IN the theory of control system, an optimal control problem

Transcription

1 IAENG Internatonal Journal of Appled Mathematcs, 46:3, IJAM_46_3_3 An Iteratve Non-overlappng Doman Decomposton Method for Optmal Boundary Control Problems Governed by Parabolc Equatons Wenyue Lu, Keyng Ma Abstract In ths paper, we consder a numercal method for solvng optmal boundary control problems governed by parabolc equatons. In order to avod large amounts of calculaton produced by tradtonal numercal methods, we establsh an teratve non-overlappng doman decomposton method. The whole doman s dvded nto many non-overlappng subdomans, and the optmal boundary control problem s decomposed nto local problems n these subdomans. Robn condtons are used to communcate the local problems on the nterfaces between subdomans. We buld the teratve scheme for solvng these local problems, and prove the convergence of the scheme. Fnally, we present a numercal example to verfy the valdty of the teratve scheme. Index Terms Parabolc equatons, optmal boundary control, non-overlappng doman decomposton method, teratve method, Robn condtons I. INTRODUCTION IN the theory of control system, an optmal control problem s to fnd a control model (.e. the control varable) admtted by the system to make the state varable tend to a target state n the process of optmzng (maxmzng/ mnmzng) the objectve functonal. If the state and control varable are subjected to partal dfferental equatons, the optmal control problem s called the optmal control problem governed by partal dfferental equatons (PDEs). In the feld of scence and engneerng, many problems, such as the Stefan-Boltzmann radaton law, the Lotka- Volterra model n populaton dynamcs, can be descrbed by optmal control problems governed by partal dfferental equatons. As well known, reference [] dscussed systematcally the theory and numercal methods of optmal control problems governed by PDEs. References []-[5] made further studes. Among the numercal methods to solve the optmal control problem governed by PDEs, an effectve one s the fnte element method,.e. buldng the fnte element space for the state varable and the control varable respectvely; establshng the dscretzed schemes for the governng PDEs; then developng the dscrete algebrac equaton systems to be solved. If the doman s large, there needs a large amount of calculaton, whch can be settled by the method of parallel computaton. Manuscrpt receved November 4, 5; revsed February, 6. Ths work was supported by the Natural Scence Foundaton of Chna under Grant 73 and 33. The correspondng author K.Y. Ma s n School of Mathematcs, Shandong Unversty, Jnan 5, Chna, e-mal: makeyng@sdu.edu.cn W.Y. Lu s n School of Mathematcs, Shandong Unversty, Jnan 5, Chna. A natural way n parallel computaton s non-overlappng doman decomposton method. Ths method can dvde the whole doman nto many subdomans, and decompose the optmal control problem nto many local problems, whch are ndependent ones on subdomans and can be calculated parallel. Hence, ths method can reduce much the amount of computaton. Untl now, there have been a lot of artcles consderng the applcaton of ths method to dfferent types of partal dfferental equatons, such as references [6]-[9]. References []-[3] dscussed some teratve non-overlappng doman decomposton methods for optmal boundary control problems governed by PDEs. The mportant character of these methods s how to buld nternal boundary condtons of state/co-state varables to communcate the local problems on the nterfaces between subdomans. Reference [4] presented an teratve another non-overlappng doman decomposton method for optmal boundary control problems governed by hyperbolc equatons and proposed an nternal boundary condton (called as Robn condton). The author proved the convergence of the method. But we should pont out that artcle [4] only consdered the case n whch the control varable s defned n the nteror of the doman, but not on the