A Regional Asymptotic Analysis of the Compensation Problem in Disturbed Systems

Transcription

1 Applied Mathematical Sciences, Vol. 1, 27, no. 54, A Regional Asymptotic Analysis of the Compensation Problem in Disturbed Systems Larbi Afifi Faculty of Sciences, University Hassan II Ain Chock B.P.5366-Maârif, Casablanca, Morocco l.afifi@fsac.ac.ma, larbi afifi@yahoo.fr Mohamed Bahadi Ecole Royale Navale, Boulevard Sour Jdid, Casablanca, Morocco m.bahadi@fsac.ac.ma Abdelhakim Chafiai Faculty of Sciences, University Hassan II Ain Chock B.P.5366-Maârif, Casablanca, Morocco a.chafiai@fsac.ac.ma Abstract In this work, we present a regional asymptotic analysis of the problem of space compensation for a class of disturbed systems. It is an extension of previous works developed on the finite time remediability problem. We define and we give characterization results of the notion of regional asymptotic remediability and we study its relationship with the regional asymptotic notions of controllability, stability and stabilizability. Then we show how to find the optimal control ensuring the asymptotic compensation, in the considered region, of a known or unknown disturbance. A characterization of the set of asymptotically remediable disturbances in this region is also presented. The cases of multi-actuators and multi-sensors are examined and applications are given. We particularly show that in the regional asymptotic case, a system may be remediable without being controllable, stable or stabilizable. Other situations are also examined and illustrative examples are given. Keywords: Asymptotic regional analysis, remediability, controllability, stability, disturbed systems, region, actuators, sensors

2 266 L. Afifi, M. Bahadi and A. Chafiai 1 Introduction In this paper, we consider a class of linear disturbed systems and we study, with respect to the output (observation), the possibility of regional asymptotic compensation of a known or unknown disturbance. This work is an extension to the regional asymptotic case of previous works on the finite time remediability problem for linear parabolic, hyperbolic, continuous or discrete systems [1,2,3]. Concerning the asymptotic aspect, one know the great importance of the asymptotic analysis, particularly the notions of stability and stabilizability, in control theory and their direct relation with the spectral analysis in the case of linear systems. A natural question is to consider the asymptotic version of the compensation problem and to study a possible extension of the developed approaches as well as the results obtained in the finite time case. Hence, by analogy with the relation between the remediability and the controllability examined in the finite time case, it is then natural to study the relationship between the asymptotic remediability and the asymptotic notions of controllability, stability and stabilizability. Let us note that for finite dimension systems, it is well known that if a linear system is controllable, then it is stabilizable. For distributed systems, the situation is not obvious and needs more precautions. The regional aspect [3,7,9...] of the considered problem is motivated by the fact that a system can be asymptotically remediable in a region ω Ω but not on the whole domain Ω, and even if it is asymptotically remediable in Ω, the cost is reduced if we are interested only in a subregion ω Ω. This paper is organized as follows: We recall in paragraph 2, the notions of exact and weak remediability in the finite time case. In paragraph 3 and under convenient hypothesis, we introduce and we characterize the regional asymptotic version of these notions first in the general case and then in the case of multi-actuators and multi-sensors. Regionally asymptotic efficient actuators ensuring the regional weak asymptotic compensation of any disturbance are particularly examined. The problem of regional asymptotic remediability with minimum energy is studied in paragraph 4. We show how to find the optimal control ensuring the regional asymptotic compensation of a disturbance. Then we characterize the set of disturbances which are asymptotically remediable on a region ω of Ω. In the fifth paragraph, we define and we characterize the notions of weak and exact regional asymptotic controllability and regionally asymptotic strategic actuators. Then we study their relationship with the weak and exact regional asymptotic remediability and regionally asymptotic efficient actuators. Indeed, we show that also in the asymptotic case, regionally strategic actuators are regionally efficient and that the converse is not true.

3 Regional asymptotic analysis 2661 In the last paragraph, we study the nature of the relation between regional asymptotic remediability and the notions of stability and stabilizability. We particularly show that a system can be asymptotically remediable without being stable or stabilizable, but this relation depend, as it will be shown, depend on the choice of the sensors and the actuators. Applications and illustrative examples are given. 2 Preliminaries Let Ω be an open and bounded subset of R n with a sufficiently regular boundary Γ = Ω, and let ω be a subregion of Ω. We consider without loss of generality, a class of linear distributed systems described by the following state equation: { ż(t) =Az(t)+Bu(t)+f(t) (S) (1) z() = z where A generates a strongly continuous semi-group (s.c.s.g.) (S(t)) t ; B L(U, Z); U is the control space and Z = L 2 (Ω) is the state space, U is supposed to be a Hilbert space. The state of the system at time t is given by: z(t) =S(t)z + t S(t s)bu(s)ds + t o S(t s)f(s)ds (2) The system (1) is augmented by the following regional output equation: (E ω ) y ω (t) =Ci ω p ω z(t) (3) where p ω be is the restriction operator defined by: p ω : L 2 (Ω) L 2 (ω) z p ω z = z ω (4) i ω is the adjoint operator of p ω, it is defined by:. i ω : L 2 (ω) L 2 (Ω) z i ω z = { z in ω otherwise and C L(Z, Y ), Y is the observation space, a Hilbert space. The observation is given by:

