Sensors structures and regional detectability of parabolic distributed systems

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1 Sensors and Actuators A ) 163±171 Sensors structures and regional detectability of parabolic distributed systems R. Al-Saphory *, A. El-Jai Systems Theory Laboratory, University of Perpignan, 52 Avenue de Villeneuve, 6660Perpignan Cedex, France Received November 2000; received in revised form 10 January 2001; accepted 3 February 2001 Abstract In this paper, we deal with the linear in nite dimensional systems in a Hilbert space where the dynamics of the system is governed by strongly continuous semi-groups. We study the concept of asymptotic exponential) regional detectability in connection with the structures of sensors for a class of parabolic distributed parameter systems. For different sensors structures, we give the characterization of the asymptotic exponential) regional detectability in order that asymptotic exponential) regional observability be achieved. Furthermore, we apply these results to the regional observer for distributed parameter diffusion systems. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Regional detectability; Regional observers; Diffusion system; Sensors 1. Introduction Many real problems in the control and asymptotic observation of distributed parameter systems can be reformulated as problems of in nite-dimensional systems, in a domain [1±4]. An extension of detectability, is that of asymptotic respectively exponential) regional o-detectability. This concept, was introduced recently by Al-Saphory and El Jai [5], and was focused on the state asymptotic respectively exponential) detection in a given part o of the domain. The purpose of this paper is to give some results related to the link between the regional detectability and sensors structures. We consider a class of distributed systems and we explore various results connected with different types of measurements, domains and boundary conditions. The main reason for introducing the concept of asymptotic respectively exponential) regional o-detectability, is the possibility to construct an asymptotic respectively exponential) regional observer for the considered system. Section 2concerns the class of considered systems and the characterization of regional strategic sensors. Section 3 devotes to the introduction of regional detectability problem. We discuss this notion with regional observability and structures of sensors. In Section 4, we give an application to various situations of sensors locations and in the last section, we study the relation between the regional observer and regional detectability and we give an application of regional observer to a diffusion system. 2. The class of considered systems Let the following assumptions be given: be a regular bounded open set of R n with 0; TŠ, T > 0 a time measurement interval. o be a non-empty given subregion of with boundary G o. We denote Q ˆ Š0; T ; Q ˆ Š0; 1 ; 1 and S T. X; U; be separable Hilbert spaces where X is the state space, U the control space and the observation space. We consider X ˆ L 2, U ˆ L 2 0; 1; R p and ˆ L 2 0; 1; R q, where p and q hold for the number of actuators and sensors. The considered system is described by the following parabolic distributed parameter system t ˆAx x; t Bu t Q x x; 0 ˆx 0 x ; x ; t ˆ0; 2.1) * Corresponding author. Tel.: ; fax: address: saphory@univ-perp.fr R. Al-Saphory). augmented with the output function z ; t ˆCx ; t 2.2) /01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S )

2 164 R. Al-Saphory, A. El-Jai / Sensors and Actuators A ) 163±171 where A is a second order linear differential operator which generates a strongly continuous semi-group S A t t0 on the Hilbert space X ˆ L 2 and is self-adjoint with compact resolvent. The operators B 2 L U; X and C 2 L X; depend on the structure of actuators and sensors. Under the given assumptions, the system 2.1) has a unique solution given by x x; t ˆS A t x 0 x t 0 S A t t Bu t dt 2.3) The measurements can be obtained by the use of zone, pointwise or lines sensors which may be located in [4]. A sensor is defined by any couple D; f where D, a nonempty closed subset of, is the spatial support of the sensor and f 2 L 2 D defines the spatial distribution