Strategic Sensors and Regional Exponential Observability

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.


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 concretereal 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.
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.

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: 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 L 2 Ω and is self-adjoint with compact resolvent.The operators B ∈ 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 L 2 0, ∞, R p is the control space, and O L 2 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: 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.

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 ∈ L 2 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 Sensors may also be pointwise when D i {b i } and f i δ b i x − b i where δ b i is Dirac mass concentrated in b i .Then, the output function 2.2 can be given by the form 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: ii Define the operator which is in the case of internal zone sensors is linear and bounded 16 .The adjoint operator K * of K is defined by iii For the region ω of the domain Ω, the operator χ ω is defined by 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 The concept of ω-strategic has been extended to the regional boundary case as in 17 .
Assume that the set ϕ n j of eigenfunctions of L 2 Ω orthonormal in L 2 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

2.11
where sup r 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 follows rank G n r n , ∀n.

2.12
For Proposition 2.1.,we need only to hold for rank G n r n , for all n, n 1, . . ., J.

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 L 2 ω or ω E -stable if, for every initial state x • • ∈ L 2 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 L 2 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 ,

2.13
If S A t t≥0 is ω E -stable, then, for all x • • ∈ L 2 Ω , the solution of autonomous system associated with 2.1 satisfies where F ω generates a strongly continuous semigroup S F ω 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 2 χ ω T 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:

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.

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:

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 x 1 ξ, t x 2 ξ, t tr where x 1 ξ, t is the state component of the unstable part of the system 2.1 and may be written in the form and x 2 ξ, t is the component state of the part of the system 2.1 given by x 2 η, t 0 .

3.3
The operator A 1 is represented by matrix of order J n 1 r n , J n 1 r n given by

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 1 ω such that A 1 − H 1 ω C which satisfies the following: ω t and, then, we have Since the semigroup generated by the operator A 2 is ω E -stable, there exists M 2 ω , α 2 ω > 0 such that and therefore x ξ, t L 2 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

3.7
Since the system 2.1 -2.2 is ω E -detectable, there exists an operator H ω ∈ L R q , L 2 ω , such that the operator A − H ω C generates exponentially regionally stable, strongly continuous semigroup S H ω t t≥0 on L 2 ω which satisfies the following relations: 3.8 Finally, we have 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 ω 1 -observable, in every subset ω 1 of ω, 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 Ω 0, 1 and b i ∈ Ω are the location of sensors b i , δ b i as in Figure 2 .The operator A Δ 1 generates a strongly continuous semigroup S A t t≥0 on the Hilbert space L 2 ω 15 .Consider the dynamical system 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 b i 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 .Now, we consider the region ω 0, β ⊂ 0, 1 and the dynamical system where 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 .

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 η 1 , η 2 , t 0 4.1 together with output function by 2.4 , 2.5 .Let ω α 1 , β 1 × α 2 , β 2 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

4.3
The following results give information on the location of internal zone or pointwise ω-strategic sensors.

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 where D ⊂ Ω is location of zone sensor and f ∈ L 2 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 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

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: where b b 1 , b 2 is the location of pointwise sensor as defined in Figure 4.If ∈Q, then m 1 and one sensor b, δ b may be sufficient for ω E -observability.Then, the dynamical system is given by

4.10
Thus, we obtain the following.

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.

Figure 1 :
Figure 1: The domain of Ω, the subregion ω, and the sensors locations.

Figure 2 :
Figure 2: The domain Ω, the subregion ω, and locations b i of internal pointwise sensors.

Figure 5 :
Figure 5: Rectangular domain and location σ of internal filament sensors.
The system 2.16 is said to be ω E -observer for the system 2.1 -2.2 if X Z and χ ω T I ω .In this case, we have F ω A − G ω C and H ω B. Then, the dynamical system 2.16 becomes 2.17 2 χ ω TA F ω χ ω T G ω C and H ω χ ω TB, 3 the system 2.16 defines an ω E -observer.