4 2662 L. Afifi, M. Bahadi and A. Chafiai y ω (t) = Ci ω p ω S(t)z + t Ci ωp ω S(t s)bu(s)ds + t Ci ωp ω S(t s)f(s)ds Obviously in (5), if u = and f =,y ω ( ) =Ci ω p ω S( )z, the observation is then normal. But if f and u, generally y ω ( ) Ci ω p ω S( )z. The finite time problem consists to study the existence of an input operator B (actuators) with respect to the output operator C (sensors), ensuring the compensation at the final time T, of any disturbance, i.e. For f L 2 (,T; Z), there exists u L 2 (,T; U) such that: (5) or equivalently y ω (T )=Ci ω p ω S(T )z CH ω T u + R ω T f = (6) where H T : L 2 (,T; U) Z u H T u = T S(T t)bu(t)dt (7) H T : L 2 (,T; Z) Z f T S(T t)f(t)dt (8) and R ω T = Ci ωp ω H T, H ω T = i ωp ω H T. The definitions can be formulated as follows: Definition 2.1 (i) We say that (S) augmented by (E ω ), (or (S)+(E ω ))isω exactly remediable on [,T], if for every f L 2 (,T; Z), there exists u L 2 (,T; U) such that CH ω T u + Rω T f = (ii) We say that (S)+(E ω ) is ω weakly remediable on [,T], if for every f L 2 (,T; Z) and ε>, there exists u L 2 (,T; U) such that: CH ω T u + R ω T f <ε

5 Regional asymptotic analysis 2663 Characterization results on the exact and weak remediability in finite time are developed in [1,2,3]. It is also shown that the regional remediability is a weaker notion than the regional controllability. The cases where the input and the output are respectively given by actuators and sensors, are considered and illustrating examples are presented. 3 Regional asymptotic compensation 3.1 Problem statement In this case, we consider the system (1) augmented by the output equation (3) with f L 2 (, + ; Z) and u L 2 (, + ; U). Let us consider consider the following operators H ω = i ω p ω H (9) where R ω f = Ci ω p ω H f (1) H : L 2 (, + ; U) Z u H u = S(t)Bu(t)dt (11) and H : L 2 (, + ; Z) Z f S(t)f(t)dt (12) The considered problem, so called regional asymptotic remediability problem, consists to study, with respect to the output operator C and the region ω, the existence of an input one B ensuring regionally the asymptotic compensation of any disturbance, i.e. For every f L 2 (, + ; Z), there exists u L 2 (, + ; U) such that: Let us note that if CH ω u + R ω f = (13)

6 2664 L. Afifi, M. Bahadi and A. Chafiai M ω (.) L 2 (, + ; R + ) such that p ω S(t) M ω (t) ; t (14) the operators and p ω H u p ω S(t)Bu(t)dt p ω H f p ω S(t)f(t)dt are well defined, then CH ω and R ω are also well defined. Remark If (S(t)) t is ω exponentially stable [4,5,8], i.e. M ω > and α ω > such that p ω S(t) M ω e αωt ; t (15) we have (14), consequently CH ω and R ω are well defined. 2- In fact, we are concerned by the operators K C,ω and R C,ω defined by: KC,ω u = Ci ω p ω S(t)Bu(t)dt and RC,ω f = Ci ω p ω S(t)f(t)dt then we need a weaker hypothesis than (14). Indeed, we suppose that there exists a function k ω (.) L 2 (, + ; R + ) such that: Ci ω p ω S(t) k ω (t) ; t (16) In this case, K C,ω and R C,ω are well defined and (13) becomes: K C,ω u + R C,ω f = (17) and as it will be shown later in this paper, the regional exponential stability is not necessary. Under hypothesis (16), the notions of exact and weak regional asymptotic remediability can be formulated as follows:

7 Regional asymptotic analysis 2665 Definition 3.2 We say that: 1- (S) +(E ω ) is exactly ω remediable asymptotically, if for every f L 2 (, + ; Z), there exists u L 2 (, + ; U) such that: K C,ωu + R C,ωf = 2- (S) +(E ω ) is weakly ω remediable asymptotically, if for every f L 2 (, + ; Z) and every ε>, there exists u L 2 (, + ; U) such that K C,ω u + R C,ω f <ε Let us note that for T>; f L 2 (, + ; Z) and u L 2 (, + ; U), the operators H T and R T defined by: and verifies H T u T R T ωf T T T S(t)Bu(t)dt Ci ω p ω S(t)f(t)dt H T u T = H T v T and R T ω f T = R ω T g T where u T and f T are respectively the restrictions of the functions f and u to the interval [,T]; v T (t) =u T (T t) and g T (t) =f T (T t). Moreover, under hypothesis (16) K C,ω u + R C,ω f = Ci ωp ω H T u T + R T ωf T + Ci T ω p ω S(t)Bu(t)dt + Ci T ω p ω S(t)f(t)dt = CH ω T v T + R ω T g T +[ε 1 (T )+ε 2 (T )] with ε 1 (T )+ε 2 (T ) when T +, then for any f L 2 (, + ; Z) and u L 2 (, + ; U), we have lim T + (CHω T v T + R ω T g T )=K C,ωu + R C,ωf

