Optimal Control of Dissolved Oxygen in Shallow Water Flow

Transcription

1 Kawahara Lab. 21 Mar Optmal Control of Dssolved Oxygen n Shallow Water Flow Toshk SEKINE Department of Cvl Engneerng, Chuo Unversty, Kasuga , Bunkyo-ku, Tokyo , Japan E-mal : d34115@educ.kc.chuo-u.ac.jp Abstract The purpose of ths paper s to present a control method of dssolved oxygendo usng the frst order adjont method and Newton based method. In the optmal control theory, the control varable s obtaned by the mnmzaton of the performance functon. The gradent of performance functon updates the control varable. The control varable s computed so as to mnmze the performance functon under the constrant of the state equatons and boundary condtons. About Newton based method, ths technque s used by the second order adjont equaton. The gradent of performance functon s obtaned by frst order adjont method. The gradent of the second order adjont equaton whch can be obtaned by calculatng an extended perturbaton s smlar to product of the Hessan matrx and perturbaton vector of control varable. The Hessan matrx s approxmately updated by BFGS method Broyden-Fletcher-Goldfarb-Shanno method or DFP Davdon-Fletcher-Powell method. Performance functon s calculated by updatng control varable. In ths research, velocty of rver s defned as control varable. As the state equaton, the coupled the shallow water equaton and the advecton dffuson equaton s appled. DO whch s dffused by the shallow water flow s controlled by the nput water flow. In the numercal study, example of optmal control of DO s shown usng Teganuma rver. As a numercal result, comparson of BFGS method and DFP method s shown. Key words : Fnte Element Method, Newton Based Method, BFGS method, DFP method, Second Order Adjont Equaton, Water Purfcaton Control, 1 Introducton In recent years, the water state of long rvers become recovered wth techncal mprovement of the water mantenance n Japan. However, ths s hardly seen n small rvers. Water polluton problem assumes serous proportons. In ths research, dssolved oxygen DO s pcked up as one of the ndces of water state. There are several water ndces as examples of BOD Bochemcal Oxygen Demand, COD Chemcal Oxygen Demand, and so on. The reason why I chose DO s because ths contamnant concentraton s appled to all type of water area such as rver, lakes, seas and so on. BOD s not appled to lake because t takes long tme to measure data n lake. COD s appled to lake and sea. However, t s not appled to rver. The water velocty n Teganuma rver s too small. Teganuma rver has property as not only the rver but also the lake. Therefore, DO whch s appled to all type of water area s reason for choce. The Teganuma rver s located n Chba prefecture n Japan. Ths rver s pcked up as a practcal phenomenon because of serous polluton problem. Recently, the water conductng project s performed n part of upstream of ths rver. However, t s neffectve for ths rver because water hardly flow. The optmal velocty s needed to 1

2 dffuse clean water for ths rver. If the optmal velocty s found, the hgher DO can be dffusng n ths rver. Then rver can be cleaned up. Therefore, optmal control of DO s carred out to fnd the optmal velocty. In ths study, water velocty s calculated as the control varable. The control varable s computed so as to mnmze the performance functon under the constrant of the state equaton and boundary condtons. The performance functon s defned the square sum of dfference between the computed and the target DO. The mnmzaton of the performance functon means that computed value becomes as close as possble to the target value. The frst order adjont equaton can be derved by the state equaton and the boundary condtons. The frst order adjont equaton s solved by the Lagrange multpler method. The Lagrange multpler method s sutable for the mnmzaton problem wth constrant condton. In ths research, two types of soluton method are appled as mnmzaton technque. One s weghted gradent method. It has been used wdely for mnmzaton technque. The theory s that the frst order adjont equaton s solved by Lagrange multpler method. Control varable s updated by the gradent whch s obtaned by the frst order adjont equaton. Another technque s the Newton based method. It s constructed by the frst and the second order adjont equaton. The frst order adjont equaton s generated by the state equaton and the boundary condtons. The second order adjont equaton s generated by perturbaton equaton and boundary condtons. The perturbaton equaton for the state varables can be derved by the formulaton that the state equaton s extended by the perturbatons. The BFGS method s one of the Newton type mnmzaton technque. Ths method can be approxmately obtaned the product between the Hessan matrx and control varable. The Hessan matrx s updated by the dfference of gradent provded by the frst order adjont equaton and the Newton based method. The more exact Newton drecton as gradent of performance functon can be obtaned by ths method. The more exact control varable can be obtaned by ts gradent. As the state equaton, the coupled the shallow water equaton wth the advecton dffuson equaton s appled. As the dscretzaton technque, the Crank-Ncolson method n the temporal drecton and the the fnte element method usng stablzed bubble functon n the space drecton are appled. The ndcal notaton and summaton conventon wth repeated ndces are used. 2 State Equaton The non-lnear shallow water equaton s used as the state equaton. The non-lnear shallow water equaton can be wrtten as follows: u u j u,j gη H ξ, νu,j u j,,j fu =0, 1 η {η Hu, =0, 2 where u,g,η,ξ and H are water velocty, gravtatonal acceleraton, water elevaton, bed elevaton and water depth, respectvely. The coeffcent of knematc eddy vscosty ν and the bottom frcton f are expressed as follows: z ν = k l 6 u η H, f = u η H. 3 η Η g u y ξ x Fg.1 X-Y-Z Coordnate System 2

