Strategic Sensors and Regional Exponential Observability

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 𝜔_𝐸-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 𝑅^𝑛, 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,𝑥𝑡)(=𝜉,0𝐴)𝑥=(𝜉𝑥,_∘𝑡)+Bu(𝑡)Θ𝑥(𝜂,𝑡)=0(𝜉)Ω(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 (𝑆_𝐴(𝑡))_𝑡≥0 on the Hilbert space 𝑋=𝐿²(Ω) and is self-adjoint with compact resolvent. The operators𝐵∈𝐿(𝑅^𝑝,𝑋) and 𝐶∈𝐿(𝑅^𝑞,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,∞,𝑅^𝑝) is the control space, and 𝑂=𝐿²(0,∞,𝑅^𝑞) 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)=𝑆_𝐴(𝑡)𝑥_∘(𝜉)+^𝑡0𝑆_𝐴(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 (𝐷_𝑖,𝑓_𝑖)_{1≤𝑖≤𝑞} where 𝐷_𝑖 denote closed subsets of Ω, which is spatial supports of sensors and 𝑓_𝑖∈𝐿²(𝐷_𝑖) define the spatial distributions of measurements on 𝐷_𝑖.

According to the choice of the parameters 𝐷_𝑖 and 𝑓_𝑖, we have various types of sensors. These sensors may be types of zones when 𝐷_𝑖⊂Ω. The output function (2.2) can be written in the form 𝑦(⋅,𝑡)=𝐶𝑥(⋅,𝑡)=_𝐷𝑖𝑥(𝜉,𝑡)𝑓_𝑖(𝜉)𝑑𝜉.(2.4) Sensors may also be pointwise when 𝐷_𝑖={𝑏_𝑖} and 𝑓_𝑖=𝛿_𝑏𝑖(𝑥−𝑏_𝑖) where 𝛿_𝑏𝑖 is Dirac mass concentrated in 𝑏_𝑖. Then, the output function (2.2) can be given by the form 𝑦(⋅,𝑡)=𝐶𝑥(⋅,𝑡)=_Ω𝑥(𝜉,𝑡)𝛿_𝑏𝑖𝜉−𝑏_𝑖𝑑𝜉.(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)=𝑆_𝐴(𝑡)𝑥_∘(𝜉).(2.6)(ii)Define the operator 𝐾∶𝑋→𝑂, 𝑥⟶𝐶𝑆_𝐴(⋅)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 𝜒_𝜔 is defined by 𝜒_𝜔𝑥∶⟶𝐿²(Ω𝜒_𝜔)⟶𝐿²(𝜔)𝑥=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 Im𝜒_𝜔𝐾^∗=𝐿²(𝜔)resp.Im𝜒_𝜔𝐾^∗(⋅)=𝐿²(𝜔).(2.10)(v)A sequence of sensors (𝐷_𝑖,𝑓_𝑖)_{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 (𝜑_𝑛𝑗) of eigenfunctions of 𝐿²(Ω) orthonormal in 𝐿²(𝜔) is associated with eigenvalues 𝜆_𝑛 of multiplicity 𝑟_𝑛 and suppose that the system (2.1) has 𝐽 unstable modes. Then, we have the following result.

Proposition 2.1. The sequence of sensors (𝐷_𝑖,𝑓_𝑖)_{1≤𝑖≤𝑞} is 𝜔-strategic if and only if (1)𝑞≥𝑟, (2)rank𝐺_𝑛=𝑟_𝑛,forall𝑛,𝑛=1,…,𝐽 with 𝐺_𝑛=𝐺_𝑛_𝑖𝑗=⎧⎪⎨⎪⎩𝜑^𝑛𝑗𝜑,𝑓_𝑖(⋅)_{𝐿2(𝐷𝐼)},inthezonecase,_𝑛𝑗𝑏_𝑖,inthepointwisecase,(2.11) where sup𝑟_𝑛=𝑟 and 𝐽=1,…,𝑟_𝑛.

