𝑃𝑥∘‖‖𝐿2(𝜔).(3.5) Since the semigroup generated by the operator 𝐴2 is 𝜔𝐸-stable, there exists 𝑀2𝜔,𝛼2𝜔>0 such that ‖‖𝑥2(‖‖⋅,𝑡)𝐿2(𝜔)≤𝑀1𝜔𝑒−𝛼1𝜔‖‖(𝐼−𝑃)𝑥∘2(‖‖⋅)𝐿2(𝜔)+𝑡0𝑀2𝜔𝑒−𝛼2𝜔(𝑡−𝜏)‖‖(𝐼−𝑃)𝑥∘2(‖‖⋅)𝐿2(𝜔)(‖𝑢𝜏)‖𝑑𝜏(3.6) and therefore ‖𝑥(𝜉,𝑡)‖𝐿2(𝜔)→0 when 𝑡→∞. Finally, the system (2.1)-(2.2) is 𝜔𝐸-detectable.
Step 3. Let 𝑒(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 𝜕𝑒𝜕𝑡(𝜉,𝑡)=𝜕𝑥𝜕𝑡(𝜉,𝑡)−𝜕𝑧𝜕𝑡(𝜉,𝑡)=𝐴𝑥(𝜉,𝑡)+Bu(𝑡)−𝐴𝑧(𝜉,𝑡)−Bu(𝑡)+𝐻𝜔=𝐶(𝑧(𝜉,𝑡)−𝑥(⋅,𝑡))𝐴−𝐻𝜔𝐶𝑒(𝜉,𝑡).(3.7)
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: ∃𝑀𝜔,𝛼𝜔‖‖𝜒>0suchthat𝜔𝑆𝐻𝜔‖‖(𝑡)𝐿2(𝜔)≤𝑀𝜔𝑒−𝛼𝜔𝑡.(3.8) Finally, we have ‖𝑒(⋅,𝑡)‖𝐿2(𝜔)≤‖‖𝜒𝜔𝑆𝐻𝜔‖‖(𝑡)𝐿2(𝜔)‖‖𝑒∘‖‖(⋅)≤𝑀𝜔𝑒−𝛼𝜔𝑡‖‖𝑒∘‖‖(⋅)(3.9) 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 exactly 𝜔-observable is exponentially 𝜔-observable,(2)a system which is exponentially observable is exponentially 𝜔-observable,(3)a system which is exponentially 𝜔-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 𝜕𝑥𝜕𝑡(𝜉,𝑡)=Δ𝑥(𝜉,𝑡)+𝑥(𝜉,𝑡)Θ𝑥(𝜉,0)=𝑥∘(𝜉)Ω𝑧(𝜂,𝑡)=0(3.10) augmented with the output function 𝑦(𝑡)=Ω𝑥(𝜉,𝑡)𝛿𝜉−𝑏𝑖𝑑𝜉,(3.11) 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 𝜕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∘𝑧(𝜉)(0,1),(0,𝑡)=𝑧(1,𝑡)=0𝑡>0,(3.12) 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 domain Ω , the subregion 𝜔 , and locations 𝑏 𝑖 of internal pointwise sensors.

Now, we consider the region 𝜔=[0,𝛽]⊂(0,1) and the dynamical system 𝜕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∘𝑧(𝜉)(0,1),(0,𝑡)=𝑧(1,𝑡)=0𝑡>0,(3.13) 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 on Ω=(0,1)×(0,1) by the form 𝜕𝑥𝜉𝜕𝑡1,𝜉2𝜉,𝑡=Δ𝑥1,𝜉2𝑡𝑥𝜉+Bu(𝑡)Θ1,𝜉2,0=𝑥∘𝜉1,𝜉2Ω𝑥𝜂1,𝜂2,𝑡=0(4.1) 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 𝜑𝑖𝑗𝜉1,𝜉2=2𝛽1−𝛼1𝛽2−𝛼2𝜉sin𝑖𝜋1−𝛼1𝛽1−𝛼1𝜉sin𝑗𝜋2−𝛼2𝛽2−𝛼2(4.2) associated with eigenvalues 𝜆𝑖𝑗𝑖=−2𝛽1−𝛼12+𝑗2𝛽2−𝛼22.(4.3)

The following results give information on the location of internal zone or pointwise 𝜔-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 𝑦(𝑡)=𝐷𝑥𝜉1,𝜉2𝑓𝜉,𝑡1,𝜉2𝑑𝜉1𝑑𝜉2,(4.4) 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 𝛽1−𝛼12𝛽2−𝛼22∉𝑄,(4.5) 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 𝜕𝑧𝜉𝜕𝑡1,𝜉2𝜉,𝑡=Δ𝑧1,𝜉2𝜉,𝑡+𝑧1,𝜉2,𝑡+Bu(t)−𝐻𝜔<𝑥(⋅,𝑡),𝑓𝑖𝑧𝜉(⋅)>−𝐶𝑧(𝜉,𝑡)Θ1,𝜉2,0=𝑧∘𝜉1,𝜉2Ω𝑧𝜂1,𝜂2.,𝑡=0(4.6) Let the measurement support be rectangular with 𝜉𝐷=1−𝑙1,𝜉1+𝑙2×𝜉2−𝑙2,𝜉2+𝑙2∈Ω,(4.7) then we have the following result.


Figure 3 
Domain Ω , subregion 𝜔 , and location 𝐷 of internal zone sensor.

Corollary 4.1. If 𝑓1 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 𝑖𝜉∘1−𝛼1𝛽1−𝛼1,𝑖𝜉∘2−𝛼2𝛽2−𝛼2∉Nforsome𝑖=1,2,…,𝐽.(4.8)

4.2. Internal Pointwise Sensor

Let us consider the case of pointwise sensor located inside of Ω. The system (4.1) is augmented with the following output function: 𝑥𝜉𝑦(𝑡)=1,𝜉2𝛿𝜉,𝑡1−𝑏1,𝜉2−𝑏2𝑑𝜉1𝑑𝜉2,(4.9) where 𝑏=(𝑏1,𝑏2) is the location of pointwise sensor as defined in Figure 4.


Figure 4 
Rectangular domain Ω , 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 𝜕𝑧𝜉𝜕𝑡1,𝜉2𝜉,𝑡=Δ𝑧1,𝜉2𝜉,𝑡+𝑧1,𝜉2,𝑡+Bu(𝑡)+𝐻𝜔𝑥𝑏1,𝑏2Θ𝑧𝜉,𝑡−𝑦(𝑡)1,𝜉2,0=𝑧∘𝜉1,𝜉2Ω𝑧𝜂1,𝜂2.,𝑡=0(4.10)

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 curve 𝜎=Im(𝛾) with 𝛾∈𝐶1(0,1) (see Figure 5), then we have the following.


Figure 5 
Rectangular domain and location 𝜎 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 domain Ω=(𝐷,1) and 𝜔=(0,𝑟𝜔) where 𝜔⊂Ω and 0<𝑟𝜔<1 [10],(3)case of boundary sensors where 𝐶∉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.