3 The boundary condtons are gven as follows: u t =û on Γ D, 4 η t =ˆη on Γ D, 5 u t =u n =û on Γ N, 6 u t =U on Γ C, 7 where the boundares Γ D, Γ N and Γ C are the Drchlet, Neumann and control boundary, respectvely and U s control varable. The ntal condtons are gven as follows: The advecton dffuson equaton can be wrtten as follows: u =û n, 8 η =ˆη n, 9 ċ u c, κc, =0, 10 where c and κ are concentraton of dssolved oxygen and the dffuson coeffcent, respectvely. The boundary condtons are gven as follows: The ntal condtons s gven as follows: c t =ĉ on Γ D, 11 b t =κc, n = ˆb on Γ N, 12 c t =ĉ on Γ C, 13 c =ĉ n Dscretzaton Technque 3.1 Temporal Dscretzaton As for the temporal dscretzaton of the basc equatons, the Crank-Ncolson method s appled: where, u n1 u n t u j u n 1 2,j gη n 1 2 H ξ, νu n 1 2,j u n 1 2 j,,j fu n 1 2 =0, 15 η n1 η n t c n1 c n t u η n 1 2, ηu n 1 2, =0, 16 u c n 1 2, cu n 1 2, κc n 1 2, =0, 17 u n 1 2 = 1 2 un1 u n, η n = 2 ηn1 η n c n = c n1 c n

4 3.2 Spatal Dscretzaton Bubble Functon Interpolaton As for the spatal dscretzaton of the basc equatons, the fnte element method based on the bubble functon nterpolaton s appled: u =Φ 1 u 1 Φ 2 u 2 Φ 3 u 3 Φ 4 ũ 4, 19 ũ 4 = u u 1 u 2 u 3, 20 η =Φ 1 η 1 Φ 2 η 2 Φ 3 η 3 Φ 4 η 4, 21 η 4 = η η 1 η 2 η 3, 22 c =Φ 1 c 1 Φ 2 c 2 Φ 3 c 3 Φ 4 c 4, 23 c 4 = c c 1 c 2 c 3, 24 Φ 1 = L 1, Φ 2 = L 2, Φ 3 = L 3, Φ 4 =27L 1 L 2 L Fg.2 Bubble Functon Element Stablzed Form In the bubble functon, the numercal stablzaton s not enough. Therefore the stablzed parameter s used for stablzaton. The stablzed parameter of the shallow water equaton and the advecton dffuson equaton τ ebu, τ ebξ and τ ebc can be wrtten as follows: for the momentum equaton of shallow water equaton, for the contnuty equaton, τ ebu = φ e, 1 2 e A 1 e 1 t φ e 2 e 1 2 ν ν 2 φ e, 2 e f φ e 2 e, 26 and for the advecton dffuson equaton, τ ebη = φ e, 1 2 e A 1 e 1 t φ e 2 e 1 2 ν jj φ e,j 2 e, 27 φ e, 1 2 τ ebc = e A 1 e 1 t φ e 2 e 1 2 κ ν φ e, 2, 28 where, ν s stablzng parameter. τ ebu, τ ebη and τ ebc are determned to be equvalent to τ es usng the stablzed fnte element method: 4

