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 [1ā€“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 [7ā€“9]. If a system is defined on a domain Ī© and represented by the model as in (Figure 1), then we are interested in the regional state on šœ” of the domain Ī©.

The concept of regional asymptotic analysis was introduced recently by Al-Saphory and El Jai in [10ā€“12], consisting in studying the behaviour of the system not in all the domain Ī© but only on particular region šœ” 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 (šœ”-strategic sensor, šœ”-detectability, and šœ”-observer). Section 3 is related to the characterization notion of šœ”šø-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 Ī© be an open bounded subset of š‘…š‘›, with boundary šœ•Ī© and let [0,š‘‡], š‘‡>0 be a time measurement interval. Suppose that šœ” be a nonempty given subregion of Ī©. We denote Ī˜=Ī©Ć—(0,āˆž) and āˆ=šœ•Ī©Ć—(0,āˆž). The considered distributed parameter systems is described by the following parabolic systems: šœ•š‘„šœ•š‘”(šœ‰,š‘”)=š“š‘„(šœ‰,š‘”)+Bu(š‘”)Ī˜š‘„(šœ‰,0)=š‘„āˆ˜ī‘(šœ‰)Ī©š‘„(šœ‚,š‘”)=0(2.1) augmented with the output function š‘¦(ā‹…,š‘”)=š¶š‘„(ā‹…,š‘”),(2.2) where š“ is a second-order linear differential operator, which generates a strongly continuous semigroup (š‘†š“(š‘”))š‘”ā‰„0 on the Hilbert space š‘‹=šæ2(Ī©) and is self-adjoint with compact resolvent. The operatorsšµāˆˆšæ(š‘…š‘,š‘‹) and š¶āˆˆšæ(š‘…š‘ž,š‘‹) depend on the structures of actuators and sensors [13, 14]. The spaces š‘‹,š‘ˆ, and š‘‚ are separable Hilbert spaces where š‘‹ is the state space, š‘ˆ=šæ2(0,āˆž,š‘…š‘) is the control space, and š‘‚=šæ2(0,āˆž,š‘…š‘ž) is the observation space, where š‘ and š‘ž are the numbers of actuators and sensors. Under the given assumption [15], the system (2.1) has a unique solution: š‘„(šœ‰,š‘”)=š‘†š“(š‘”)š‘„āˆ˜(ī€œšœ‰)+š‘”0š‘†š“(š‘”āˆ’šœ)Bu(šœ)š‘‘šœ.(2.3)

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

2.2. šœ”-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 šœ”. (i)Sensors are any couple (š·š‘–,š‘“š‘–)1ā‰¤š‘–ā‰¤š‘ž where š·š‘– denote closed subsets of Ī©, which is spatial supports of sensors and š‘“š‘–āˆˆšæ2(š·š‘–) define the spatial distributions of measurements on š·š‘–.

According to the choice of the parameters š·š‘– and š‘“š‘–, we have various types of sensors. These sensors may be types of zones when š·š‘–āŠ‚Ī©. The output function (2.2) can be written in the form ī€œš‘¦(ā‹…,š‘”)=š¶š‘„(ā‹…,š‘”)=š·š‘–š‘„(šœ‰,š‘”)š‘“š‘–(šœ‰)š‘‘šœ‰.(2.4) Sensors may also be pointwise when š·š‘–={š‘š‘–} and š‘“š‘–=š›æš‘š‘–(š‘„āˆ’š‘š‘–) where š›æš‘š‘– is Dirac mass concentrated in š‘š‘–. Then, the output function (2.2) can be given by the form ī€œš‘¦(ā‹…,š‘”)=š¶š‘„(ā‹…,š‘”)=Ī©š‘„(šœ‰,š‘”)š›æš‘š‘–ī€·šœ‰āˆ’š‘š‘–ī€øš‘‘šœ‰.(2.5) In the case of internal pointwise sensors, the operator š¶ 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: š‘„(šœ‰,š‘”)=š‘†š“(š‘”)š‘„āˆ˜(šœ‰).(2.6)(ii)Define the operator š¾āˆ¶š‘‹ā†’š‘‚, š‘„āŸ¶š¶š‘†š“(ā‹…)š‘„(2.7) which is in the case of internal zone sensors is linear and bounded [16]. The adjoint operator š¾āˆ— of š¾ is defined by š¾āˆ—ī€œš‘¦=š‘”0š‘†āˆ—š“(š‘ )š¶āˆ—š‘¦(š‘ )š‘‘š‘ .(2.8)(iii)For the region šœ” of the domain Ī©, the operator šœ’šœ” is defined by šœ’šœ”āˆ¶šæ2(Ī©)āŸ¶šæ2(šœ”)š‘„āŸ¶šœ’šœ”š‘„=š‘„|šœ”,(2.9) where š‘„|šœ” is the restriction of š‘„ to šœ”.(iv)An autonomous system associated to (2.1)-(2.2) is exactly (resp., weakly) šœ”-observable if Imšœ’šœ”š¾āˆ—=šæ2ī‚€(šœ”)resp.Imšœ’šœ”š¾āˆ—(ā‹…)=šæ2ī‚(šœ”).(2.10)(v)A sequence of sensors (š·š‘–,š‘“š‘–)1ā‰¤š‘–ā‰¤š‘ž is šœ”-strategic if the system (2.1)-(2.2) is weakly šœ”-observable [5].