of the sensing measurements on D. According to the choice of the parameters D and f, we have various types of sensors. In the case of a pointwise sensor, D is reduced to the location fbg with b 2 and f ˆ d b where d is the Dirac mass concentrated in b. In the case of q sensors, we shall consider D i ; f i 1iq where D i or D and f i 2 L 2 T D i, with D i Dj ˆ;if 1 i 6ˆ j q. The output function 2.2) may be written in the form z ; t ˆhx ; t ; f i i L 2 D i 2.4) The sensors may be pointwise, denoted by b i ; d bi where b i 2, the Eq. 2.2) then becomes z ; t ˆx b i ; t 2.5) Finally the sensors may be pointwise or zone) with D In the case of boundary pointwise sensors, the output function is similar to 2.5) with b i and in the case with D i ˆ G i are zones subset The output is given by z ; t ˆhx ; t ; f i i L2 G i 2.6) The sensors D i ; f i 1iq are said to be o-strategic if the system 2.1) together with the output function 2.2) is weakly regionally observable in o [6]. 3. Asymptotic exponential) regional detectability The detectability is in some sense a dual notion of stabilizability. In El Jai and Pritchard [4] this notion was considered in the whole domain. In this section, we shall extend these results to the regional case by considering o as subregion of Fig. 1). Regional detectability characterization needs some assumptions which are related to the asymptotic behaviour of the system, i.e. stability Definitions and characterizations We recall that the problem of stability is one of the most important in the analysis of control systems. Fig. 1. Considered domain and subregion o. The semi-group S A t t0 is said to be asymptotically respectively exponentially) stable if for every initial state x 0 2 L 2 the corresponding solution of the autonomous system of 2.1) x x; t converges asymptotically respectively exponentially) to zero as t tends to 1. Itis easy to see that the system 2.1) is exponentially stable if and only if for some positive constants M and a jjs A t jj L2 Me at ; t 0 If S A t t0 is an asymptotically respectively exponentially) stable semi-group, then for all x 0 2 L 2 the solution of the associated autonomous system satisfies lim jjx ; t jj t!1 L 2 ˆ lim jjs A t x 0 jj t!1 L 2 ˆ 0 3.1) The system 2.1) is said to be asymptotically respectively exponentially) stable, if the operator A generates a semigroup which is asymptotically respectively exponentially) stable. In the finite dimensional linear systems, the concept of exponential stability is equivalent to asymptotic stability. It is not the case if the state space X is infinite dimensional. The system 2.1) together with the output 2.2) is said to be asymptotically respectively exponentially) detectable if there exists an operator H :! X such that A HC generates a strongly continuous semi-group S H t t0 which is asymptotically exponentially) stable. If a system is asymptotically respectively exponentially) detectable then it is possible to construct an asymptotic respectively exponential) observer for the original system. If we consider the x; t ˆAy x; t Bu t H z ; t Cy x; t Q y x; 0 ˆy 0 x ; y ; 0 ˆ0; 3.2) then y x; t estimates asymptotically respectively exponentially) the state x x; t because e x; t ˆx x; t y x; t x; t ˆ A HC e x; t with e x; 0 ˆx 0 x; t y 0 x; t and if the system is asymptotically respectively exponentially) detectable, it is possible to choose H which realizes lim t!1 jje ; t jj L 2 ˆ 0: The system 2.1) together with the output function 2.2) is said to be asymptotically respectively exponentially)

3 R. Al-Saphory, A. El-Jai / Sensors and Actuators A ) 163± observable if there exists a dynamical system which is an asymptotic respectively exponential) observer estimator) for the systems 2.1)± 2.2). Remark 3.1. In this paper we only need the relation 3.1) to be true on a given subdomain o lim jjx ; t jj t!1 L 2 o ˆ 0 3.3) We may refer to this as asymptotic respectively exponential) regional o-stability which is equivalent for the considered class of systems to the asymptotically respectively exponential) stability. Definition 3.2. The system 2.1) is said to be asymptotically respectively exponentially) regionally o-stable, if the operator A generates a semi-group which is regionally asymptotically respectively exponentially) o-stable. Definition 