Proof. The proof of this proposition is similar to the rank condition in [16]; the main difference is that the rank condition is as followsrank𝐺_𝑛=𝑟_𝑛,∀𝑛.(2.12) For Proposition 2.1., we need only to hold for rank 𝐺_𝑛=𝑟_𝑛,forall𝑛,𝑛=1,…,𝐽.

2.3. 𝜔_𝐸-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 𝜔_𝐸-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 𝜔_𝐸-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 𝜔_𝐸-stable if and only if, for some positive constants 𝑀_𝜔 and 𝛼_𝜔,‖‖𝜒_𝜔𝑆_𝐴(⋅)‖‖_𝐿2(𝜔)≤𝑀_𝜔𝑒^{−𝛼𝜔𝑡}𝑡≥0.(2.13) If (𝑆_𝐴(𝑡))_𝑡≥0 is 𝜔_𝐸-stable, then, for all 𝑥_∘(⋅)∈𝐿²(Ω), the solution of autonomous system associated with (2.1) satisfies ‖‖𝑥_∘(𝑡)‖‖_𝐿2(𝜔)=‖‖𝜒_𝜔𝑆_𝐴(⋅)𝑥_∘‖‖_𝐿2(𝜔)≤𝑀_𝜔𝑒^{−𝛼𝜔𝑡}‖‖𝑥_∘‖‖_𝐿2(𝜔)(2.14) and then lim_𝑡→∞‖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 𝜔_𝐸-detectable) if there exists an operator 𝐻_𝜔∶𝑅^𝑞→𝐿²(𝜔) such that (𝐴−𝐻_𝜔𝐶) generates a strongly continuous semigroup (𝑆_𝐻𝜔(𝑡))_𝑡≥0 which is 𝜔_𝐸-stable.

Definition 2.5. Consider the system (2.1)-(2.2) together with the dynamical system 𝜕𝑧𝜕𝑡(𝜉,𝑡)=𝐹^𝜔𝑧𝑥(𝜉(,𝜉0,)𝑡)=+𝑧𝐺_𝜔∘𝑢(𝑡)+𝐻_𝜔𝑦(𝑡)Θ𝑧(𝜂,𝑡)=0(𝜉)Ω,(2.16) where 𝐹_𝜔 generates a strongly continuous semigroup (𝑆_𝐹𝜔(𝑡))_𝑡≥0 which is stable on Hilbert space 𝑍,𝐺_𝜔∈𝐿(𝑅^𝑝,𝑍) and 𝐻_𝜔∈𝐿(𝑅^𝑞,𝑍). The system (2.16) defines an 𝜔_𝐸-estimator for 𝜒_𝜔𝑇𝑥(𝜉,𝑡) if(1)lim_𝑡→∞‖𝑧(⋅,𝑡)−𝜒_𝜔𝑇𝑥(⋅,𝑡)‖_𝐿2(𝜔)=0, (2)𝜒_𝜔𝑇 maps 𝐷(𝐴) in 𝐷(𝐹_𝜔) where 𝑧(𝜉,𝑡) is the solution of the system (2.16).

Definition 2.6. The system (2.16) specifies an 𝜔_𝐸-observer for the system (2.1)-(2.2) if the following conditions hold:(1)there exist 𝑀_𝜔∈𝐿(𝑅^𝑞,𝐿²(𝜔)) and 𝑁_𝜔∈𝐿(𝐿²(𝜔)) such that 𝑀_𝜔𝐶+𝑁_𝜔𝜒_𝜔𝑇=𝐼_𝜔,(2.17)(2)𝜒_𝜔𝑇𝐴+𝐹_𝜔𝜒_𝜔𝑇=𝐺_𝜔𝐶 and 𝐻_𝜔=𝜒_𝜔𝑇𝐵,(3)the system (2.16) defines an 𝜔_𝐸-observer.