The concept of šœ”-strategic has been extended to the regional boundary case as in [17]. Assume that the set (šœ‘š‘›š‘—) of eigenfunctions of šæ2(Ī©) orthonormal in šæ2(šœ”) is associated with eigenvalues šœ†š‘› of multiplicity š‘Ÿš‘› and suppose that the system (2.1) has š½ unstable modes. Then, we have the following result.

Proposition 2.1. The sequence of sensors (š·š‘–,š‘“š‘–)1ā‰¤š‘–ā‰¤š‘ž is šœ”-strategic if and only if (1)š‘žā‰„š‘Ÿ, (2)rankšŗš‘›=š‘Ÿš‘›,forallš‘›,š‘›=1,ā€¦,š½ with šŗš‘›=ī€·šŗš‘›ī€øš‘–š‘—=āŽ§āŽŖāŽØāŽŖāŽ©ī«šœ‘š‘›š‘—,š‘“š‘–ī¬(ā‹…)šæ2(š·š¼)šœ‘,inthezonecase,š‘›š‘—ī€·š‘š‘–ī€ø,inthepointwisecase,(2.11) where supš‘Ÿš‘›=š‘Ÿ and š½=1,ā€¦,š‘Ÿš‘›.

Proof. The proof of this proposition is similar to the rank condition in [16]; the main difference is that the rank condition is as followsrankšŗš‘›=š‘Ÿš‘›,āˆ€š‘›.(2.12) For Proposition 2.1., we need only to hold for rank šŗš‘›=š‘Ÿš‘›,forallš‘›,š‘›=1,ā€¦,š½.

2.3. šœ”šø-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 šæ2(šœ”) (or šœ”šø-stable) if, for every initial state š‘„āˆ˜(ā‹…)āˆˆšæ2(Ī©), the solution of the autonomous system associated with (2.1) converges exponentially to zero when š‘”ā†’āˆž.

Definition 2.3. The system (2.1) is said to be exponentially stable on šœ” (or šœ”šø-stable) if the operator š“ generates a semigroup which is exponentially stable in šæ2(šœ”). It is easy to see that the system (2.1) is šœ”šø-stable if and only if, for some positive constants š‘€šœ” and š›¼šœ”,ā€–ā€–šœ’šœ”š‘†š“ā€–ā€–(ā‹…)šæ2(šœ”)ā‰¤š‘€šœ”š‘’āˆ’š›¼šœ”š‘”š‘”ā‰„0.(2.13) If (š‘†š“(š‘”))š‘”ā‰„0 is šœ”šø-stable, then, for all š‘„āˆ˜(ā‹…)āˆˆšæ2(Ī©), the solution of autonomous system associated with (2.1) satisfies ā€–ā€–š‘„āˆ˜ā€–ā€–(š‘”)šæ2(šœ”)=ā€–ā€–šœ’šœ”š‘†š“(ā‹…)š‘„āˆ˜ā€–ā€–šæ2(šœ”)ā‰¤š‘€šœ”š‘’āˆ’š›¼šœ”š‘”ā€–ā€–š‘„āˆ˜ā€–ā€–šæ2(šœ”)(2.14) and then limš‘”ā†’āˆžā€–š‘„(š‘”)ā€–šæ2(šœ”)=0.(2.15)

Definition 2.4. The system (2.1) together with output (2.2) is said to be exponentially detectable on šœ” (or šœ”šø-detectable) if there exists an operator š»šœ”āˆ¶š‘…š‘žā†’šæ2(šœ”) such that (š“āˆ’š»šœ”š¶) generates a strongly continuous semigroup (š‘†š»šœ”(š‘”))š‘”ā‰„0 which is šœ”šø-stable.