boundary. Invoked by the work of [4], we wll dscuss an teratve non-overlappng doman decomposton method for optmal boundary control problems governed by parabolc equatons. The structure of ths artcle s as follows: n Secton II, we gve an optmal boundary control problem governed by parabolc equatons, and buld the co-state equatons and optmal boundary condtons; In Secton III, we set up the teratve non-overlappng doman decomposton scheme by usng Robn condtons and prove the convergence; In Secton IV, we present an numercal example, and verfy the valdty of the teratve scheme. We make some conclusons n Secton V. II. MODEL Let Ω R be a bounded convex doman wth an smooth boundary Ω and [, T ] be a tme nterval. Throughout the paper, we adopt the standard notatons for Sobolev spaces on Ω. We wll take the state space L (, T ; V ) wth V = H (Ω) and the control space L (, T ; U) wth U L ( Ω). We consder the followng optmal boundary control problems governed by parabolc equatons: mn J(u, y(u)) () u U (Advance onlne publcaton: 6 August 6)

2 IAENG Internatonal Journal of Appled Mathematcs, 46:3, IJAM_46_3_3 subject to y(x, t) y(x, t) = f(x, t), (x, t) n Ω (, T ), y(x, t) = u(x, t) + g(x, t), (x, t) on Γ N (, T ), ν y(x, t) = y D (x, t), (x, t) on Γ D (, T ), y(x, ) = y (x), x n Ω, () where, the state varable y(u) L (, T ; V ) and the control varable u L (, T ; U), Γ N and Γ D are Neuman and Drchlet boundary, respectvely, Ω = Γ N Γ D, Γ N Γ D =, ν s an unt outer normal vector, f(x, t), g(x, t), y (x) and y D (x, t) are known functons. Let the objectve functonal be J(u, y(u)) = { γ y z d dxdt + α u dsdt }. Ω (,T ) Γ N (,T ) (3) Here, z d (x, t) s the desred state varable, the constants α > and γ > play the roles of balancng the contrbutons of the state varable y and control varable u. Accordng to references [], [3], we can derve the adjont equaton of () p(x, t) p(x, t) = γ(y(x, t) z d (x, t)), (x, t) n Ω (, T ), p(x, t) =, (x, t) on Γ N (, T ), (4) ν p(x, t) =, (x, t) on Γ D (, T ), p(x, T ) =, x n Ω, where p(x, t) s the co-state varable of y(x, t). And we know that when the objectve functonal J gets ts optmum, the control varable u L (, T ; U) should satsfy J (u)(v u), v L (, T ; U). (5) Accordng to the defnton of the drectonal dervatve of the objectve functonal and references [], [3], we can deduce that the nequalty (5) equals to J ( ) (u)(v u) = αu + p (v u)dsdt, (6) Γ N (,T ) v L (, T ; U). Ths nequalty s called as the optmalty condton. Then, the optmal boundary control problems ()-() are equvalent to an optmalty system composed of the state equaton (), the co-state equaton (4) and the optmalty condton (6). We can get the soluton of problems ()-() by solvng the optmalty system (), (4) and (6). III. ITERATIVE NON-OVERLAPPING DOMAIN DECOMPOSITION In ths secton, we wll buld an teratve non-overlappng doman decomposton scheme for the system (), (4) and (6), and prove the convergence. A. Iteratve Doman Decomposton Frst, we dvde Ω nto several non-overlappng subdomans Ω, =,,, N, Ω = N Ω, Ω Ωj =, j. Let Γ D, = Γ D Ω, Γ N, = Γ N Ω, Γ D,, Γ N,, Σ j = Ω Ωj be the nternal boundary between Ω and Ω j, and Σ j = Σ j. Let ν s the unt outer normal vector on Ω. We suppose that ths decomposton holds the regularty to guarantee the global and local equatons wth good propertes. Then, we decompose the system of (),(4) and (6) nto several local problems on subdomans and use the teratve method to solve them. Take the local problem on the subdoman Ω for an example,.e., the doman Ω and boundares Γ N, Γ D are replaced by Ω, Γ N,, Γ D,, respectvely. We defne the local soluton at step k + on subdoman Ω s (y k+, p k+, u k+ ). Hence, the local problem s and y k+ y k+ y k+ = f, n Ω (, T ), ν = u k+ + g, on Γ N, (, T ), = y D, on Γ D, (, T ), y k+ y k+ (x, ) = y (x), n Ω, pk+ p k+ = γ(y k+ z d ), n Ω (, T ), p k+ =, ν on Γ N, (, T ), p k+ =, on Γ D, (, T ), p k+ (T, x) =, n Ω, Γ N, (,T ) (7) (8) (p k+ + αu k+ )(v u k+ )dsdt, (9) v L (, T ; U ), where U s a local control space and just the restrcton of the space U on Ω,.e. u U, u Ω = u U. () For the later use, we defne the followng nner products and norms: (y, y ) = yy dxdt, y = (y, y), Ω (,T ) < y, y > j = yy dxdt, y j =< y, y > j, Σ j (,T ) < y, y > N, = yy dsdt, y N, =< y, y > N,. Γ N, (,T ) () (Advance onlne publcaton: 6 August 6)