8 2666 L. Afifi, M. Bahadi and A. Chafiai 3.2 Characterization Let us note that for f = Bu, we have RC,ω f = K C,ωu, consequently Im(KC,ω ) Im(R C,ω ) (18) We have the following characterization result: Proposition 3.3 Under hypothesis (16): (i) (S)+(E ω ) is exactly ω remediable asymptotically if and only if this is equivalent to: Im(R C,ω ) Im(K C,ω ) (ii) γ ω > such that θ Y, we have S (.)i ω p ω C θ L 2 (,+ ;Z ) γ ω B S (.)i ω p ω C θ L 2 (,+ ;U ) (19) Proof: (i) derives from the definition. (i) (ii) derives from the following lemma [4,5] Lemma 3.4 Let X, Y, Z be Banach reflexive spaces, P L(X, Z) and Q L(Y,Z). There is equivalence between: and Im(P ) Im(Q) γ > such that for any z Z, we have P z X γ Q z Y Concerning the weak regional asymptotic remediability, we have the following result. Proposition 3.5 Under hypothesis (16): (i) (S)+(E ω ) is weakly ω remediable asymptotically if and only if Im(R C,ω) Im(K C,ω ) (2)

9 Regional asymptotic analysis 2667 or equivalently (ii) Ker[B (R C,ω ) ]=Ker[(R C,ω ) ] (21) Proof: (i) derives from the definition. (ii) (iii) is proved by considering the orthogonal spaces, and using (18) and the fact that (K C,ω ) = B (R C,ω ) (22) We examine hereafter the case where the system is exited by actuators and the observation is given by sensors [6,7]. 3.3 Case of actuators and sensors In the case of p actuators (Ω k,g k ) 1 k p, we have U = IR p, Z = L 2 (Ω) and B : IR p L 2 (Ω) u(t) Bu(t) = p k=1 g ku k (t) (23) where u =(u 1,,u p ) tr L 2 (, + ; IR p ) and g k L 2 (Ω) ; Ω k = supp(g k ) Ω, we have B z =( g 1,z,, g p,z ).,. is the inner product in L 2 (Ω). It is easy to show the following result: Corollary 3.6 (S) +(E ω ) is exactly ω remediable asymptotically, if there exists γ ω > such that: θ Y, we have S (t)i ω p ω C θ 2 Z dt γ ω p ( g k,s (t)i ω p ω C θ ) 2 dt (24) Now, if the output is given by q sensors (D l,h l ) 1 l q ; h l L 2 (Ω); D l = supp(h l ) and for l j, D l D j =, Y = IR q and the operator C is defined by: k=1 C : L 2 (Ω) IR q z Cz =( h 1,z,, h q,z ) tr (25)

10 2668 L. Afifi, M. Bahadi and A. Chafiai we have C θ = q θ l h l for θ =(θ 1,,θ q ) IR q (26) l=1 We have the following characterization result: Corollary 3.7 (S)+(E ω ) is exactly ω remediable asymptotically if and only if, there exists γ ω > such that θ =(θ 1,,θ q ) IR q q 2 S (t)i ω p ω θ l h l dt γ ω l=1 p ( k=1 q g k,s (t)i ω p ω θ l h l ) 2 dt (27) 3.4 Regional asymptotically efficient actuators We introduce hereafter the notion of regional asymptotically efficient actuators [1,2,3] and we give characterization results in the case of a class of linear systems. Definition 3.8 Actuators (Ω k,g k ) 1 k p are ω -efficient asymptotically ( or just ω efficient), if the corresponding system (S)+(E ω ) is weakly ω remediable asymptotically. For the characterization, we consider without loss of generality, the system (S) with a dynamics A of the form l=1 Az = + λ n n=1 r n j=1 z, ϕ nj ϕ nj (28) where λ 1,λ 2,... are reals such that λ 1 >λ 2 >λ 3 >..., {ϕ nj,n 1; j = 1,r n }, is an orthonormal basis of Z, r n is the multiplicity of the eigenvalue λ n. It is well known that A generates a s.c.s.g. (S(t)) t given by S(t)z = Obviously, if + e λnt n=1 r n j=1 z, ϕ nj ϕ nj (29) sup λ n = λ 1 < (3) n 1

