#### 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 and represented by the model as in (Figure 1), then we are interested in the regional state on of the domain .

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 but only on particular region 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 (-strategic sensor, -detectability, and -observer). Section 3 is related to the characterization notion of -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 be an open bounded subset of , with boundary and let , be a time measurement interval. Suppose that be a nonempty given subregion of . We denote and . The considered distributed parameter systems is described by the following parabolic systems: augmented with the output function where is a second-order linear differential operator, which generates a strongly continuous semigroup on the Hilbert space and is self-adjoint with compact resolvent. The operators and depend on the structures of actuators and sensors [13, 14]. The spaces , and are separable Hilbert spaces where is the state space, is the control space, and is the observation space, where and are the numbers of actuators and sensors. Under the given assumption [15], the system (2.1) has a unique solution:

The problem is that how to observe exponentially the current state in a given subregion (see Figure 1), using convenient sensors and to give a sufficient condition for the existence of a regional exponential observability.

##### 2.2. -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 . (i)Sensors are any couple where denote closed subsets of , which is spatial supports of sensors and define the spatial distributions of measurements on .

According to the choice of the parameters and , we have various types of sensors. These sensors may be types of zones when . The output function (2.2) can be written in the form Sensors may also be pointwise when and where is Dirac mass concentrated in . Then, the output function (2.2) can be given by the form In the case of internal pointwise sensors, the operator 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: (ii)Define the operator , which is in the case of internal zone sensors is linear and bounded [16]. The adjoint operator of is defined by (iii)For the region of the domain , the operator is defined by where is the restriction of to .(iv)An autonomous system associated to (2.1)-(2.2) is exactly (resp., weakly) -observable if (v)A sequence of sensors is -strategic if the system (2.1)-(2.2) is weakly -observable [5].

The concept of -strategic has been extended to the regional boundary case as in [17]. Assume that the set of eigenfunctions of orthonormal in is associated with eigenvalues of multiplicity and suppose that the system (2.1) has unstable modes. Then, we have the following result.

Proposition 2.1. *The sequence of sensors is -strategic if and only if *(1)*,
*(2)* with
where and .*

*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
For Proposition 2.1., we need only to hold for rank .

##### 2.3. -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 (or -stable) if, for every initial state , the solution of the autonomous system associated with (2.1) converges exponentially to zero when .

*Definition 2.3. *The system (2.1) is said to be exponentially stable on (or -stable) if the operator generates a semigroup which is exponentially stable in . It is easy to see that the system (2.1) is -stable if and only if, for some positive constants and ,
If is -stable, then, for all , the solution of autonomous system associated with (2.1) satisfies
and then

*Definition 2.4. *The system (2.1) together with output (2.2) is said to be exponentially detectable on (or -detectable) if there exists an operator such that generates a strongly continuous semigroup which is -stable.

*Definition 2.5. *Consider the system (2.1)-(2.2) together with the dynamical system
where generates a strongly continuous semigroup which is stable on Hilbert space and . The system (2.16) defines an -estimator for if(1),
(2) maps in where is the solution of the system (2.16).

*Definition 2.6. *The system (2.16) specifies an -observer for the system (2.1)-(2.2) if the following conditions hold:(1)there exist and such that
(2) and ,(3)the system (2.16) defines an -observer.

*Definition 2.7. *The system (2.16) is said to be -observer for the system (2.1)-(2.2) if and . In this case, we have and . Then, the dynamical system (2.16) becomes

*Definition 2.8. *The system (2.1)-(2.2) is -observable if there exists a dynamical system which is exponential -observer, for the original system. Now, the approach which is observed is that the current state exponentially is given by the following result.

#### 3. Strategic Sensors and -Observability

In this section, we give an approach which allows to construct an -estimator of . This method avoids the consideration of initial state [6]; it enables to observe exponentially the current state in without needing the effect of the initial state of the considered system.