Definition 2.5. Consider the system (2.1)-(2.2) together with the dynamical system šœ•š‘§šœ•š‘”(šœ‰,š‘”)=š¹šœ”š‘„(šœ‰,š‘”)+šŗšœ”š‘¢(š‘”)+š»šœ”š‘¦(š‘”)Ī˜š‘§(šœ‰,0)=š‘§āˆ˜ī‘,(šœ‰)Ī©š‘§(šœ‚,š‘”)=0(2.16) where š¹šœ” generates a strongly continuous semigroup (š‘†š¹šœ”(š‘”))š‘”ā‰„0 which is stable on Hilbert space š‘,šŗšœ”āˆˆšæ(š‘…š‘,š‘) and š»šœ”āˆˆšæ(š‘…š‘ž,š‘). The system (2.16) defines an šœ”šø-estimator for šœ’šœ”š‘‡š‘„(šœ‰,š‘”) if(1)limš‘”ā†’āˆžā€–š‘§(ā‹…,š‘”)āˆ’šœ’šœ”š‘‡š‘„(ā‹…,š‘”)ā€–šæ2(šœ”)=0, (2)šœ’šœ”š‘‡ maps š·(š“) in š·(š¹šœ”) where š‘§(šœ‰,š‘”) is the solution of the system (2.16).

Definition 2.6. The system (2.16) specifies an šœ”šø-observer for the system (2.1)-(2.2) if the following conditions hold:(1)there exist š‘€šœ”āˆˆšæ(š‘…š‘ž,šæ2(šœ”)) and š‘šœ”āˆˆšæ(šæ2(šœ”)) such that š‘€šœ”š¶+š‘šœ”šœ’šœ”š‘‡=š¼šœ”,(2.17)(2)šœ’šœ”š‘‡š“+š¹šœ”šœ’šœ”š‘‡=šŗšœ”š¶ and š»šœ”=šœ’šœ”š‘‡šµ,(3)the system (2.16) defines an šœ”šø-observer.

Definition 2.7. The system (2.16) is said to be šœ”šø-observer for the system (2.1)-(2.2) if š‘‹=š‘ and šœ’šœ”š‘‡=š¼šœ”. In this case, we have š¹šœ”=š“āˆ’šŗšœ”š¶ and š»šœ”=šµ. Then, the dynamical system (2.16) becomes šœ•š‘§šœ•š‘”(šœ‰,š‘”)=š“š‘§(šœ‰,š‘”)+Bu(š‘”)āˆ’šŗšœ”ī‘.(Cz(šœ‰,š‘”)āˆ’š‘¦(ā‹…,š‘”))Ī˜š‘§(šœ‰,0)=0Ī©š‘§(šœ‚,š‘”)=0(2.18)

Definition 2.8. The system (2.1)-(2.2) is šœ”šø-observable if there exists a dynamical system which is exponential šœ”šø-observer, for the original system. Now, the approach which is observed is that the current state š‘„(šœ‰,š‘”) exponentially is given by the following result.

3. Strategic Sensors and šœ”šø-Observability

In this section, we give an approach which allows to construct an šœ”šø-estimator of š‘„(šœ‰,š‘”). This method avoids the consideration of initial state [6]; it enables to observe exponentially the current state in šœ” without needing the effect of the initial state of the considered system.