Definition 2.7. The system (2.16) is said to be 𝜔_𝐸-observer for the system (2.1)-(2.2) if 𝑋=𝑍 and 𝜒_𝜔𝑇=𝐼_𝜔. In this case, we have 𝐹_𝜔=𝐴−𝐺_𝜔𝐶 and 𝐻_𝜔=𝐵. Then, the dynamical system (2.16) becomes 𝜕𝑧𝜕𝑡(𝜉,𝑡)=𝐴𝑧(𝜉,𝑡)+Bu(𝑡)−𝐺^𝜔𝑧𝑧(𝜂(,𝜉𝑡,)0=)=00Ω(C.z(𝜉,𝑡)−𝑦(⋅,𝑡))Θ(2.18)

Definition 2.8. The system (2.1)-(2.2) is 𝜔_𝐸-observable if there exists a dynamical system which is exponential 𝜔_𝐸-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 𝜔_𝐸-Observability

In this section, we give an approach which allows to construct an 𝜔_𝐸-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 (𝐷_𝑖,𝑓_𝑖)_{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 𝜔_𝐸-observable by the following dynamical system: 𝜕𝑧𝜕𝑡(𝜉,𝑡)=𝐴𝑧(𝜉,𝑡)+Bu(𝑡)−𝐺^𝜔𝑧(𝜉,0)=𝑧_∘𝐶(𝑧(𝜉,𝑡)−𝑦(⋅,𝑡))Θ𝑧(𝜂,𝑡)=0(𝜉)Ω.(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)=𝐴¹𝑥𝑥₁(𝜉,𝑡)+𝑃Bu(𝑡)Θ₁𝑥(𝜉,0)=𝑥_∘1(𝜉)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)=𝐴²𝑥𝑥₂(𝜉,𝑡)+(𝐼−𝑃)Bu(𝑡)Θ₂𝑥(𝜉,0)=𝑥_∘2(𝜉)Ω₂(𝜂,𝑡)=0.(3.3) The operator 𝐴₁ is represented by matrix of order (∑^𝐽𝑛=1𝑟_𝑛,∑^𝐽𝑛=1𝑟_𝑛) given by 𝐴₁=diagP𝜆B₁,=…,𝐺𝜆_11tr,𝜆₂,…,𝜆₂,…,𝜆_j,…,𝜆_j,,𝐺^2tr,…,𝐺^𝐽tr.(3.4)
Step 2. Since the sequence suite of sensors (𝐷_𝑖,𝑓_𝑖)_{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 𝜔_𝐸-detectable and hence there exists an operator 𝐻^1𝜔 such that 𝐴₁−𝐻^1𝜔𝐶 which satisfies the following: ∃𝑀^1𝜔,𝛼^1𝜔>0 such that ‖𝑒^{(𝐴1−𝐻1𝜔𝐶)𝑡}‖≤𝑀^1𝜔𝑒^{−𝛼1𝜔𝑡} and, then, we have ‖‖𝑥₁(⋅,𝑡)‖‖_𝐿2(𝜔)≤𝑀^1𝜔𝑒^{−𝛼1𝜔𝑡}‖‖𝑃𝑥_∘‖‖_𝐿2(𝜔).(3.5) Since the semigroup generated by the operator 𝐴₂ is 𝜔_𝐸-stable, there exists 𝑀^2𝜔,𝛼^2𝜔>0 such that ‖‖𝑥₂(⋅,𝑡)‖‖_𝐿2(𝜔)≤𝑀^1𝜔𝑒^−𝛼1𝜔‖‖(𝐼−𝑃)𝑥_∘2(⋅)‖‖_𝐿2(𝜔)+^𝑡0𝑀^2𝜔𝑒^{−𝛼2𝜔(𝑡−𝜏)}‖‖(𝐼−𝑃)𝑥_∘2(⋅)‖‖_𝐿2(𝜔)‖𝑢(𝜏)‖𝑑𝜏(3.6) and therefore ‖𝑥(𝜉,𝑡)‖_𝐿²_(𝜔)→0 when 𝑡→∞. Finally, the system (2.1)-(2.2) is 𝜔_𝐸-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)==𝐴𝜕𝜕𝑥𝑥𝑡((𝜉𝜉,,𝑡𝑡))−+B𝜕𝜕𝑧u𝑡((𝑡𝜉),−𝑡)𝐴𝑧(𝜉,𝑡)−Bu(𝑡)+𝐻_𝜔=𝐴−𝐻_𝜔𝐶(𝑧(𝜉,𝑡)−𝑥(⋅,𝑡))𝐶𝑒(𝜉,𝑡).(3.7)