Theorem 3.1. *Suppose that the sequence of sensors is -strategic and the spectrum of contain eigenvalues with nonnegative real parts. Then, the system (2.1)-(2.2) is -observable by the following dynamical system:
*

*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 and on two parts, unstable and stable. The state vector may be given by where is the state component of the unstable part of the system (2.1) and may be written in the form
and is the component state of the part of the system (2.1) given by
The operator is represented by matrix of order () given by
*Step 2. *Since the sequence suite of sensors is -strategic for the unstable part of the system (2.1). The subsystem (3.2) is weakly -observable [5], and since it is of finite dimensional, it is exactly -observable [2]. Therefore, it is -detectable and hence there exists an operator such that which satisfies the following: such that and, then, we have
Since the semigroup generated by the operator is -stable, there exists such that
and therefore when . Finally, the system (2.1)-(2.2) is -detectable.*Step 3. *Let where is solution of the system (3.1). Driving the above equation and using (2.1) and (3.1), we obtain

Since the system (2.1)-(2.2) is -detectable, there exists an operator , such that the operator generates exponentially regionally stable, strongly continuous semigroup on which satisfies the following relations:
Finally, we have
with and therefore converges exponentially to zero as . Thus, the dynamical system (3.1) observes exponentially the regional state of the system original system and (2.1)-(2.2) is -observable.

*Remark 3.2. *We can deduce that(1)a system which is exactly -observable is exponentially -observable,(2)a system which is exponentially observable is exponentially -observable,(3)a system which is exponentially -observable is exponentially -observable, in every subset of , but the converse is not true. This may be proven in the following example.

*Example 3.3. *Consider the system
augmented with the output function
where and are the location of sensors as in (Figure 2). The operator generates a strongly continuous semigroup on the Hilbert space [15]. Consider the dynamical system
where is the Hilbert space, and is linear operator. If , then the sensors are not strategic for the unstable subsystem (3.10) [1] and therefore the system (3.10)-(3.11) is not exponentially detectable in [14]. Then, the dynamical system (3.12) is not observer and then (3.10)-(3.11) is not exponentially observable [16].

Now, we consider the region and the dynamical system where . If , then the sensors are -strategic for the unstable subsystem of (3.10) [7] and then the system (3.10)-(3.11) is -detectable. Therefore, the system (3.10)-(3.11) is -observable by -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 by the form together with output function by (2.4), (2.5). Let be the considered region which is subset of (0,1) × (0,1). In this case, the eigenfunctions of system (4.1) are given by associated with eigenvalues

The following results give information on the location of internal zone or pointwise -strategic sensors.

##### 4.1. Internal Zone Sensor

Consider the system (4.1) together with output function (2.2) where the sensor supports are located in . The output (2.2) can be written by the form where is location of zone sensor and . In this case of Figure 3, the eigenfunctions and the eigenvalues are given by (4.2) and (4.3). However, if we suppose that then and one sensor may be sufficient to achieve -observability [18]. In this case, the dynamical system (3.1) is given by Let the measurement support be rectangular with then we have the following result.

Corollary 4.1. *If is symmetric about and is symmetric about , then the system (4.1)–(4.4) is -observable by the dynamical system (4.6) if
*

##### 4.2. Internal Pointwise Sensor

Let us consider the case of pointwise sensor located inside of . The system (4.1) is augmented with the following output function: where is the location of pointwise sensor as defined in Figure 4.

If Q, then and one sensor may be sufficient for -observability. Then, the dynamical system is given by

Thus, we obtain the following.

Corollary 4.2. *The system (4.1)–(4.9) is not -observable by the dynamical system (4.10) if and , for every .*

##### 4.3. Internal Filament Sensor

Consider the case of the observation on the curve with (see Figure 5), then we have the following.

Corollary 4.3. *If the observation recovered by filament sensor such that it is symmetric with respect to the line , then the system (4.1)–(4.9) is not -observable by (4.10) if and for all .*

*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 and where and [10],(3)case of boundary sensors where ; 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.