11 Regional asymptotic analysis 2669 (S(t)) t is exponentially stable and then ω exponentially stable. system (S) is augmented by the regional output equation The (E ω ) y ω = Ci ω p ω z and the operator B is given by (23), i.e. p Bu(t) = g k u k (t) k=1 For n 1, let M n be the matrix defined by M n =( g k,ϕ nj ) 1 k p;1 j rn (31) We have the following characterization result. Proposition 3.9 The actuators (Ω k,g k ) k=1,p are ω efficient if and only if Ker(p ω C )= Ker(M n fn ω ) n 1 where f ω n is defined by f ω n : θ Y f ω n (θ) =(<C θ, ϕ n1 > ω,..., < C θ, ϕ nrn > ω ) tr IR rn (32) Y is the dual of Y and <.,.> ω is the inner product in L 2 (ω). Proof: For such systems and using the analyticity property, the proof is similar to that established in the finite time case. Indeed, (S)+(E ω ) is weakly ω remediable asymptotically if and only if Ker[B (RC,ω) ]=Ker[(RC,ω) ] For θ Y, we have (R C,ω ) θ = n 1 by analyticity, we have e λnt r n j=1 <i ω p ω C θ, ϕ nj >ϕ nj (R C,ω ) θ = r n j=1 p ω C θ = <i ω p ω C θ, ϕ nj >ϕ nj = ; n 1

12 267 L. Afifi, M. Bahadi and A. Chafiai then On the other hand Ker[(R C,ω) ]=Ker[p ω C ] (33) [B (R C,ω ) θ](t) = ( e λnt r n and also by analyticity, we obtain <g k,ϕ nj >< i ω p ω C θ, ϕ nj > ) tr n 1 j=1 k=1,p B (R C,ω ) θ = r n j=1 <g k,ϕ nj >< C θ, ϕ nj > ω =; n 1, k =1,p M n fn ω (θ) =; n 1 then Ker[B (R C,ω) ]= n 1 Ker(M n f ω n ) and hence Ker[p ω C ]= n 1 Ker(M n f ω n ) Now, we assume that the output is given by q zone sensors (D l,h l ) 1 l q with h l L 2 (D l ), D l = supp(h l ) Ω and measure(d l ω) > for 1 l q. In this case, the functions (h l ) l=1,q are linearly independent, because D l D j = for l j, and measure (D l ω) >, then and using (33), we have On the other hand Ker[p ω C ]={} Ker[(R C,ω ) ]={} B (R C,ω ) θ = ( e λnt r n n 1 j=1 <g k,ϕ nj > ) tr q θ l <h l,ϕ nj > ω l=1 k=1,p

13 Regional asymptotic analysis 2671 consequently B (R C,ω ) θ = r n j=1 <g k,ϕ nj > q θ l <h l,ϕ nj > ω =; k =1,p; n 1 l=1 M n G tr n,ω θ =; n 1 where G n,ω is the matrix defined by then G n,ω =(<h l,ϕ nj > ω ) l =1,q j=1,r n Ker[B (R C,ω ) ]= n 1 Ker(M n G tr n,ω ) and using (21), we have the following characterization result. Proposition 3.1 The actuators (Ω k,g k ) k=1,p are ω efficient if and only if n 1 It is easy to deduce the following corollary. Ker(M n G tr n,ω )={} (34) Corollary 3.11 If there exists n 1 such that or such that rank(m n G tr n,ω )=q (35) and rank(g tr n,ω)=q (36) then the actuators (Ω k,g k ) k=1,p are ω efficient. rank(m n )=r n (37)

14 2672 L. Afifi, M. Bahadi and A. Chafiai 4 Regional asymptotic remediability with minimum energy In this part and under hypothesis (16), we consider the following optimal control problem: For f L 2 (, + ; Z), does exists a control u L 2 (, + ; U) such that: If u exists, is it optimal? Let K C,ωu + R C,ωf =? D ω = { u L 2 (, + ; U) such that KC,ω u + R C,ω f =} D ω is supposed to be a non empty set. We consider the function: The problem becomes J ω (u) = K C,ω u + R C,ω f 2 Y + u 2 L 2 (,+ ;U) Min u Dω J ω (u) and will be resolved using an extension of the Hilbert Uniqueness Method (H.U.M.). For θ Y Y, we consider θ Fω =( B S (t)i ω p ω C θ 2 U dt) 1 2. Fω is a semi-norm. We suppose that it is a norm, this is equivalent to assume that (S)+(E ω ) is weakly ω remediable asymptotically. Let F ω = Y. Fω F ω is a Hilbert space with the inner product θ, τ Fω = B S (t)i ω p ω C θ, B S (t)i ω p ω C τ dt; θ, τ F ω Let Λ C,ω be the operator defined by Λ C,ω = K C,ω (K C,ω ) Λ C,ω has a unique extension as an isomorphism F ω F ω such that: Λ C,ωθ, τ Y = θ, τ Fω ; θ, τ F ω