3.3. The system 2.1) together with the output function 2.2) is said to be asymptotically respectively exponentially) regionally o-detectable if there exists an operator H o :! L 2 o such that A H o C generates a strongly continuous semigroup S Ho t t0 which is asymptotically respectively exponentially) regionally o-stable. It is clear that: 1. A system which is asymptotically exponentially) detectable, is regionally asymptotically exponentially) o-detectable, 2. A system which is exponentially regionally o- detectable, is asymptotically regionally o-detectable, 3. A system which is asymptotically exponentially) regionally o-detectable, is asymptotically exponentially) regionally o 1 -detectable, for every subset o 1 of o Regional detectability and regional observability It has been shown that a system which is exactly observable is detectable [4]. This interesting result remains nonconstructive because it is related to the choice of the operator H in 3.2). The autonomous system associated to 2.1) with output function 2.2) may be written by the form x; t ˆAx x; t Q x x; 0 ˆx 0 x ; 3.4) x ; t ˆ0; S z ; t ˆCx ; t ; Q where x x; 0 is supposed to be unknown. Define now the operator K : x 2 X! Kx ˆ CS A t x 2 3.5) then z ; t ˆK t x 0. We denote by K :! X the adjoint of K given by K z ; t ˆ t 0 S s C z ; s ds 3.6) Consider a subdomain o of and let w o be the function defined by w o : L 2!L 2 o 3.7) x! w o ˆ x jo where x jo is the restriction of the state x to o: The system 3.4) is said to be exactly respectively weakly) regionally observable in o if Im w o K ˆ L 2 o respectively Im w o K ˆ L 2 o If the system 3.4) is weakly regionally observable in o then x x; 0 is given by x x; 0 ˆ K K 1 K z ; t where K K 1 K is the pseudo-inverse of the operator K [6,7]. These definitions have been extended to regional boundary case [±10]. However, one can easily deduce the following important results. Corollary 3.4. A system which is exactly observable, then it is asymptotically exponentially) observable. Corollary 3.5. If the system 2.1) together with output function 2.2) is exactly regionally o-observable, then it is regionally asymptotically respectively exponentially) o-detectable. From this result, we can easily deduce that there exists g > 0 such that jjcs A t x ; t jj L 2 0;T; gjjw os A t x ; t jj L 2 o ; x 2 L 2 o Thus the notion of regional detectability is far less restrictive than that of exact regional observability in o Sensors and regional detectability As in El Jai and Pritchard [4], we shall develop a characterization result that links the regional detectability and sensors structures. For that purpose, let us consider the set j i of functions of L 2 orthonormal in L 2 o associated with the eigenvalues l i of multiplicity m i and suppose that the system 2.1) has J unstable modes. Thus the suf cient condition of asymptotic respectively exponential) regional o-detectability is given by the following theorem. Theorem 3.6. Suppose that there are q sensors D i ; f i 1iq and the spectrum of A contains J eigenvalues with nonnegative real parts. The system 2.1) together with output function 2.2) is asymptotically respectively exponentially)

4 166 R. Al-Saphory, A. El-Jai / Sensors and Actuators A ) 163±171 regionally o-detectable if and only if 1. q m 2. rank G i ˆ m i ; i; i ˆ 1;...; J with G ˆ G ij < hj j ; f i i L2 D i ; ˆ j j b i ; : hj j ; f i i L2 G i ; for zone sensors for pointwise sensors for boundary zone sensors where sup m i ˆ m < 1 and j ˆ 1;...; 1. Proof. The proof is limited to the case of zone sensors. 1. Under the assumptions of Section 2, 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 x; t ˆ x 1 x; t x 2 x; t Š tr where x 1 x; t is the state component of the unstable part of the system 2.1), may be written in the form x; t ˆA 1x 1 x; t PBu t ; Q 3.) x 1 x; 0 ˆx 10 x ; x 1 ; t ˆ0; and x 2 x; t is the component state of the stable part of the system 2.1) given by x; t ˆA 2x 2 x; t I P Bu t ; Q 3.9) x 2 x; 0 ˆx 20 x ; x 2 ; t ˆ0; The operator A 1 is represented by a matrix of order P J iˆ1 m i; P J iˆ1 m i given by A 1 ˆ diag l 1 ;...; l 1 ;...; l J ;...; l J Š and PB ˆ G tr 1 ; Gtr 2 ;...; Gtr J Š. By using the condition 2) of this theorem, we deduce that the suite D i ; f i 1iq of sensors