Since the system (2.1)-(2.2) is 𝜔_𝐸-detectable, there exists an operator 𝐻_𝜔∈𝐿(𝑅^𝑞,𝐿²(𝜔)), such that the operator (𝐴−𝐻_𝜔𝐶) generates exponentially regionally stable, strongly continuous semigroup (𝑆_𝐻𝜔(𝑡))_𝑡≥0 on 𝐿²(𝜔) which satisfies the following relations: ∃𝑀_𝜔,𝛼_𝜔>0suchthat‖‖𝜒_𝜔𝑆_𝐻𝜔(𝑡)‖‖_𝐿2(𝜔)≤𝑀_𝜔𝑒^{−𝛼𝜔𝑡}.(3.8) Finally, we have ‖𝑒(⋅,𝑡)‖_𝐿²_(𝜔)≤‖‖𝜒_𝜔𝑆_𝐻𝜔(𝑡)‖‖_𝐿2(𝜔)‖‖𝑒_∘(⋅)‖‖≤𝑀_𝜔𝑒^{−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 𝜔_𝐸-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,𝑡𝑥)(=𝜉,Δ0)𝑥=(𝜉𝑥,𝑡_∘)+𝑥(𝜉,𝑡)Θ𝑧(𝜂,𝑡)=0(𝜉)Ω(3.10) augmented with the output function 𝑦(𝑡)=_Ω𝑥(𝜉,𝑡)𝛿𝜉−𝑏_𝑖𝑑𝜉,(3.11) where Ω=(0,1) and 𝑏_𝑖∈Ω are the location of sensors (𝑏_𝑖,𝛿_𝑏𝑖) as in (Figure 2). The operator 𝐴=(Δ+1) generates a strongly continuous semigroup (𝑆_𝐴(𝑡))_𝑡≥0 on the Hilbert space 𝐿²(𝜔) [15]. Consider the dynamical system 𝜕𝑧𝜕𝑡(𝜉,𝑡)=Δ𝑧(𝜉,𝑡)+𝑧𝑧((𝜉𝜉,,𝑡0))−=𝐻𝑧𝐶_∘(𝑧(𝜉,𝑡)−𝑥(𝜉,𝑡))(0,1),𝑡>0,𝑧(0,𝑡)=𝑧(1,𝑡()𝜉)=0(0,𝑡1>),0,(3.12) where 𝐻∈𝐿(𝑅^𝑞,𝑍),𝑍 is the Hilbert space, and 𝐶∶𝑍→𝑅^𝑞 is linear operator. If 𝑏_𝑖∈𝑄, then the sensors (𝑏_𝑖,𝛿_𝑏𝑖) 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 𝑏𝑖 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)−𝐻^𝜔𝑧(𝜉,0)=𝑧_∘𝐶(𝑧(𝜉,𝑡)−𝑥(𝜉,𝑡))(0,1),t>0,𝑧(0,𝑡)=𝑧(1,𝑡()𝜉)=0(0,𝑡1>),0,(3.13) where 𝐻_𝜔∈𝐿(𝑅^𝑞,𝐿²(𝜔)). If 𝑏_𝑖/𝛽∉𝑄, then the sensors (𝑏_𝑖,𝛿_𝑏𝑖) are 𝜔-strategic for the unstable subsystem of (3.10) [7] and then the system (3.10)-(3.11) is 