Pointwise observation of the state given by parabolic system with boundary condition involving multiple time delays

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1 1.1515/acsc Archives of Cotrol Scieces Volue 26(LXII), 216 No. 2, pages Poitwise observatio of the state give by parabolic syste with boudary coditio ivolvig ultiple tie delays ADAM KOWALEWSKI Various optiizatio probles for liear parabolic systes with ultiple costat tie delays are cosidered. I this paper, we cosider a optial distributed cotrol proble for a liear parabolic syste i which ultiple costat tie delays appear i the Neua boudary coditio. Sufficiet coditios for the existece of a uique solutio of the parabolic equatio with the Neua boudary coditio ivolvig ultiple tie delays are proved. The tie horizo T is fixed. Makig use of the Lios schee [13], ecessary ad sufficiet coditios of optiality for the Neua proble with the quadratic cost fuctio with poitwise observatio of the state ad costraied cotrol are derived. Key words: distributed cotrol, parabolic syste, tie delays, poitwise observatio. 1. Itroductio Ifiite diesioal distributed paraeter systes with delays ca be used to describe ay pheoea i the real world. As is well kow, heat coductio, properties of elastic-plastic aterial, fluid dyaics, diffusio-reactio processes, trasissio of the sigals at the certai distace by usig electric log lies, etc., all lie withi this area. The object that we are studyig (teperature, displaceet, cocetratio, velocity, etc.) is usually referred to as the state. We are iterested i the case where the state satisfies proper differetial equatios that are derived fro basic physical laws, such as Newto s law, Fourier s law etc. The space i which the state exists is called the state space, ad the equatio that the state satisfies is called the state equatio. Cosequetly, by a ifite diesioal syste we ea oe whose correspodig state space is ifiite diesioal. I particular, we are iterested i the cases where the state equatios are oe of the followig types: partial differetial equatios, itegro-differetial equatios, or abstract evolutio equatios. The Author is with AGH Uiversity of Sciece ad Techology, Istitute of Autoatics ad Bioedical Egieerig, 3-59 Cracow, al. Mickiewicza 3, Polad, e-ail: ako@agh.edu.pl. The research preseted here was carried out withi the research prograe AGH Uiversity of Sciece ad Techology, No Received

2 19 A. KOWALEWSKI Very iportat optiizatio probles associated with the optial cotrol of distributed parabolic systes with tie delays appearig i the boudary coditios have bee studied recetly i Refs. [1] - [12] ad [15], [16]. I this paper, we cosider a optial distributed cotrol proble for a liear parabolic syste i which ultiple costat tie delays appear i the Neua boudary coditio. Such systes costitute i a liear approxiatio, a uiversal atheatical odel for the plasa cotrol process [12]. Sufficiet coditios for the existece of a uique solutio of such parabolic equatios with the Neua boudary coditios ivolvig ultiple tie delays are proved. I this paper, we restrict our cosideratios to the case of the distributed cotrol for the Neua proble. Cosequetly, we forulate the followig optial cotrol proble. We assue that the cost fuctio has the quadratic for with poitwise observatio of the state. Moreover, the tie horizo is fixed i our optiizatio proble. Fially, we ipose soe costraits o the distributed cotrol. Makig use of the Lios fraework [13] ecessary ad sufficiet coditios of optiality with the quadratic cost fuctio with poitwise observatio of the state ad costraied cotrol are derived for the Neua proble. 2. Existece ad uiqueess of solutios Cosider ow the distributed-paraeter syste described by the followig parabolic delay equatio y + A(t)y = v x Ω,t (,T ) (1) t y = η A y(x,) = y (x) x Ω (2) y(x,t h i ) + u x Γ,t (,T ) (3) y(x,t ) = Ψ (x,t ) x Γ,t [ h,) (4) where: Ω R is a bouded, ope set with boudary Γ, which is a C - aifold of y diesio 1. Locally, Ω is totally o oe side of Γ. is a oral derivative at Γ, η A directed towards the exterior of Ω, y y(x,t;v), v v(x,t), = Ω (,T ), = Ω [,T ], Σ = Γ (,T ), Σ = Γ [ h,)