15 Regional asymptotic analysis 2673 and Λ C,ω θ F = θ Fω ; θ F ω ω We show hereafter how to find the optimal control ensuring the regional asymptotic compensation of a disturbance f. Proposition 4.1 If R C,ω f F ω, then there exists a unique θ f in F ω such that Λ C,ωθ f = R C,ωf and the control u θf =(K C,ω ) θ f verifies K C,ωu θf + R C,ωf = Moreover, u θf is optimal with uθf L = θ 2 (,+ ;U) f Fω Proof: We have Λ C,ω θ f = Ci ω p ω S(t)BB S (t)i ω p ω C θ f dt = K C,ω u θ f = R C,ω f D ω is closed, convex and non empty. For u D ω, we have J ω (u) = u 2 L 2 (,+ ;U) J ω is strictly convex on D ω, then it admits a unique minimum in u D ω characterized by If v D ω, we have u,v u ; v D ω u θf,v u θf = (KC,ω ) θ f,v (KC,ω ) θ f = θ f,kc,ω v K C,ω (K C,ω ) θ f = θ f,kc,ω v Λ C,ω θ f = Since u is unique, then u = u θf and u θf is optimal with uθf 2 = (K C,ω ) θ f 2 = θ f, Λ C,ω θ f = θ f 2 F ω

16 2674 L. Afifi, M. Bahadi and A. Chafiai Remark 4.2 This result can be extended to the case where the observation is not exact, i.e. is as follows z ω (t) =y ω (t)+e ω (t) where y ω (t) is the exact observation given by (3) and e ω (t) is a measurement error. The result is similar. Now, we give hereafter a characterization of the set E ω of disturbances f which are exactly ω remediable asymptotically, i.e. E ω = { f L 2 (, + ; Z) : u L 2 (, + ; U) such that K C,ωu + R C,ωf = } Proposition 4.3 E ω is the inverse image of F ω by the operator R C,ω, i.e. R C,ω E ω = F ω Proof: For y F ω, there exists a unique θ in F ω such that Λ C,ωθ = y, then Let u be the control defined by K C,ω (K C,ω ) θ = y u =(K C,ω ) θ we have K C,ω u = y, and for f = Bu L2 (, + ; L 2 (Ω)), we have K C,ω u = R C,ω f = y, then y R C,ω E ω. Conversely, let y R C,ω E ω, then there exists f L 2 (, + ; L 2 (Ω)) such that y = R C,ω f and K C,ω u + R C,ω f = with u L2 (, + ; U). If we identify KC,ω u with the linear mapping L ω : θ Y KC,ω u, θ, we have: L ω (θ) = K C,ω u, θ = u, (K C,ω ) θ Then L ω (θ) u L 2 (,+ ;U). θ F ω and consequently L ω is continuous on Y for the topology of F ω and can be extended continuously, in a unique way, to the space F ω. Then L ω F and KC,ω u = R C,ω f = y F ω.

17 Regional asymptotic analysis Regional asymptotic remediability and regional asymptotic controllability In the finite time case, it is shown that the controllability is stronger than the remediability. In the asymptotic one, this relation is not obvious and needs more mathematical precautions. Hence, as it will be seen in the next paragraph (example 2), a system can be asymptotically remediable even if the asymptotic controllability problem is not well defined or has no sense. In this part, with a convenient choice of spaces and operators and under convenient hypothesis, we define and we characterize the notion of regional asymptotic controllability and we study its relationship with the regional asymptotic remediability. More precisely, we show that also in the regional asymptotic case, the controllability remain stronger than the remediability. The case of multi-actuators is examined and an application with various illustrative situations is presented. 5.1 Regional asymptotic controllability We consider the system described by the following equation { ż(t) = Az(t)+Bu(t) ; t> (S ) z() = z We suppose that A generates a s.c.s.g.(s(t)) t o such that: M ω (.) L 2 (, + ; R + ) such that p ω S(t) M ω (t) ; t (38) In this case, the notion of regional asymptotic controllability introduced hereafter is well defined. Definition 5.1 The system (S ) is said to be exactly ( resp. weakly ) ω controllable asymptotically if Im(p ω H )=L 2 (ω) ( resp. Im(p ω H )=L 2 (ω) ) Using Lemma 3.4, it is easy to show that the system (S ) is exactly ω controllable asymptotically if and only if γ ω > such that z Z γ ω (p ω H ) z L 2 (,+ ;U ); z L 2 (ω) or equivalently

18 2676 L. Afifi, M. Bahadi and A. Chafiai z Z γ ω B S (.)i ω z L 2 (,+ ;U ); z L 2 (ω) Concerning the weak regional asymptotic controllability, the system (S ) is weakly ω-controllable asymptotically, if and only if This is equivalent to Ker[(p ω H ) ]={} The operator = p ω H (H ) i ω is positive definite In the case of an operator A given by (28): Az = n 1 r n λ n j=1 and (S ) excited by p zone actuators, i.e. p Bu(t) = g k u k (t) k=1 For n 1, let M n be the matrix defined by <z,ϕ nj >ϕ nj (39) M n =( g k,ϕ nj ) 1 k p;1 j rn (4) and γ n (ω) defined by with γ n (ω) = γ n,1 (ω)... γ n,rn (ω) γ n,j (ω) =(γ nj,km (ω) {k 1;m=1,rn} and γ nj,km (ω) = ϕ nj,ϕ km ω Then, using the analyticity property, (S ) is weakly ω controllable asymptotically if and only if