Theorem 3.1. Suppose that the sequence of sensors (š·š‘–,š‘“š‘–)1ā‰¤š‘–ā‰¤š‘ž is šœ”-strategic and the spectrum of š“ contain š½ eigenvalues with nonnegative real parts. Then, the system (2.1)-(2.2) is šœ”šø-observable by the following dynamical system: šœ•š‘§šœ•š‘”(šœ‰,š‘”)=š“š‘§(šœ‰,š‘”)+Bu(š‘”)āˆ’šŗšœ”š¶(š‘§(šœ‰,š‘”)āˆ’š‘¦(ā‹…,š‘”))Ī˜š‘§(šœ‰,0)=š‘§āˆ˜ī‘.(šœ‰)Ī©š‘§(šœ‚,š‘”)=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 š‘ƒ and š¼āˆ’š‘ƒ on two parts, unstable and stable. The state vector may be given by š‘„(šœ‰,š‘”)=[š‘„1(šœ‰,š‘”)+š‘„2(šœ‰,š‘”)]tr where š‘„1(šœ‰,š‘”) is the state component of the unstable part of the system (2.1) and may be written in the form šœ•š‘„1šœ•š‘”(šœ‰,š‘”)=š“1š‘„1š‘„(šœ‰,š‘”)+š‘ƒBu(š‘”)Ī˜1(šœ‰,0)=š‘„āˆ˜1(š‘„šœ‰)Ī©1ī‘(šœ‚,š‘”)=0(3.2) and š‘„2(šœ‰,š‘”) is the component state of the part of the system (2.1) given by šœ•š‘„2šœ•š‘”(šœ‰,š‘”)=š“2š‘„2š‘„(šœ‰,š‘”)+(š¼āˆ’š‘ƒ)Bu(š‘”)Ī˜2(šœ‰,0)=š‘„āˆ˜2(š‘„šœ‰)Ī©2ī‘.(šœ‚,š‘”)=0(3.3) The operator š“1 is represented by matrix of order (āˆ‘š½š‘›=1š‘Ÿš‘›,āˆ‘š½š‘›=1š‘Ÿš‘›) given by š“1īšœ†=diag1,ā€¦,šœ†1,šœ†2,ā€¦,šœ†2,ā€¦,šœ†j,ā€¦,šœ†jīž,ī€ŗšŗPB=1tr,šŗ2tr,ā€¦,šŗš½trī€».(3.4)
Step 2. Since the sequence suite of sensors (š·š‘–,š‘“š‘–)1ā‰¤š‘–ā‰¤š‘ž is šœ”-strategic for the unstable part of the system (2.1). The subsystem (3.2) is weakly šœ”-observable [5], and since it is of finite dimensional, it is exactly šœ”-observable [2]. Therefore, it is šœ”šø-detectable and hence there exists an operator š»1šœ” such that š“1āˆ’š»1šœ”š¶ which satisfies the following: āˆƒš‘€1šœ”,š›¼1šœ”>0 such that ā€–š‘’(š“1āˆ’š»1šœ”š¶)š‘”ā€–ā‰¤š‘€1šœ”š‘’āˆ’š›¼1šœ”š‘” and, then, we have ā€–ā€–š‘„1ā€–ā€–(ā‹…,š‘”)šæ2(šœ”)ā‰¤š‘€1šœ”š‘’āˆ’š›¼1šœ”š‘”ā€–ā€–š‘ƒš‘„āˆ˜ā€–ā€–šæ2(šœ”).