is o-strategic for the unstable part of the system 2.1), the subsystem 3.) is weakly regionally observable in o, and since it is finite dimensional, then it is exactly regionally observable in o. Therefore it is asymptotically respectively exponentially) regionally o-detectable, and hence there exists an operator Ho 1 such that A 1 Ho 1 C which satisfies the following: 9M 1 o ; a1 o > 0 such that jje A 1 H 1 o C t jj L2 o M1 o e a1 o t and then we have jjx 1 ; t jj L 2 o M1 o e a1 o t jjpx 0 jj L 2 o Since the semi-group generated by the operator A 2 is asymptotically respectively exponentially) regionally o-stable, then there exist Mo 2 ; a2 o > 0 such that jjx 2 ; t jj L2 o M2 o e a2 o t jj I P x 0 jj L2 o t 0 Mo 2 e a2 o t t jj I P x 0 jj L2 o jju t jj dt and therefore x x; t!0 when t!1: Finally, the systems 2.1)± 2.2) is asymptotically respectively exponentially) regionally detectable in o. 2. If the system 2.1) together with the output function 2.2) is asymptotically respectively exponentially) regionally detectable in o, there exists an operator H o 2 L L 2 0; 1; R q ; L 2 o, such that A H o C generates an asymptotically respectively exponentially) regionally stable, strongly continuous semi-group S Ho t t0 on the space L 2 o which satisfies the following 9M o ; a o > 0 such that jjw o S Ho t jj L 2 o M oe a o t Thus the unstable subsystem 3.) is asymptotically respectively exponentially) regionally detectable in o. Since this subsystem is of finite dimensional, then it is exactly regionally observable. Therefore 3.) is weakly regionally observable and hence it is o-strategic, i.e. Kw o x ; t ˆ0 ) x ; t ˆ0Š [6]. For x ; t 2L 2 o, we have Kw o x ; t ˆ X J jˆ1 e l jt hj j ; x ; t i L 2 o hj j ; f i i L 2! 1iq If the unstable system 3.) is not o-strategic, there exists x ; t 2L 2 o, such that Kw o x ; t ˆ0; this leads X J jˆ1 hj j ; x ; t i L2 o hj j ; f i i L2 ˆ 0 The state vectors x i may be given by x i ; t ˆ hj 1 ; x ; t i L 2 o hj J ; x ; t i L 2 o Štr 6ˆ 0 we then obtain G i x i ˆ 0 for all i, i ˆ 1;...; J and therefore rank G i 6ˆ m i. 4. Application to measurements structures In this section we give the speci c results related to some examples of sensors structures and we apply these results to different situations of the domain, which usually follow from symmetry considerations. We consider the two-dimensional system de ned on ˆŠ0; 1 Š0; 1 by the x 1; x 2 ; t ˆ@2 x 2 x 1 ; x 2 ; x 1 2 x 1 ; x 2 ; t x x 1 ; x 2 ; t Q 2 x 1 ; 2 ; t ˆ0 t > 0 x x 1 ; x 2 ; 0 ˆx 0 x 1 ; x 2 4.1) together with output function is given by 2.4), 2.5) or 2.6). Let o ˆŠa 1 ; b 1 Ša 2 ; b 2 be the considered region is subset

5 of Š0; 1 Š0; 1. In this case the eigenfunctions of the system 4.1) for Dirichlet boundary conditions are given by 1=2 4 j ij x 1 ; x 2 ˆ b 1 a 1 b 2 a 2 x1 a 1 sin ip sin jp x 2 a 2 4.2) b 1 a 1 b 2 a 2 associated with eigenvalues! i 2 l ij ˆ b 1 a 1 2 j 2 b 2 a 2 2 p 2 4.3) In the case of Neumann or mixed boundary conditions, we have different functions. We illustrate some practical examples of the linear parabolic system 4.1) Case of a zone sensor Consider the system 4.1) together with output function 2.2) where the sensor supports D are located in Internal zone sensor We discuss this case with different domains. Rectangular domain: the output function 2.2) can be written by the form z t ˆ x x 1 ; x 2 ; t f x 1 ; x 2 dx 1 dx 2 4.4) D where D is the location of the zone sensor and f 2 L 2 D. In the case of Fig. 2), the eigenfunctions and the eigenvalues are given in the Eqs. 4.2) and 4.3). However, if we suppose that b 1 a 1 2 b 2 a 1 2 =2Q 4.5) then m ˆ 1 and one sensor may be sufficient for asymptotic respectively exponential) regional o-detectability. The measurement supports is rectangular with D ˆ x 10 l 1 ; x 10 l 1 Š x 20 l 2 ; x 20 l 2 Š. We then have the result. Corollary 4.1. Suppose that f 1 is symmetric about x 1 ˆ x 10 and f 2 is symmetric with respect to x 2 ˆ x 20. The system 4.1) together with the output function 4.4) is not asymptotically respectively exponentially) regionally o-detectable if i x 10 a 1 = b 1 a 1 and i x 20 a 2 = b 2 a 2 2N for some i; i ˆ 1;...; J. Circular domain: consider the systems 4.1) and 2.4). So, this system may be given by the following form R. Al-Saphory, A. El-Jai / Sensors and Actuators A ) 163± Fig. 2. Domain, subdomain o and location D of internal zone sensor. and with output function z i t ˆ x r i ; y i ; t f r i ; y i dr i dy i ; D i 0 y i r; y; t ˆ@2 2 r; y; x r; y; t x r; y; t Q x r; y; 0 ˆx 0 r; y x 1; y; t ˆ0 y 2 0; 2pŠ; t > r i o < r i < 1 2 ; 2 i q 4.7) where ˆ D 0; 1 is defined as in Fig. 3). So, the eigenfunctions and eigenvalues concerning the region o ˆ D 0; r o ; r o 2Š0; 1 are defined by l ij ˆ b 2 ij ; i 0; j 1 4.) where b ij are the zeros of the Bessel functions J i and j 0j r; y ˆJ 0 b 0j r ; j 1 j ij1 r; y ˆJ i b ij1 r cos iy ; i; j 1 1 j ij2 r; y ˆJ i b ij2 r sin iy ; i; j ) with multiplicity m i ˆ 2for all i; j 6ˆ 0 and m i ˆ 1 for all i; j ˆ 0. In this case, the asymptotic respectively exponential) regional detectability in o is required at least two zone sensors D i ; f i 2iq where D i ˆ r i ; y i 2iq [4]. Thus we have the result. Corollary 4.2. Suppose that f i and D i ; are symmetric with respect to y ˆ y i, for all i; 2 i q. Then the system 4.6)± 4.7) is not regionally asymptotically respectively exponentially) o-detectable if i 0 y 1 y 2 =p 2 N for some i 0 ; i 0 ˆ 1;...; J Boundary zone sensor We consider the system 4.1) with the Dirichlet boundary conditions and output function 2.6). We study this case with different geometrical domains. The domain Š0; 1 Š0; 1 : now the output function 2.2) is given by z t 1; 2 ; t f 1 ; 2 d 1 d ) G 4.6)

6 16 R. Al-Saphory, A. El-Jai / Sensors and Actuators A ) 163±171 Fig. 3. Domain, subdomain o and locations D 1 ; D 2 of internal zone sensors. Fig. 5. Circular domain and locations G 1 ; G 2 of boundary zone sensors. where is the support of the boundary sensor and f 2 L 2 G. The sensor D; f may be located on the boundary in G ˆ 10 l; 10 lšf1g, then we have. Corollary 4.3. If the function f is symmetric with respect to 1 ˆ 10, then the system 4.1) together with the output function 4.10) is not regionally asymptotically respectively exponentially) o-detectable if i 10 a 1 = b 1 a 1 2N for some i; 1 i J. When the sensor is located in G ˆ 10 l 1 ; 1Š f0g S S f1g 0; 20 l 2 ŠˆG 1 G2 where see Fig. 4). We obtain the following result. Corollary 4.4. Suppose that the function f jg1 is symmetric with respect to 1 ˆ 10 ; and the function f jg2 is symmetric about 2 ˆ 20. Then the system 4.1) together with the output function 4.10) is not regionally asymptotically respectively exponentially) o-detectable if 10 a 1 = b 1 a 1 and 20 a 2 = b 2 a 2 2N for some i; 1 i J. The domain D 0; 1 : here, we consider the system 4.6) with output function z i t ˆ G 1; y i; t f 1; y i dy i ; 4.11) 0 y i 2p; t > 0 In this case, it is necessary to have at least two boundary zone sensors G i ; f i 2iq with G i ˆ 1; y i 2iq and if the function f jgi is symmetric with respect to y ˆ yi 2iq as in Fig. 5. So, we have. Corollary 4.5. The system 4.6) together with the output function 4.11) is not regionally asymptotically respectively exponentially) o-detectable if i y 1 y 2 =p 2 N for some i; 1 i J Case of a pointwise sensor Let us consider the case of pointwise sensor located inside of or on the boundary Internal pointwise sensor We can discuss the following. Case of Fig. 6: the system 4.1) is augmented with the following output z t ˆ x x 1 ; x 2 ; t d x 1 b 1 ; x 2 b 2 dx 1 dx ) where b ˆ b 1 ; b 2 is the location of pointwise sensor in is defined asinfig. 6.If b 1 a 1 = b 2 a 1 =2Q then m ˆ 1 and one sensor b; d b may be sufficient for asymptotic respectively exponential) regional o-detectability. Then we obtain. Corollary 4.6. The systems 4.1)± 4.12) is regionally asymptotically respectively exponentially) o-detectable if i b 1 a 1 = b 1 a 1 and i b 2 a 2 = b 2 a 2 =2N, for every i; i ˆ 1;...; J. Case of Fig. 7: consider the system 4.6) with the function de ned by form z i t ˆ x r i ; y i ; t f i r i ; y i dr i dy i 4.13) where 2 i q; 0 y i 2p; 1=2 r io < r i < 1=2 ; t > 0: The sensors may be located in p 1 ˆ r 1 ; y 1 and p 2 ˆ r 2 ; y 2 2 see Fig. 7). Corollary The system 4.6)± 4.13) is regionally asymptotically respectively exponentially) o-detectable if i y 1 y 2 = p=2n for all i ˆ 1;...; J. Fig. 4. Rectangular domain and locations G; G of boundary zone sensors. Fig. 6. Rectangular domain and location b of internal pointwise sensor.