3 IAENG Internatonal Journal of Appled Mathematcs, 46:3, IJAM_46_3_3 Accordng to these defntons, we establsh the teratve scheme of (7)-(8) on Ω ( yk+, v) + ( y k+, v) = (f, v) + < u k+ + g, v > Γ N, j,σ j < yk+, v > j, ν ( pk+, v) + ( p k+, v) = γ(y k+ z d, v) < pk+, v > j, ν j,σ j () where v H,ΓD, = {v H (Ω ) v =, when x on Γ D, }. At the same tme, we should put forward the boundary condtons on the nterfaces between the subdomans. These condtons are taken the form of Robn condton, where the state and co-state varables are skew-symmetrcally coupled [4]: y k+ + βp k+ = yk j + βp k j, on Σ j, ν ν j p k+ βy k+ = pk j βyj k, on Σ j, ν ν (3) where the constant β >. These condtons are called as the transmsson condtons, because they can strengthen the contnuty of the solutons of local problems and thers drectonal dervatve on Σ j at successve teraton steps. So the global problem can be composed by the local problems. The value of β wll be selected n Secton IV. B. Proof of convergence In the secton, we defne the local error between the global and local soluton n Ω (ȳ, p, ū ) = (y, p, u) (y, p, u ). (4) It s easy to see that ths error (ȳ, p, ū ) also satsfes the coupled equatons (7)-(9) and ()-(3), where f =, g =, y =, y d =. We use the followng sequence of energes on the nterfaces between subdomans to prove the convergence E k+ = j, Σ j { ȳ k+ j + β p j + pk+ ν ν + βȳ k+ } j. (5) Now, f we take the place of (y k+, p k+, u k+ ) by (ȳ k+, p k+, ū k+ ) n ()-(3), then (5) becomes E k+ = E k β j,σ j { ȳ k+ <, p k+ > j ν < pk+, ȳ k+ > j + < ȳk, p k > j ν ν < pk ν, ȳ k > j }. j (6) We take the place of (y k+, p k+, u k+ ) by (ȳ k+, p k+, ū k+ ) n (7)-(8). These equatons are multpled by p k+ and ȳ k+, and ntegrated by parts n spaces-tme doman, respectvely. Then, we can get the followng two equatons and (ȳ k+, pk+ ) + ( ȳ k+, p k+ =< ū k+, p k+ > ΓN, + < ȳk+, p k+ > Ω/Γ ν N,, ( pk+, ȳ k+ ) + ( p k+, ȳ k+ = γ(ȳ k+, ȳ k+ ) + < pk+, ȳ k+ > Ω. ν ) ) (7) (8) Subtractng these above results (7)-(8), we obtan, for all, j, Σ j {< ȳk+, p k+ ν > j < pk+, ȳ k+ > j } ν = γ ȳ k+ + < p k+, ū k+ > ΓN,. (9) Now, we use the global and local nequalty (6) and (9). Under the assumptons (), we take v = u k+ n (6) and v = u n (9), respectvely. Subtractng the two nequaltes, we can get the estmate < p k+, ū k+ > ΓN, α ū k+ Γ N, () Combnng (9) and () together, we can obtan the followng decrease law for the energes: E k+ E k β {γ ȳ k+ + α ū k+ +γ ȳ k + α ū k Γ N, }. Γ N, () In a word, we can obtan that the sequence {E k } s bounded and monotone decreasng, then the lmt of {E k } exsts. The followng result of convergence on each subdoman Ω can be derved ȳ k+ k, p k+ k, ū k+ ΓN, k. () Hence, the convergence of the scheme (7)-(9) s proven. IV. NUMERICAL EXAMPLE In ths secton, we present an example to prove the valdty of the teratve non-overlappng doman decomposton method mentoned n the above secton. We consder the model ()-(4) by choosng Ω = [, ] [ /, /] and T =. Let Γ N = Γ l Γ r, Γ D = Γ u Γ d, where Γ l and Γ r are the left and rght edges of Ω, respectvely; Γ u and Γ d are the upsde and downsde edges. To compare wth the numercal solutons well, we suppose the exact solutons of the model are: (x, x ) Ω, t [, T ] y = (T t) cos(πx ) cos(πx ), p = (T t) cos(πx ) cos(πx ), u = (T t) cos(πx ), z d = ((π + )(T t) + ) cos(πx ) cos(πx ), f = (π (T t) ) cos(πx ) cos(πx ). (3) We dvde Ω nto two non-overlappng subdomans: Ω = Ω Ω, Ω = [, ] [ /, /], Ω = [, ] (Advance onlne publcaton: 6 August 6)