3 POINTWISE OBSERVATION OF THE STATE GIVEN BY PARABOLIC SYSTEM WITH BOUNDARY CONDITION INVOLVING MULTIPLE TIME DELAYS 191 h i are specified positive ubers represetig tie delays such that h 1 < h 2 <... < h for i = 1,...,, Ψ is a iitial fuctio defied o Σ. The operator A(t) has the for A(t)y = i, j=1 x i ad the fuctios a i j (x,t) satisfy the coditio i, j=1 a i j (x,t)φ i Φ j α ( a i j (x,t) y(x,t) ) x j (5) Φ 2 i α >, (6) (x,t), Φ i R where: a i j (x,t) are real C fuctios defied o (closure of ). The equatios (1) - (4) costitute a Neua proble. First we shall prove sufficiet coditios for the existece of a uique solutio of the ixed iitial-boudary value proble (1) - (4) for the case where v L 2 (). For this purpose, for ay pair of real ubers r,s, we itroduce the Sobolev space H r,s () ( [14], Vol. 2, p.6) defied by H r,s () = H (,T ;H r (Ω)) H s (,T ;H (Ω)) (7) which is a Hilbert space ored by y(t) 2 H r (Ω) dt+ y 2 H s (,T ;H (Ω)) 1 2 (8) where: the spaces H r (Ω) ad H s (,T ;H (Ω)) are defied i Chapter 1 ( [14], Vol.1) respectively. Cosequetly, soe properties ad cetral theores for the fuctios y H r,s () are give i [8], [11] ad [14]. The existece of a uique solutio for the ixed iitial-boudary value proble (1) - (4) o the cylider ca be proved usig a costructive ethod, i.e., first, solvig (1) - (4) o the subcylider 1 ad i tur o 2, etc. util the procedure covers the whole cylider. I this way the solutio i the previous step deteries the ext oe. For siplicity, we itroduce the followig otatios: E j = (( j 1)λ, jλ) where λ = ihi, j = Ω E j, Σ j = Γ E j for j = 1,... K. Usig the results of Sectio 14 ([13], pp ) we ca prove the followig result.

4 192 A. KOWALEWSKI Lea 1 Let v L 2 () (9) y j 1 (,( j 1)λ) L 2 (Ω) (1) where: q j (x,t) = q j H 1/2,1/4 (Σ j ) (11) y j 1 (x,t h i ) + u(x,t) The, there exists a uique solutio y j H 2,1 ( j ) for the ixed iitial-boudary value proble (1), (3), (1). Proof We observe that for j = 1, y j 1 Σ (x,t h i ) = Ψ (x,t h i ). The the assuptios (1) ad (11) are fulfilled if we assue that Ψ H 1/2,1/4 (Σ ), u H 1/2,1/4 (Σ) ad y L 2 (Ω). These assuptios are sufficiet to esure the existece of a uique solutio y 1 H 2,1 ( 1 ). Next for j = 2 we have to verify that y 1 (,λ) L 2 (Ω) ad q 2 H 1/2,1/4 (Σ 2 ). Really, fro the Theore 3.1 ( [14], Vol.1, p.19) we ca prove that y 1 H 2,1 ( 1 ) iplies that the appig t y 1 (,t) is cotiuous fro [,λ] H 1 (Ω) L 2 (Ω), hece y 1 (,λ) L 2 (Ω). The usig the Trace Theore ( [14], Vol. 2, p.9) we ca verify that y 1 H 2,1 Σ1 ( 1 ) iplies that y 1 y 1 is a liear, cotiuous appig of H 2,1 ( 1 ) H 1/2,1/4 (Σ). Assuig that u H 1/2,1/4 (Σ), the coditio q 2 H 1/2,1/4 (Σ 2 ) is fulfilled. We shall ow suarize the foregoig result for ay j, j = 3,...,K. Theore 1 Let y,ψ,u ad v be give with y L 2 (Ω),Ψ H 1/2,1/4 (Σ ), u H 1/2,1/4 (Σ) ad v L 2 (). The, there exists a uique solutio y H 2,1 () for the ixed iitial-boudary value proble (1) - (4). Moreover, y(, jλ) L 2 (Ω) for j = 1,...,K. 3. Proble forulatio. Optiizatio theores We shall ow forulate the optial distributed cotrol proble for the Neua proble. Let us deote by U = L 2 () the space of cotrols. The tie horizo T is fixed i our proble. Let x 1,...,x be poits of Ω. We assue that the observatio is {y(x j,t;v)}, 1 j - provided we ca attach a eaig to this.