19 Regional asymptotic analysis 2677 Ker[M n γ n (ω)] = {} (41) n 1 Such actuators are said to be asymptotically ω strategic. We have the following result showing that equally in the regional asymptotic case, the controllability is stronger than the remediability. Proposition 5.2 If (S ) is exactly (respectively weakly) ω controllable asymptotically, then (S)+(E ω ) is exactly (respectively weakly) ω remediable asymptotically. Proof: 1) For θ Y, we have S (.)i ω p ω C θ 2 L 2 (,+ ;L 2 (Ω)) = S (t)i ω p ω C θ 2 L 2 (Ω) dt S (t)i ω 2 dt p ω C θ 2 L 2 (ω) M ω p ω C θ 2 L 2 (ω) with M ω > Since (S ) is exactly ω controllable asymptotically, there exists γ ω > such that then p ω C θ 2 L 2 (ω) γ ω B S (.)i ω p ω C θ 2 L 2 (,+ ;U ) S (.)i ω p ω C θ 2 L 2 (,+ ;L 2 (Ω)) γ B S (.)i ω p ω C θ 2 L 2 (,+ ;U ) with γ = M ω γω 2 >. The result is then given by proposition ) (S)+(E ω ) weakly ω remediable asymptotically is equivalent to i.e. Ker[B (R C,ω ) ]=Ker[(R C,ω ) ] this is equivalent to Ker[B (R C,ω ) ] Ker[(R C,ω ) )]

20 2678 L. Afifi, M. Bahadi and A. Chafiai Ker[(K C,ω ) ] Ker[(R C,ω ) ] because (K C,ω ) = B (R C ).Forθ Ker[(K C,ω ) ], we have (K C,ω ) θ =, then (p ω C )θ = because Ker[(p ω H ) ]={}, then θ Ker[p ω C ]. Since Ker(p ω C ) Ker[(R C,ω ) ], we have the result. Remark 5.3 The converse is not true, this is illustrated in the following paragraph. 5.2 Example 1 We consider the following diffusion system with a Dirichlet boundary condition: (S 1 ) z(x,t) t =Δz(x, t)+ p k=1 g k(x)u k (t)+f(x, t) inω ], + [ z(x, ) = z (x) inω z(x, t) =in Ω ], + [ (42) augmented by the following output equation given by q zone sensors (E ω 1 ) yω =( h 1,z ω,, h q,z ω ) tr In the one dimension case with Ω = ], 1[, the Laplacian operator Δ admits an orthonormal basis of eigenfunctions defined by ϕ n (ξ) = 2 sin(nπξ); n 1 the corresponding eigenvalues are simple and given by λ n = n 2 π 2 ; n 1 Δ generates a self adjoint s.c.s.g. (S(t)) t defined by: S(t)z = + n=1 e n2 π 2t z, ϕ n ϕ n (43) and which is exponentially stable. stable and the operators (S(t)) t is then ω exponentially

21 Regional asymptotic analysis 2679 and H u = p + k=1 n=1 e n2 π 2t u k (t)dt g k,ϕ n ϕ n H f = + n=1 e n2 π 2t f(., t),ϕ n ϕ n dt are well defined and (S 1 )+(E1 ω )isω exactly remediable asymptotically if and only if, there exists γ ω > such that: θ =(θ 1,,θ q ) IR q, we have 1 q 2n 2 π ( θ 2 l h l,ϕ n ω ) 2 γ ω n 1 l=1 p k=1 ( n 1 e n2 π 2t g k,ϕ n l=1 q θ l h l,ϕ n ω ) 2 dt In the case of one sensor and one actuator, this inequality becomes: n 1 1 2n 2 π 2 [θ h, ϕ n ω ] 2 γ ω or equivalently n 1 1 2n 2 π 2 [ h, ϕ n ω ] 2 γ ω [ e n2 π 2t g, ϕ n θ h, ϕ n ω ] 2 dt; θ IR n 1 [ n 1 e n2 π 2t g, ϕ n h, ϕ n ω ] 2 dt this is verified for example for ω =Ω,g = h = ϕ n with n 1. But the corresponding system (S 1 ) is not exactly Ω -controllable asymptotically because it is not weakly Ω controllable asymptotically. Concerning the weak asymptotic regional remediability, the general characterization result is given in proposition 3.1. In the case of an actuator (Ω 1,g 1 ) and a sensor (D, h), we have p = q =1. Let n 1 such that h, ϕ n, then rank(g tr n,ω )=1. The actuator (Ω 1,g 1 )isω efficient asymptotically if g 1,ϕ n, i.e. g 1 (ξ)sin(nπξ)dξ Ω 1 For example, if g 1 = ϕ n,(ω 1,g 1 )isω efficient asymptotically but not Ω strategic asymptotically, because the condition 1 sin(nπξ)sin(n πξ)dξ ;n 1