4 IAENG Internatonal Journal of Appled Mathematcs, 46:3, IJAM_46_3_3 [ /, /], the nner boundary Γ = {} [ /, /]. In the process of calculaton, the numercal solutons are computed on the trangular meshes, wth dfferent mesh sze h =.,.,.5 sequentally. The state varable y and costate varable p are approxmated by pecewse lnear fnte elements, whle the control varable u s approxmated by pecewse constant fnte elements. Usng Backward-Euler scheme to approxmate the tme dervatve, we get the followng fully dscrete teratve scheme of (7)-(8) y n t = (f n+, v) + < u n+,k+ ( yn+,k+ ( pn+,k+ j,σ j, v) + ( y n+,k+, v) p n,k+ + g n+, v > ΓN, < yn+,k+, v > j, ν t = γ(y n+,k+ z n+ d, v) j,σ j, v) + ( p n,k+, v) < pn,k+, v > j, ν (4) where the frst subscrpt n + means at tme t n+ = (n + ) t, whle the second subscrpt k + s for the teratve step, and the tme step sze s t =.. Reference [4] showed that some egenvalues of the dscrete teraton operator are close to and sometmes even exceed because of numercal errors. Hence, they suggested to use an underrelaxed verson of the transmsson condtons nstead of (3). Followng ther deas to our example, we take the followng form y ν n+,k+ + βp n+,k+ = ρ ( yn+,k j ν j +( ρ) ( y n+,k ν p n+,k+ ν βy n+,k+ = ρ ( pn+,k j ν j +( ρ) ( p n+,k ν + βp n+,k ) j + βp n+,k ), on Σj, βy n+,k ) j βy n+,k ), on Σj. (5) Here, the parameter belongs to (, ) and s always chosen as ρ = /. It s easy to see that the smlar convergence proof as Secton III can also be establshed wth (5). The parameter β has a decsve nfluence on the speed of convergence. We choose β = /h for each case of our numercal calculatons. We present the followng numercal results at t =.5 for examples. Tables I and II show L -norm error and convergence rate of varables y, p and u n subdoman Ω and Ω, respectvely. We choose four ponts on Γ as examples to show the effect of the computatons. Table III consders for the state TABLE III THE COMPARISON OF STATE VARIABLE y ON THE INTERFACE Γ (x, x ) y y y y y (,.4) e 4 (,.5) e 4 (,.) e 4 (,.35) e 5 TABLE IV THE COMPARISON OF CO-STATE VARIABLE p ON THE INTERFACE Γ (x, x ) y y y y y (,.4) e 5 (,.5) e 4 (,.) e 4 (,.35) e 5 varable, where y s the exact soluton, y and y are the approxmate solutons n Ω and Ω, respectvely. And smlarly, Table IV shows for the co-state varable p. Takng the case of h =.5 for an example, Fgures -6 below present the fgure of the exact and approxmate soluton for varables y, p and u, respectvely. For the case of h =.5, Fgure 7 presents the trends of objectve functonal J(u) n Ω and Ω, when choosng α =. and α =. dfferently. The teraton numbers of J(u) change nsgnfcantly as long as the decreasng of the value of α, snce the value of α Γ N (,T ) u ds has a small nfluence on the whole value of J(u). V. CONCLUSION We have consdered an teratve non-overlappng doman decomposton method for solvng optmal boundary control problems governed by parabolc equatons. The teratve scheme was establshed. The whole doman was dvded nto many non-overlappng subdomans, and Robn condtons were used to communcate the local problems on the nterfaces between subdomans. We proved the convergence of the teratve scheme and presented a numercal example to verfy the valdty of the teratve scheme. In ths paper, the parabolc equatons were lnear, and the objectve functonal was defned over the whole tme nterval [, T ]. We can extend our method to the case of nonlnear parabolc equatons wth the objectve functonal at the fnal state. The results for ths case wll be presented n a forthcomng paper. ACKNOWLEDGMENT The authors would lke to thank the referees for ther constructve comments leadng to an mproved presentaton of ths paper. REFERENCES [] J. L. Lons, Optmal control of systems governed by partal dfferental equatons, Sprnger-Verlag, New York, 97. [] P. Nettaanmäk and D. Tba, Optmal control of nonlnear parabolc systmes, Theroy, Algorthms and Applcatons, Marcel Dekker, INC., New York, 994. [3] W.B. Lu and N.N. Yan, Adaptve fnte element method for optmal control governed by PDEs, Scence Press, Bejng, 8. (Advance onlne publcaton: 6 August 6)