(3.5) Since the semigroup generated by the operator š“2 is šœ”šø-stable, there exists š‘€2šœ”,š›¼2šœ”>0 such that ā€–ā€–š‘„2(ā€–ā€–ā‹…,š‘”)šæ2(šœ”)ā‰¤š‘€1šœ”š‘’āˆ’š›¼1šœ”ā€–ā€–(š¼āˆ’š‘ƒ)š‘„āˆ˜2(ā€–ā€–ā‹…)šæ2(šœ”)+ī€œš‘”0š‘€2šœ”š‘’āˆ’š›¼2šœ”(š‘”āˆ’šœ)ā€–ā€–(š¼āˆ’š‘ƒ)š‘„āˆ˜2(ā€–ā€–ā‹…)šæ2(šœ”)(ā€–š‘¢šœ)ā€–š‘‘šœ(3.6) and therefore ā€–š‘„(šœ‰,š‘”)ā€–šæ2(šœ”)ā†’0 when š‘”ā†’āˆž. Finally, the system (2.1)-(2.2) is šœ”šø-detectable.
Step 3. Let š‘’(šœ‰,š‘”)=š‘„(šœ‰,š‘”)āˆ’š‘§(šœ‰,š‘”) where š‘§(šœ‰,š‘”) is solution of the system (3.1). Driving the above equation and using (2.1) and (3.1), we obtain šœ•š‘’šœ•š‘”(šœ‰,š‘”)=šœ•š‘„šœ•š‘”(šœ‰,š‘”)āˆ’šœ•š‘§šœ•š‘”(šœ‰,š‘”)=š“š‘„(šœ‰,š‘”)+Bu(š‘”)āˆ’š“š‘§(šœ‰,š‘”)āˆ’Bu(š‘”)+š»šœ”=ī€·š¶(š‘§(šœ‰,š‘”)āˆ’š‘„(ā‹…,š‘”))š“āˆ’š»šœ”š¶ī€øš‘’(šœ‰,š‘”).(3.7)
Since the system (2.1)-(2.2) is šœ”šø-detectable, there exists an operator š»šœ”āˆˆšæ(š‘…š‘ž,šæ2(šœ”)), such that the operator (š“āˆ’š»šœ”š¶) generates exponentially regionally stable, strongly continuous semigroup (š‘†š»šœ”(š‘”))š‘”ā‰„0 on šæ2(šœ”) which satisfies the following relations: āˆƒš‘€šœ”,š›¼šœ”ā€–ā€–šœ’>0suchthatšœ”š‘†š»šœ”ā€–ā€–(š‘”)šæ2(šœ”)ā‰¤š‘€šœ”š‘’āˆ’š›¼šœ”š‘”.(3.8) Finally, we have ā€–š‘’(ā‹…,š‘”)ā€–šæ2(šœ”)ā‰¤ā€–ā€–šœ’šœ”š‘†š»šœ”ā€–ā€–(š‘”)šæ2(šœ”)ā€–ā€–š‘’āˆ˜ā€–ā€–(ā‹…)ā‰¤š‘€šœ”š‘’āˆ’š›¼šœ”š‘”ā€–ā€–š‘’āˆ˜ā€–ā€–(ā‹…)(3.9) with š‘’āˆ˜(ā‹…)=š‘„āˆ˜(ā‹…)āˆ’š‘§āˆ˜(ā‹…) and therefore š‘’(šœ‰,š‘”) converges exponentially to zero as š‘”ā†’āˆž. Thus, the dynamical system (3.1) observes exponentially the regional state š‘„(šœ‰,š‘”) of the system original system and (2.1)-(2.2) is šœ”šø-observable.