22 268 L. Afifi, M. Bahadi and A. Chafiai is not satisfied. On the other hand, for ω = ], 1 2[, h = ϕ1 and g = ϕ 2, the actuator (Ω,g) is not Ω efficient asymptotically because g, ϕ n h, ϕ n Ω =; n 1 But (Ω,g)isω efficient asymptotically, because for n = 2, we have g, ϕ 2 h, ϕ 2 ω = ϕ 1,ϕ 2 ω Then the relation (35) in corollary 3.11 is verified. Hence, also in the asymptotic case, a system can be regionally remediable without being it on the whole domain. 6 Asymptotic remediability and stabilizability In this section, we study regionally the relation between the asymptotic compensation and the notions of stability and stabilizability. We particularly show that the problem of regional asymptotic compensation may be well defined even if the system is not regionally stable and that a non stable system may be remediable without being stabilizable. The nature of this relation depend on the choice of the actuators, the sensors and the other parameters of the considered system. To show this, we consider without loss of generality, the case where ω = Ω. We assume that (S ) is not exponentially stable and that the unstable part is a finite dimension subspace of the state space Z. These properties and other situations are illustrated with more details by considering as second example, a diffusion system with a Neuman boundary condition. 6.1 Stabilizability and actuators First, let us recall the notion of stabilizability. Definition 6.1 The system (S ) is said to be exponentially stabilizable if there exists a feedback control u = Fz such that the operator A BF generates a s.c.s.g. (S F (t)) t o which is exponentially stable.

23 Regional asymptotic analysis 2681 With the same notations, we consider the case of actuators (Ω i,g i ) i=1,p and an operator A defined by (39) and generating a non stable s.c.s.g. (S(t)) t o. We have the following characterization result showing the relation between the unstable part and the choice of actuators stabilizing the system [8,9] Proposition 6.2 We assume that the system (S ) is not exponentially stable and that there exist a finite number J 1 of non negative eigenvalues noted λ 1,..., λ J. Then the system (S ) excited by p actuators (Ω i,g i ) i=1,p is stabilizable if and only if i) p sup r n 1 n J ii) rankm n = r n for 1 n J, with M n defined in (31). Proof: Let z = ( zu where z u and z s are respectively the projections of the state z on the unstable and the stable parts. We have z s ) { (Su ) ż u (t) = A u z u (t)+pbu(t) (S s ) ż s (t) = A s z s (t)+(i P )Bu(t) (44) P is the projection operator on the unstable part and A u is the diagonal matrix given by A u = λ 1... λ 1... λ J... (45) λ J J J where λ i appears r i times for i =1,J. The order of A u is then ( r i, r i ). On the other hand, we have i=1 i=1

24 2682 L. Afifi, M. Bahadi and A. Chafiai PB = M T 1 M T 2. M T J (46) Mi T is the transposal matrix of M i. Under the condition ii) in the proposition, the finite dimension system (S u ) is controllable and hence is stabilizable. Consequently, there exists a control such that Then and u = Fz u e (Au PBF)t βe αt with β,α > and t z u (t) Pz βe αt u(t) F Pz βe αt Concerning the system (S s ), the operator A s generates a s.c.s.g. which is exponentially stable, then there exist β,α >, with α<α, such that z s (t) β (I P )z e αt + β (I P )z β (I P )z e αt + t e α(t τ) u(τ)dτ ββ F Pz (I P )z We then have the result. t e αt e (α α)τ dτ If a non stable system is stabilizable, we have rankm n = r n for 1 n J, i.e. kerm n = {} ;1 n J then for convenient sensors ( satisfying (34) in proposition 3.1 ), the system is also weakly asymptotically remediable.

25 Regional asymptotic analysis 2683 On the other hand, a non stable system may be asymptotically remediable but not stabilizable. This is illustrated by the following example. 6.2 Example 2 We consider, without loss of generality, the following one dimension system with Ω = ], 1[ and a Neumann boundary condition: (S2) z(x, t) t = Δz(x, t)+f(x, t)+ z(x, ) = z (x) in], 1[ p g i (x)u i (t) in], 1[ ], + [ i=1 z(,t) x In this case, we have = z(1,t) x = in ], + [ S(t)z = n e n2 π 2t z, ϕ n ϕ n with ϕ n (ξ) = 2 cos(nπξ); n 1 and ϕ 1 The eigenvalues are given by λ n = n 2 π 2 ; n 1 and λ = (S(t)) t is not exponentially stable and the number of non negative eigenvalues is J = 1. The operators H u = p i=1 n e n2 π 2t u i (t)dt g i,ϕ n ϕ n H f = e n2 π 2t f,ϕ n ϕ n dt n and hence the asymptotic controllability problem are not generally well defined. The system (S 2 ) is augmented by the output equation: (E 2 ) y =( h 1,z,, h q,z ) tr with h 1,,h q orthogonal to ϕ, i.e. to the unstable part: h i,ϕ =; 1 i q