7 R. Al-Saphory, A. El-Jai / Sensors and Actuators A ) 163± Fig. 7. Circular domain and locations p 1 ; p 2 of internal pointwise sensors. Fig. 9. Rectangular domain and location b of boundary pointwise sensor. 2. If r 1 ˆ r 2, the system 4.6)± 4.13) is regionally asymptotically respectively exponentially) o-detectable if i y 1 y 2 =p=2n for all i ˆ 1;...; J Filament sensors Consider the case where ˆŠ0; 1 Š0; 1 and o ˆŠ a 1 ; b 1 Ša 2 ; b 2. Suppose that the observation on the curve s ˆ Im g with g 2 C 1 0; 1 Fig. ), then we have. Corollary 4.. If the observation recovered by the filament sensor s; d s such that s is symmetric with respect to the line x ˆ x 0. The system 4.1)± 4.12) is regionally asymptotically respectively exponentially) o-detectable if i x 10 a 1 = b 1 a 1 and i x 20 a 2 = b 2 a 2 =2N, for every i; i ˆ 1;...; J Boundary pointwise sensor Let us consider the system 4.1) with Dirichlet boundary condition, so, we can study the following. Case of Fig. 9: in this case the sensor b; d b is located Fig. 9). The output function is given by z 1; 2 ; t d 1 b 1 ; 2 b 2 d 1 d ) Then we can obtain. Corollary 4.9. The system 4.1)± 4.14) is regionally asymptotically respectively exponentially) o-detectable if i b 1 a 1 = b 1 a 1 and i b 2 a 2 = b 2 a 2 =2N, for every i; i ˆ 1;...; J. The case of Fig. 10: in this case we consider the system 4.6) with z i t 1; y i; t f 1; y i dy i ; y i 2 0; 2pŠ; t > 4.15) When the pointwise sensors at the polar coordinates p i ˆ 1; y i where y i 2 0; 2pŠ and 2 i q. We have the following result. Corollary Then the system 4.6)± 4.15) is regionally asymptotically respectively exponentially) o-detectable if i y 1 y 2 =p=2n for every i; 1 i J. These results can be extended with boundary Neumann conditions. 5. Regional observer and regional detectability In this section, we give an approach which allows to determine a regional asymptotic exponential) estimator of Tx x; t in o, based on the regional asymptotic exponential) o-detectability. This approach derives from Luenberger observer type as introduced by Gressang and Lamont [11]. For that purpose, we recall that some de nitions concerns the regional observer and the regional detectability Definitions and characterizations Definition 5.1. Suppose that there exists a dynamical system with state y x; t 2 a Hilbert space) x; t ˆF oy x; t G o u t H o z ; t ; Q 5.1) y x; 0 ˆy 0 x ; y ; t ˆ0; where F o generates a strongly continuous semi-group which is asymptotically respectively exponentially) stable on the space ; G o 2 L U; and H o 2 L ;. The system Fig.. Rectangular domain and location s of internal filament sensors. Fig. 10. Circular domain and locations p 1 ; p 2 sensors. of boundary pointwise

8 170 R. Al-Saphory, A. El-Jai / Sensors and Actuators A ) 163± ) defines a regional asymptotic respectively exponential) estimator for w o Tx x; t if 1. lim t!1 w o Tx x; t y x; t Š ˆ 0; x 2 o, 2. w o T maps D A into D F o where x x; t and y x; t are the solutions of 2.1)± 2.2) and 5.1). Definition 5.2. The system 5.1) specifies an asymptotical respectively exponential) observer for the system 2.1)± 2.2) if it satisfies the following conditions: 1. There exists M o 2 L ; L 2 o and N o 2 L L 2 o such that M o C N o w o T ˆ I o. 2. w o TA F o w o T ˆ G o C and H o ˆ w o TB. 3. The system 5.1) determines a regional asymptotic respectively exponential) estimator for w o Tx x; t. Definition 5.3. The system 5.1) is said to be an identity asymptotic respectively exponential) regional observer for the system 2.1)± 2.2) if w o T ˆ I o and X