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

Example 3.3. Consider the system šœ•š‘„šœ•š‘”(šœ‰,š‘”)=Ī”š‘„(šœ‰,š‘”)+š‘„(šœ‰,š‘”)Ī˜š‘„(šœ‰,0)=š‘„āˆ˜ī‘(šœ‰)Ī©š‘§(šœ‚,š‘”)=0(3.10) augmented with the output function ī€œš‘¦(š‘”)=Ī©ī€·š‘„(šœ‰,š‘”)š›æšœ‰āˆ’š‘š‘–ī€øš‘‘šœ‰,(3.11) where Ī©=(0,1) and š‘š‘–āˆˆĪ© are the location of sensors (š‘š‘–,š›æš‘š‘–) as in (Figure 2). The operator š“=(Ī”+1) generates a strongly continuous semigroup (š‘†š“(š‘”))š‘”ā‰„0 on the Hilbert space šæ2(šœ”) [15]. Consider the dynamical system šœ•š‘§šœ•š‘”(šœ‰,š‘”)=Ī”š‘§(šœ‰,š‘”)+š‘§(šœ‰,š‘”)āˆ’š»š¶(š‘§(šœ‰,š‘”)āˆ’š‘„(šœ‰,š‘”))(0,1),š‘”>0,š‘§(šœ‰,0)=š‘§āˆ˜š‘§(šœ‰)(0,1),(0,š‘”)=š‘§(1,š‘”)=0š‘”>0,(3.12) where š»āˆˆšæ(š‘…š‘ž,š‘),š‘ is the Hilbert space, and š¶āˆ¶š‘ā†’š‘…š‘ž is linear operator. If š‘š‘–āˆˆš‘„, then the sensors (š‘š‘–,š›æš‘š‘–) are not strategic for the unstable subsystem (3.10) [1] and therefore the system (3.10)-(3.11) is not exponentially detectable in Ī© [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 šœ”=[0,š›½]āŠ‚(0,1) and the dynamical system šœ•š‘§šœ•š‘”(šœ‰,š‘”)=Ī”š‘§(šœ‰,š‘”)+š‘§(šœ‰,š‘”)āˆ’š»šœ”š¶(š‘§(šœ‰,š‘”)āˆ’š‘„(šœ‰,š‘”))(0,1),t>0,š‘§(šœ‰,0)=š‘§āˆ˜š‘§(šœ‰)(0,1),(0,š‘”)=š‘§(1,š‘”)=0š‘”>0,(3.13) where š»šœ”āˆˆšæ(š‘…š‘ž,šæ2(šœ”)). If š‘š‘–/š›½āˆ‰š‘„, then the sensors (š‘š‘–,š›æš‘š‘–) are šœ”-strategic for the unstable subsystem of (3.10) [7] and then the system (3.10)-(3.11) is šœ”šø-detectable. Therefore, the system (3.10)-(3.11) is šœ”šø-observable by šœ”šø-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 Ī©=(0,1)Ɨ(0,1) by the form šœ•š‘„ī€·šœ‰šœ•š‘”1,šœ‰2ī€øī€·šœ‰,š‘”=Ī”š‘„1,šœ‰2š‘”ī€øš‘„ī€·šœ‰+Bu(š‘”)Ī˜1,šœ‰2ī€ø,0=š‘„āˆ˜ī€·šœ‰1,šœ‰2ī€øĪ©š‘„ī€·šœ‚1,šœ‚2ī€øī‘,š‘”=0(4.1) together with output function by (2.4), (2.5). Let šœ”=(š›¼1,š›½1)Ɨ(š›¼2,š›½2) be the considered region which is subset of (0,1)ā€‰Ć—ā€‰(0,1). In this case, the eigenfunctions of system (4.1) are given by šœ‘š‘–š‘—ī€·šœ‰1,šœ‰2ī€ø=2ī”ī€·š›½1āˆ’š›¼1š›½ī€øī€·2āˆ’š›¼2ī€øī‚µšœ‰sinš‘–šœ‹1āˆ’š›¼1š›½1āˆ’š›¼1ī‚¶ī‚µšœ‰sinš‘—šœ‹2āˆ’š›¼2š›½2āˆ’š›¼2ī‚¶(4.2) associated with eigenvalues šœ†š‘–š‘—īƒ©š‘–=āˆ’2ī€·š›½1āˆ’š›¼1ī€ø2+š‘—2ī€·š›½2āˆ’š›¼2ī€ø2īƒŖ.(4.3)

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

4.1. Internal Zone Sensor

Consider the system (4.1) together with output function (2.2) where the sensor supports š· are located in Ī©. The output (2.2) can be written by the form ī€œš‘¦(š‘”)=š·š‘„ī€·šœ‰1,šœ‰2ī€øš‘“ī€·šœ‰,š‘”1,šœ‰2ī€øš‘‘šœ‰1š‘‘šœ‰2,(4.4) where š·āŠ‚Ī© is location of zone sensor and š‘“āˆˆšæ2(š·). In this case of Figure 3, the eigenfunctions and the eigenvalues are given by (4.2) and (4.3). However, if we suppose that ī€·š›½1āˆ’š›¼1ī€ø2ī€·š›½2āˆ’š›¼2ī€ø2āˆ‰š‘„,(4.5) then š‘Ÿ=1 and one sensor may be sufficient to achieve šœ”šø-observability [18]. In this case, the dynamical system (3.1) is given by šœ•š‘§ī€·šœ‰šœ•š‘”1,šœ‰2ī€øī€·šœ‰,š‘”=Ī”š‘§1,šœ‰2ī€øī€·šœ‰,š‘”+š‘§1,šœ‰2ī€ø,š‘”+Bu(t)āˆ’š»šœ”<š‘„(ā‹…,š‘”),š‘“š‘–š‘§ī€·šœ‰(ā‹…)>āˆ’š¶š‘§(šœ‰,š‘”)Ī˜1,šœ‰2ī€ø,0=š‘§āˆ˜ī€·šœ‰1,šœ‰2ī€øĪ©š‘§ī€·šœ‚1,šœ‚2ī€øī‘.,š‘”=0(4.6) Let the measurement support be rectangular with ī€ŗšœ‰š·=1āˆ’š‘™1,šœ‰1+š‘™2ī€»Ć—ī€ŗšœ‰2āˆ’š‘™2,šœ‰2+š‘™2ī€»āˆˆĪ©,(4.7) then we have the following result.