5 POINTWISE OBSERVATION OF THE STATE GIVEN BY PARABOLIC SYSTEM WITH BOUNDARY CONDITION INVOLVING MULTIPLE TIME DELAYS 193 If we ow assue that the coefficiets of the operator A i the equatio (1) are sufficietly regular, the fro the Theore 1 it follows that y(v) H 2,1 (). (12) Hece, y(v) L 2 (,T ;H 2 (Ω)) ad y(x j,t) has eaig (ad t y(x j,t) L 2 (,T )) if H 2 (Ω) C (Ω) (13) which is true if (ad oly if) < i.e. 3. (14) Hece we ake the stadig hypothesis that the diesio is 3. The the observatio Cy(v) = {y(x j,t;v)} (L 2 (,T )). (15) The cost fuctio is ow give I(v) = λ 1 Cy(v) z d 2 (L 2 (,T )) +λ 2 (Nv)v dxdt. (16) If z d = {z d1,...,z }, I(v) = λ 1 j=1 y(x j,t;v) z d j (t) 2 dt+ λ 2 (Nv)v dxdt (17) where: λ i, λ 1 + λ 2 > ; z d j (t) are give eleets i L 2 (,T ) ad N is a positive, liear operator o L 2 () ito L 2 (). Fially, we assue the followig costrait o cotrols v U ad, where U ad is a closed, covex subset of U (18) Let y(x,t;v) deote the solutio of the ixed iitial-boudary value proble (1)- (4) at (x,t) correspodig to a give cotrol v U ad. We ote fro the Theore 1 that for ay v U ad the cost fuctio (17) is well-defied sice y(v) H 2,1 (). The solvig of the forulated optial cotrol proble is equivalet to seekig a v U ad such that I(v ) I(v) v U ad. The fro the Theore 1.3 ( [13], p. 1) it follows that for λ 2 > a uique optial cotrol v exists; oreover, v is characterized by the followig coditio I (v ) (v v ) v U ad (19)

6 194 A. KOWALEWSKI Usig the for of the cost fuctio give by (17) we ca express (19) i the followig for λ 1 (y(x j,t;v ) z d j (t))(y(x j,t;v) y(x j,t;v ))dt + j=1 +λ 2 Nv (v v )dxdt v U ad (2) To siplify (2), we itroduce the adjoit equatio ad for every v U ad, we defie the adjoit variable p = p(v) = p(x,t;v) as the solutio of the equatio p(v) t x Ω, t (,T ) + A (t)p(v) = λ 1 j=1(y(x j,t;v) z d j (t)) δ(x x j ) (21) where p(v) η A (x,t) = p(x,t ;v) = x Ω (22) p(x,t + h i ;v) x Γ,t (,T h ) (23) p(v) (x,t) = x Γ,t (T h,t ) (24) η A g(t) δ(x x j ) is the distributio, Ψ A (t)p = g(t)ψ(x j,t)dt, Ψ D() i, j=1 x j ( a i j (x,t) p ) x i The existece of a uique solutio for the proble (21) - (24) o the cylider ca be proved usig a costructive ethod. It is easy to otice that for give z d ad v, proble (21) - (24) ca be solved backwards i tie startig fro t = T, i.e., first, solvig (21) - (24) o the subcylider K ad i tur o K 1, etc. util the procedure covers the whole cylider. For this purpose, we ay apply Theore 1 (with a obvious chage of variables) to proble (21) - (24) (with reversed sese of tie, i.e., t = T t). Lea 2 Let the hypothesis of Theore 1 be satisfied. The, for give z d j (t) L 2 (,T ) ad ay v L 2 (), there exists a uique solutio p(v) H 2,1 () for the proble (25)

7 POINTWISE OBSERVATION OF THE STATE GIVEN BY PARABOLIC SYSTEM WITH BOUNDARY CONDITION INVOLVING MULTIPLE TIME DELAYS 195 (21) - (24) defied by traspositio = j=1 p(v ) ( ) Ψ t + AΨ dxdt = ( y(x j,t;v ) z d j (t) ) Ψ(x j,t)dt (26) Ψ H 2,1 (),Ψ = ad Ψ(x,T ) =. Σ Reark 1 The right had side of (26) is a cotiuous liear for o H 2,1 () if 3. Cosequetly, after trasforatios, the first copoet o the left-had side of (2) ca be rewrite as λ 1 j=1 (y(x j,t;v ) z d j (t))(y(x j,t;v) y(x j,t;v ))dt = = Substitutig (27) ito (2) we obtai p(v )(v v ) dxdt (27) (p(v ) + λ 2 Nv )(v v ) dxdt v U ad (28) Theore 2 For the proble (1) - (4) with the cost fuctio (17) with z d j (t) L 2 (,T ) ad λ 2 > ad with costraits o cotrols (18), there exists a uique optial cotrol v which satisfies the axiu coditio (28). Cosider ow the particular case where U ad = L 2 (). Thus the axiu coditio (28) is satisfied whe v = λ 1 2 N 1 p(v ) (29) We ust otice that the coditios of optiality derived above (Theore 2) allow us to obtai a aalytical forula for the optial cotrol i particular cases oly (e.g. there are o costraits o cotrols). This results fro the followig: the deteriig of the fuctio p(v ) i the axiu coditio fro the adjoit equatio is possible if ad oly if we kow y which correspods to the cotrol v. These utual coectios ake the practical use of the derived optiizatio forulas difficult. Therefore we resig fro the exact deteriig of the optial cotrol ad we use approxiatio ethods.