ˆ. Definition 5.4. The system 5.1) is said to be an asymptotic respectively exponential) regional reduced-order observer for the system 2.1)± 2.2) if X ˆ. Definition 5.5. The system 2.1)± 2.2) is asymptotically respectively exponentially) regionally observable in o if there exists a dynamical system which is asymptotic exponential) regional observer for the original system. Proposition 5.6. Suppose that the system 2.1) together with output function 2.2) is an asymptotically respectively exponentially) regionally o-detectable then, the dynamical system described by the following x; t ˆAy x; t Bu t H o Cy x; t z ; t ; Q y x; 0 ˆ0; y ; t ˆ0; 5.2) is an asymptotically respectively exponentially) regional observer of the system 2.1)± 2.2), if lim x x; t y x; t Š ˆ 0; x 2 o: t!1 Proof. Let e x; t ˆx x; t y x; t where y x; t is the solution of the system 5.2). Deriving the above equation and inserting the Eqs. 2.1) and 5.2), we x; t ˆ x; @t ˆ Ax x; t Bu t Ay x; t Bu t H o C y x; t x ; t ˆ A H o C e x; t : Since the system 2.1)± 2.2) is asymptotically respectively exponentially) regionally o-detectable, there exists an operator H o 2 L L 2 0; 1; R q ; L 2 o, such that the operator A H o C generates asymptotically respectively exponentially) regionally stable, strongly continuous semi-group S Ho t t0 on L 2 o which is satisfied the following relations. 9M o ; a o > 0 such that jjw o S Ho t jj L2 o M oe a o t Finally, we have jje ; t jj L 2 o jjw os Ho t jj L 2 o jje 0 jj M o e a o t jje 0 jj with e 0 ˆ x 0 y 0 and therefore e x; t converges to zero as t!1. Thus the dynamical system 5.2) may be considered as an identity) asymptotically respectively exponentially) regionally observer for the system 2.1)± 2.2) without needing the stability of the system 2.1). Remark 5.7. It is easily to show the following: 1. A system which is exactly regionally observable in o, is asymptotically exponentially) regionally observable in o. 2. A system which is asymptotically exponentially) observable, is asymptotically exponentially) regionally observable in o. 3. A system which is asymptotically exponentially) regionally observable in o, is asymptotically exponentially) regionally observable in every subset o 1 of o. These results can be extended to general case and reduced-order) of regional observer as in [5] Application to a regional observer for distributed parameter diffusion systems Consider the case of two dimensional distributed parameter diffusion system described by the parabolic x 1; x 2 ; t ˆ@2 x 2 x 1 ; x 2 ; x 1 2 x 1 ; x 2 ; t bx x 1 ; x 2 ; t ; Q 2 x 1 ; 2 ; t ˆ0; t > 0 x x 1 ; x 2 ; 0 ˆx 0 x 1 ; x 2 ; 5.3) where the above-stated equations in discussing heat-conduction problems [12]. Let o be a subregion of ˆŠ0; 1 Š0; 1 and suppose there exists a single sensor D 1 ; f with D 1 ˆ d 1 l 1 ; d 1 l 1 Š d 2 l 2 ; d 2 l 2 ŠŠ0; 1 Š0; 1. Thus the augmented output function is given by z t ˆ x x 1 ; x 2 ; t f x 1 ; x 2 dx 1 dx 2 5.4) D 1 The eigenfunctions of the 2 = 2 = 2 2 b for the Dirichlet boundary conditions are defined by j ij x 1 ; x 2 ˆ2sin ip d 1 sin jp d 2