Corollary 4.1. If š‘“1 is symmetric about šœ‰1=šœ‰āˆ˜1 and š‘“2 is symmetric about šœ‰2=šœ‰āˆ˜2, then the system (4.1)ā€“(4.4) is šœ”šø-observable by the dynamical system (4.6) if š‘–ī€·šœ‰āˆ˜1āˆ’š›¼1ī€øī€·š›½1āˆ’š›¼1ī€ø,š‘–ī€·šœ‰āˆ˜2āˆ’š›¼2ī€øī€·š›½2āˆ’š›¼2ī€øāˆ‰Nforsomeš‘–=1,2,ā€¦,š½.(4.8)

4.2. Internal Pointwise Sensor

Let us consider the case of pointwise sensor located inside of Ī©. The system (4.1) is augmented with the following output function: ī€œš‘„ī€·šœ‰š‘¦(š‘”)=1,šœ‰2ī€øš›æī€·šœ‰,š‘”1āˆ’š‘1,šœ‰2āˆ’š‘2ī€øš‘‘šœ‰1š‘‘šœ‰2,(4.9) where š‘=(š‘1,š‘2) is the location of pointwise sensor as defined in Figure 4.

If (š›½1āˆ’š›¼1)/(š›½2āˆ’š›¼2)āˆ‰Q, then š‘š=1 and one sensor (š‘,š›æš‘) may be sufficient for šœ”šø-observability. Then, the dynamical system is given by šœ•š‘§ī€·šœ‰šœ•š‘”1,šœ‰2ī€øī€·šœ‰,š‘”=Ī”š‘§1,šœ‰2ī€øī€·šœ‰,š‘”+š‘§1,šœ‰2ī€ø,š‘”+Bu(š‘”)+š»šœ”ī€·š‘„ī€·š‘1,š‘2ī€øī€øĪ˜š‘§ī€·šœ‰,š‘”āˆ’š‘¦(š‘”)1,šœ‰2ī€ø,0=š‘§āˆ˜ī€·šœ‰1,šœ‰2ī€øĪ©š‘§ī€·šœ‚1,šœ‚2ī€øī‘.,š‘”=0(4.10)

Thus, we obtain the following.

Corollary 4.2. The system (4.1)ā€“(4.9) is not šœ”šø-observable by the dynamical system (4.10) if š‘–(š‘1āˆ’š›¼1)/(š›½1āˆ’š›¼1) and š‘–(š‘2āˆ’š›¼2)/(š›½2āˆ’š›¼2)āˆˆš‘, for every š‘–,1ā‰¤š‘–ā‰¤š½.

4.3. Internal Filament Sensor

Consider the case of the observation on the curve šœŽ=Im(š›¾) with š›¾āˆˆš¶1(0,1) (see Figure 5), then we have the following.

Corollary 4.3. If the observation recovered by filament sensor (šœŽ,š›æšœŽ) such that it is symmetric with respect to the line šœ‰=šœ‰āˆ˜, then the system (4.1)ā€“(4.9) is not šœ”šø-observable by (4.10) if š‘–(šœ‰āˆ˜1āˆ’š›¼1)/(š›½1āˆ’š›¼1) and š‘–(šœ‰āˆ˜2āˆ’š›¼2)/(š›½2āˆ’š›¼2)āˆˆš‘ for all š‘–=1,...,š‘ž.

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 Ī©=(š·,1) and šœ”=(0,š‘Ÿšœ”) where šœ”āŠ‚Ī© and 0<š‘Ÿšœ”<1 [10],(3)case of boundary sensors where š¶āˆ‰šæ(š‘‹,š‘…š‘ž); 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 ā€œbadā€ 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.