5 IAENG Internatonal Journal of Appled Mathematcs, 46:3, IJAM_46_3_3 TABLE I L -NORM ERROR AND CONVERGENCE RATE IN SUBDOMAIN Ω h y p u error rate error rate error rate..5673e.73e 7.47e. 4.39e e e e e e.5 TABLE II L -NORM ERROR AND CONVERGENCE RATE IN SUBDOMAIN Ω h y p u error rate error rate error rate..734e.439e.885e e e e e e e (a) Exact soluton of y (b) Approxmate soluton of y Fg.. The exact and approxmate soluton of y n subdoman Ω [4] L. Ge, W.B. Lu and D.P. Yang, Adaptve fnte element approxmaton for a constraned optmal control problem va mult-meshes, Journal of Scentfc Computng, vol. 4, no., pp , 9. [5] N.N. Yan and Z.J. Zhou, A pror and a posteror error analyss of edge stablzaton Galerkn method for the optmal control problem governed by convecton-domnated dffuson equaton, Journal of Computatonal and Appled Mathematcs, vol. 3, no., pp. 98-7, 9. [6] P.E. Bjorstad and O.B. Wdlund, Iteratve methods for the soluton of ellptc problems onregonsp arttoned nto substructures, SIAM Journal on Numercal Analyss, vol. 3, no. 6, pp. 97-, 986. [7] J.H. Bramble, J.E. Pascak and A. H. Scharta, The constructon of precondng for ellptc problems by substructurng, Mathematcs of Computaton, vol. 47, no. 75, pp. 3-34, 986. [8] T.J. Sun and K.Y. Ma, Parallel Galerkn doman decomposton procedures for wave euqaton, Journal of Computatonal and Appled Mathematcs, vol. 33, pp ,. [9] K.Y. Ma and T.J. Sun, Galerkn doman decomposton procedures for parabolc equatons on rectangular doman, Internatonal Journal for Numercal Methods n Fluds, vol. 6, no. 4, pp ,. [] J.L. Lons, A. Bensoussan and R. Glownsk, Méthode de décomposton applquée au contrôle optmal de systèmes dstrbués, 5th IFIP Conference on Optmzaton Technques, Lecture Notes n Computer Scence, Sprnger Verlag, Berln, Germany, vol. 5, 973. [] M. Berggren and M. Henkenschloss, Parallel soluton of optmal control problems by tme-doman decomposton, Computatonal Scence for the st Century, edted by M.O. Brsteau, et al., Wley, New York, 997. [] G. Leugerng, Doman decomposton of optmal control problems for dynamc networks of elastc strngs, Computatonal Optmzaton and Applcatons, vol. 6, no., pp. 5-7,. [3] G. Leugerng, Dynamc doman decomposton of optmal control problems for networks of strngs and Tmoshenko beams, SIAM Journal on Control and Optmzaton, vol. 37, no. 6, pp , 999. [4] J.D. Benamou, Doman decomposton, optmal control of system governed by partal dfferental equatons, and Sysnthess of feedback laws, Journal of Optmzaton Theory and Applcatons, vol., no., pp. 5-36, 999. [5] J.D. Benamou, Décomposton de domane pour le contrôle optmal de systèmes gouvernés par des equatons d Evoluton, Comptes Rendus de l Académe des Scences de Pars, Sére I, vol. 34, pp. 65-7, 997. [6] J.D. Benamou, Doman decomposton methods wth coupled transmsson condtons for the optmal control of systems governed by ellptc partal dfferental equatons, SIAM Journal on Numercal Analyss, vol. 33, no. 6, pp. 4-46, 996. (Advance onlne publcaton: 6 August 6)

6 IAENG Internatonal Journal of Appled Mathematcs, 46:3, IJAM_46_3_ (c) Exact soluton of y (d) Approxmate soluton of y Fg.. The exact and approxmate soluton of y n subdoman Ω (e) Exact soluton of p (f) Approxmate soluton of p Fg.3. The exact and approxmate soluton of p n subdoman Ω (g) Exact soluton of p (h) Approxmate soluton of p Fg.4. The exact and approxmate soluton of p n subdoman Ω (Advance onlne publcaton: 6 August 6)

7 IAENG Internatonal Journal of Appled Mathematcs, 46:3, IJAM_46_3_ () Exact soluton of u (j) Approxmate soluton of u Fg.5. The exact and approxmate soluton of u n subdoman Ω (k) Exact soluton of u (l) Approxmate soluton of u Fg.6. The exact and approxmate soluton of u n subdoman Ω the value of objectve functonal α=. α= teraton numbers (m) n Ω the value of objectve functonal α=. α= teraton numbers (n) n Ω Fg.7. The functonal J(u) n Ω and Ω at h =.5 (Advance onlne publcaton: 6 August 6)