9 R. Al-Saphory, A. El-Jai / Sensors and Actuators A ) 163± associated with the eigenvalues l ij ˆ b i 2 j 2 p 2 The system of regional stabilizability in connection the actuator structures is x 1; x 2 ; t ˆ@2 y 2 x 1 ; x 2 ; y 1 2 x 1 ; x 2 ; t by x 1 ; x 2 ; t Bu x 1 ; x 2 ; t H o Cy x 1 ; x 2 ; t z t ; 2 y 1 ; 2 ; t ˆ0; y x 1 ; x 2 ; 0 ˆy 0 x 1 ; x 2 ; Q 5.5) together with the system 5.3)± 5.4) are equivalent to the system 2.1), 2.2)± 5.2) with the operator A 2 = 2 2 = 2 2 b. Using the Proposition 5.6 and Corollary 4.1, the system 5.5) is a regional observer for the system 5.3)± 5.4) if D 1 x x 1 ; x 2 ; t f x 1 ; x 2 dx 1 dx 2 6ˆ 0 5.6) Equivalently, if i d 1 a 1 = b 1 a 1 and i d 2 a 2 = b 2 a 2 =2N. Under this condition, there exists an operator H o 2 L L 2 0; 1; R q ; L 2 o such that A H o C generates a regionally exponentially stable, strongly continuous semi-group S Ho t t0 on L 2 o and weobtain lim x x; t y x; t Š ˆ 0; x 2 o t!1 We can easily extend this application to the case of pointwise sensor and to the case of boundary zone or pointwise) sensor with different boundary conditions. 6. Conclusion In this paper we have presented the existence of the suf cient condition of regional asymptotic respectively exponential) o-detectability and we have discussed the regional asymptotic respectively exponential) detection problem for a class of distributed parameter systems with regional observability and regional observer. Many interesting results concerning the choice of sensors structures are explored and illustrated in speci c situations. We have shown that it is possible to construct a regional observer for distributed parameter diffusion system by using the concept of regional asymptotic respectively exponential) o- detectability. An extension of these results to the problem References [1] A. El Jai, L. Berrahmoune, Localisation d'actionnaires-zones pour la controãlabiliteâ de systeámes paraboliques, C.R. Acad. Sci. Paris ). [2] A. El Jai, L. Berrahmoune, Localisation d'actionnaires ponctuels pour la controãlabiliteâ de systeámes paraboliques, C.R. Acad. Sci. Ser. 1 Paris 29 3) 194). [3] A. El Jai, L. Berrahmoune, Localisation d'actionnaires-frontieáre pour la controãlabiliteâ de systeámes paraboliques, C.R. Acad. Sci. Ser. 1 Paris 29 ) 194). [4] A. El Jai, A.J. Pritchard, Sensors and Controls in the Analysis of Distributed Systems Analysis, Ellis Horwood series in Applied Mathematics and its Applications, Wiley, New ork, 19. [5] R. Al-Saphory, A. El Jai, Asymptotic regional reconstruction, Int. J. Syst. Sci., submitted for publication. [6] A. El Jai, M.C. Simon, E. errik, Regional observability and sensor structures, Sens. Actuators 2 39) 1993) 95±102. [7] E. errik, Regional Analysis of Distributed Parameter Systems, Ph.D. Thesis, University of Rabat, Morocco, [] R. Al-Saphory, A. El Jai, Asymptotic boundary regional reconstruction, Int. J. Syst. Sci., in preparation. [9] E. errik, L. Badraoui, A. El Jai, Sensors and regional boundary state reconstruction of parabolic systems, Sens. Actuators ) 102±117. [10] E. errik, A. Boutoulout, A. El Jai, Actuators and regional boundary controllability of parabolic systems, Int. J. Syst. Sci. 1 31) 2000) 73±2. [11] R. Gressang, G.B. Lamont, bservers for systems characterized by semi-groups, IEEE Trans. Automatics Control ) 523±52. [12] S. Kitamura, S. Sakairi, M. Nishimura, bserver for distributedparameter diffusion systems, Electrical Eng. Jpn. 6 92) 1972). Biographies A. El Jai is a professor at the University of Perpignan in France. He obtained his Doctorat d'etat in Automatic Control 197) at LAAS, Toulouse. Professor El Jai wrote three books on distributed parameter systems and many papers in the area of systems analysis and control. At present, he is the head of the Systems Theory Laboratory at the University of Perpignan, France. R. Al Saphory is a researcher at LTS Systems Theory Laboratory) of University of Perpignan in France. His research area is focused on distributed parameter systems analysis and control.