8 196 A. KOWALEWSKI I the case of perforace fuctioal (17) with λ 1 > ad λ 2 =, the optial cotrol proble reduces to the iiizig of the fuctioal o a closed ad covex subset i a Hilbert space. The, the optiizatio proble is equivalet to a quadratic prograig oe (Ref. [11]) which ca be solved by the use of the well-kow algoriths, e.g. Gilbert s (Ref. [11]). The practical applicatio of Gilbert s algorith to a optial cotrol proble for a parabolic syste with boudary coditio ivolvig a tie delay is preseted i [12]. Usig Gilbert s algorith, a oe-diesioal uerical exaple of the plasa cotrol process is solved. 4. Coclusios The results preseted i the paper ca be treated as a geeralizatio of the results cocerig poitwise observatio of the state give by the parabolic equatio with the hoogeeous Dirichlet boudary coditio obtaied i [13] oto the case of parabolic equatios with the Neua boudary coditios ivolvig ultiple costat tie delays. Sufficiet coditios for the existece of a uique solutio of such parabolic equatios with the Neua boudary coditios ivolvig ultiple costat tie delays are proved (Lea 1 ad Theore 1). The optial cotrol is characterized by usig the adjoit equatio (Lea 2). The ecessary ad sufficiet coditios of optiality are derived for a liear quadratic proble (1)-(4), (17), (18) (Theore 2). We ca also obtai estiates ad a sufficiet coditio for the boudedess of solutios for such parabolic tie delay systes with specified fors of feedback cotrol. Fially, we ca cosider optial cotrol probles of tie delay hyperbolic systes with poitwise observatio of the state. The ideas etioed above will be developed i forthcoig papers. Refereces [1] G. KNOWLES: Tie-optial cotrol of parabolic systes with boudary coditios ivolvig tie delays. J. Optiiz. Theor. Applics., 25(4), (1978), [2] A. KOWALEWSKI: Optial cotrol with iitial state ot a priori give ad boudary coditio ivolvig a delay. Lecture Notes i Cotrol ad Iforatio Scieces, 95 (1987), 94-18, Spriger-Verlag, Berli-Heidelberg. [3] A. KOWALEWSKI: Boudary cotrol of distributed parabolic syste with boudary coditio ivolvig a tie-varyig lag. It. J. Cotrol, 48(6), (1988),

9 POINTWISE OBSERVATION OF THE STATE GIVEN BY PARABOLIC SYSTEM WITH BOUNDARY CONDITION INVOLVING MULTIPLE TIME DELAYS 197 [4] A. KOWALEWSKI: Feedback cotrol for a distributed parabolic syste with boudary coditio ivolvig a tie-varyig lag. IMA J. Math. Cotrol ad Iforatio, 7(2), (199), [5] A. KOWALEWSKI: Miiu tie proble for a distributed parabolic syste with boudary coditio ivolvig a tie-varyig lag. Arch. Autoatic ad Reote Cotrol, XXXV(3-4), (199), [6] A. KOWALEWSKI: Optiality coditios for a parabolic tie delay syste. Lecture Notes i Cotrol ad Iforatio Scieces, 144, (199), , Spriger- Verlag, Berli-Heidelberg. [7] A. KOWALEWSKI: Optial cotrol of parabolic systes with tie-varyig lags. IMA J. Math. Cotrol ad Iforatio, 1( 2), (1993), [8] A. KOWALEWSKI: Optial cotrol of distributed parabolic systes with ultiple tie tie-varyig lags. It. J. Cotrol, 69(3), (1998), [9] A. KOWALEWSKI: Optiizato of parabolic systes with deviatig arguets. It. J. Cotrol, 72(11), (1999), [1] A. KOWALEWSKI: Optial cotrol of tie delay parabolic systes. Optiizatio, 5(1-2), (21), [11] A. KOWALEWSKI: Optial cotrol of ifiite diesioal distributed paraeter systes with delays. Uiversity of Miig ad Metallurgy Press, Cracow, 21. [12] A. KOWALEWSKI ad J. DUDA: O soe optial cotrol proble for a parabolic syste with boudary coditio ivolvig a tie-varyig lag. IMA J. Math. Cotrol ad Iforatio, 9(2), (1992), [13] J.L. LIONS: Optial cotrol of systes govered by partial differetial equatios. Spriger-Verlag, Berli-Heidelberg, [14] J.L. LIONS ad E. MAGENES: No-hoogeeous boudary value probles ad applicatios. 1 ad 2, Spriger-Verlag, Berli-Heidelberg, [15] P.K.C. WANG: Optial cotrol of parabolic systes wih boudary coditios ivolvig tie delays. SIAM J. Cotrol, 13(2), (1975), [16] K.H. WONG: Optial cotrol coputatio for parabolic systes with boudary coditios ivolvig tie delays. J. Optiiz. Theor. Applics., 53(3), (1987),