𝜔_𝐸-detectable. Therefore, the system (3.10)-(3.11) is 𝜔_𝐸-observable by 𝜔_𝐸-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 𝜕𝑥𝜕𝑡𝜉¹𝑥,𝜉𝜉₂,𝑡=Δ𝑥𝜉₁,𝜉₂𝑡+Bu(𝑡)Θ₁𝑥,𝜉𝜂₂₁,0=𝑥_∘𝜉₁,𝜉₂Ω,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 𝜑_𝑖𝑗𝜉₁,𝜉₂=2𝛽₁−𝛼₁𝛽₂−𝛼₂sin𝑖𝜋𝜉¹−𝛼₁𝛽₁−𝛼₁sin𝑗𝜋𝜉²−𝛼₂𝛽₂−𝛼₂(4.2) associated with eigenvalues 𝜆_𝑖𝑗=−𝑖²𝛽₁−𝛼₁²+𝑗²𝛽₂−𝛼₂².(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)=_𝐷𝑥𝜉₁,𝜉₂,𝑡𝑓𝜉₁,𝜉₂𝑑𝜉₁𝑑𝜉₂,(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−𝛼₁²𝛽₂−𝛼₂²∉𝑄,(4.5) then 𝑟=1 and one sensor may be sufficient to achieve 𝜔_𝐸-observability [18]. In this case, the dynamical system (3.1) is given by 𝜕𝑧𝜕𝑡𝜉¹,𝜉₂,𝑡=Δ𝑧𝜉₁,𝜉₂,𝑡+𝑧𝑧𝜉𝜉₁,𝜉₂,𝑡+Bu(t)−𝐻_𝜔<𝑥(⋅,𝑡),𝑓_𝑖(⋅)>−𝐶𝑧(𝜉,𝑡)Θ₁𝑧,𝜉𝜂₂₁,0=𝑧_∘𝜉₁,𝜉₂Ω,𝜂₂,𝑡=0.(4.6) Let the measurement support be rectangular with 𝐷=𝜉₁−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 𝜔_𝐸-observable by the dynamical system (4.6) if 𝑖𝜉_∘1−𝛼₁𝛽₁−𝛼₁,𝑖𝜉^∘2−𝛼₂𝛽₂−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: 𝑦(𝑡)=𝑥𝜉₁,𝜉₂,𝑡𝛿𝜉₁−𝑏₁,𝜉₂−𝑏₂𝑑𝜉₁𝑑𝜉₂,(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,𝛿_𝑏) may be sufficient for 𝜔_𝐸-observability. Then, the dynamical system is given by 𝜕𝑧𝜕𝑡𝜉¹,𝜉₂,𝑡=Δ𝑧𝜉₁,𝜉₂,𝑧𝑡+𝜉𝑧𝜉₁,𝜉₂,𝑡+Bu(𝑡)+𝐻_𝜔𝑥𝑏₁,𝑏₂,𝑡−𝑦(𝑡)₁Θ𝑧,𝜉𝜂₂₁,0=𝑧_∘𝜉₁,𝜉₂Ω,𝜂₂,𝑡=0.(4.10)

Thus, we obtain the following.

Corollary 4.2. The system (4.1)–(4.9) is not 𝜔_𝐸-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,𝛿_𝜎) such that it is symmetric with respect to the line 𝜉=𝜉_∘, then the system (4.1)–(4.9) is not 𝜔_𝐸-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,𝑟_𝜔) where 𝜔⊂Ω and 0<𝑟_𝜔<1 [10],(3)case of boundary sensors where 𝐶∉𝐿(𝑋,𝑅^𝑞); 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.