26 2684 L. Afifi, M. Bahadi and A. Chafiai The operators KC and R C are then well defined and the characterization results are similar to those obtained in example 1 for a Dirichlet boundary condition. On the other hand, concerning the stabilizability, in the case of one actuator, we have p =1 r =1 and using proposition 6.2, the system is stabilizable if and only if i.e. rankm = r =1 g, ϕ For g = ϕ n with n 1, we have g, ϕ = and then the system is not stabilizable. But for example, if also h = ϕ n, we have h, ϕ = The problem of asymptotic compensation is well posed. We have g, ϕ n h, ϕ n =1 The system is then remediable asymptotically. This not means that the asymptotic remediability is weaker than the stabilizabilty. The relation between these two notions depend on the choice of the sensors and the actuators. Indeed: If g(x) =2x, we have g, ϕ = 1, the system is then stabilizable. On the other hand, for h = ϕ 1, we have h, ϕ = In this case, the problem of asymptotic compensation is well posed. Moreover, we have g, ϕ 1 h, ϕ 1 and hence, the system is also asymptotically remediable. Now, if g = ϕ, we have g, ϕ and then the system is stabilizable. But for h = ϕ n with n 1, we have g, ϕ n h, ϕ n =; n 1

27 Regional asymptotic analysis 2685 consequently, the system is not asymptotically remediable. Obviously, with a non convenient choice of the sensors and the actuators, the system may be non remediable asymptotically and non stabilizable. Hence, for g = ϕ 1, we have g, ϕ = and then the system is not stabilizable. Concerning the remediability, if h = ϕ 2, we have g, ϕ n h, ϕ n =; n 1 then, the system is not also asymptotically remediable. Conclusion In this paper, we have presented a regional asymptotic analysis of the compensation problem. Indeed, under convenient hypothesis and a convenient choice of operators and spaces, we have first introduced and characterized the notions of weak and exact regional asymptotic remediability and regionally asymptotic efficient actuators. Then, we have introduced and characterized the notions of weak and exact regional asymptotic controllability and regionally asymptotic strategic actuators. Using an extension of the Hilbert Uniqueness Method, we have equally shown how to find, with respect to the observation only, the optimal control ensuring regionally the asymptotic compensation of a known or unknown disturbance. We have also characterized the set of disturbances which are asymptotically remediable in a region ω of Ω. We have shown that also in the asymptotic case, a system can be regionally remediable but not regionally controllable or without being remediable on the whole domain Ω. The relation between the regional asymptotic compensation and the notions of stability and stabilizability is also examined. It is particularly shown that a system can be asymptotically remediable in a region without being stable or stabilizable in this region. Applications to a diffusion system are given and various other situations are considered. The results are developed essentially for a class of linear distributed systems and for zone sensors and actuators, but these results can be extended to other systems and, with a convenient choice of spaces, to the case where the input and output operators are not bounded (pointwise sensors and pointwise actuators). ACKNOWLEDGEMENTS. With our gratitude for Professor A. El Jai and Academy Hassan II of Sciences and Technics.

28 2686 L. Afifi, M. Bahadi and A. Chafiai References [1] L. Afifi, A. Chafiai and A. El Jai, Sensors and actuators for compensation in hyperbolic systems. Foorteenth International Symposium of Mathematical Theory of Networks and Systems, MTNS 2, June (2), Perpignan, France. [2] L. Afifi, A. Chafiai and A. El Jai, Spatial Compensation of boundary disturbances by boundary actuators Int. J. of Applied Mathematics and Computer Science, Vol. 11, N 4 (21), pp 92. [3] L. Afifi, A. Chafiai and A. El Jai, Regionally efficient and strategic actuators. International Journal of systems Science. Vol. 33, N. 1 (22), pp [4] R.F. Curtain and A.J. Pritchard, Infinite Dimensional Linear Systems Theory. Lecture Notes in Control and Information Sciences, vol. 8, Berlin, [5] R.F. Curtain and H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Texts in Applied Mathematics, Springer-Verlag, New York, [6] A. El Jai and A.J. Pritchard, Sensors and controls in the Analysis of Distributed Systems. Texts in Applied Mathematic Series, Wiley, [7] A. El Jai and A.J. Pritchard, M.C. Simon and E. Zerrik, Regional controllability of Distributed Systems. International Journal of Control. Vol.62 (1995), pp [8] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués. Masson, Paris, [9] E. Zerrik, Analyse régionale des systèmes distribués. Thèse d Etat, Faculté des Sciences, Rabat, Maroc, Received: April 28, 27