5 for the momentum equaton of shallow water equaton, 1 τ ebu = 2 τ 1 es u α , τ 1 2 U 4ν u es t u = h e h 2, 29 e η H for the contnuty equaton, 1 τ ebη = 2 τ 1 es η α 1 2 U , τ 1 4ν u es t η = h e h 2, 30 e η H and for the advecton dffuson equaton, where, τ ebc = 1 2 τ 1 es C α 1, τ 1 es t C = 2 UC 2 4κ h e h 2 e 2 1 2, 31 α = A e φ e 2 e φ e, 1 2, h e = 2A e, U = u 2 v 2 gη H, U C = u 2 v e Therefore, the stablzed parameter for the momentum and the contnuty equaton of the shallow water equaton and for the advecton dffuson equaton can be wrtten as follows: for the momentum equaton of shallow water equaton, for the contnuty equaton, ν ν 2 φ e, 2 e = φ e, 1 2 e τes 1 f φ e 2 A e, 33 e and for the advecton dffuson equaton, ν jj φ e,j 2 e = φ e, 1 2 e τes 1 A, 34 e κ ν φ e,j 2 e = φ e, 1 2 e τ 1 es A C, 35 e these can be effected to barycenter pont of vscosty term. e s element doman, a, b e, φ e 2 e and A e s expressed as follows: a, b e = ab d, φ e 2 e = φ e,φ e e, A e = d, 36 e e the ntegraton of bubble functon s expressed as follows: φ e, 1 φe = A e 3, φ e 2 e = A e Performance Functon The control varable s computed so as to mnmze the performance functon under the constrant of the state equaton and boundary condtons. The mnmzaton of the performance functon means that concentraton of DO becomes as close as possble to the target concentraton of DO. J = 1 c c obj T Q c c obj ddt, 38 2 where Q, c and c obj are weghtng dagonal matrx, the computed DO and the target water DO, respectvely. The superscrpt T denotes transpose. 5

6 5 Adjont Equaton Method 5.1 Extended Performance Functon The Lagrange multpler method s sutable for the mnmzaton problem wth constrant condton. The extended performance functon J s expressed as follows: J = J u u u j u,j gη H ξ, νu,j u j,,j fu ddt η η {η Hu, ddt c ċ u c, κc, ddt, 39 Where u, η and c denote the Lagrange multpler for velocty, water elevaton and DO, respectvely. The purpose s to fnd the optmal control varable so as to mnmze the performance functon under the constrant of the state equaton and boundary condton. Expanson of each varable s expressed as follows: u = u 0 u 1 u 2..., η = η 0 η 1 η 2..., c = c 0 c 1 c 2..., u = u 0 u 1 u 2..., η = η 0 η 1 η 2..., c = c 0 c 1 c 2..., 40 J s rearranged and expressed as follows when these varables substtute to extended performance functon, J = 1 T c 0 c 1 c 2... Q c 0 c 1 c 2... ddt 2 { u 0 u 1 u 2... u 0 u 1 u 2... u 0 j u 1 j u 2 j... u 0,j u1,j u2,j... g η 0 η 1 η 2... H ξ, ν u 0 j, u1 j, u2 j,... u 0,j u1,j u2,j...,j f u 0 u 1 u 2... ddt { η 0 η 1 η 2... η 0 η 1 η 2... η 0 η 1 η 2... H u 0 u 1 u 2... ddt, { c 0 c 1 c 2... ċ 0 ċ 1 ċ u 1 u 2... u c 0 c 1 c 2... κ c 0 c 1 c 2... ddt, = J 0 J 1 J 2..., 41, 6

7 where J 0 s expressed as follows: J 0 = 1 2 c 0 c obj T Q c 0 c obj ddt u 0 u 0 u 0 j u 0,j gη0 H ξ, νu 0,j u0 j,,j fu 0 ddt η 0 η 0 {η 0 Hu 0, ddt c 0 ċ 0 u 0 c 0, κc 0, ddt. 42 Eq.42 s the same form as the extended performance functon t self Eq Dervaton of Adjont Equaton Method The extended performance functon J can be expressed n the followng form wth respect to control varable U. J = J 0 g T U 1 2 U T HU... ds c dt, 43 S c where g, U and H denote the gradent, Hessan matrx and control varable, respectvely. The control varable U s also expanded as follows: U = U 1 U 2..., 44 then, J s rearranged and expressed as follows: J = J 0 g T U U 1T HU 1 Eq.45 can be rewrtten as follows: where S c g T U U 1T HU U 2T HU U 2T HU 2... ds c dt. 45 J = J 0 J 1 J 2... = J 0 J 1 1 J 1 2 J 2 1 J 2 2 J , 46 J 1 1 = J 1 2 = 1 2 J 2 1 = J 2 2 = 1 2 J 2 3 = 1 2 consderng the statonary condton, f g T U 1 ds c dt, 47 S c U 1T HU 1 ds c dt, 48 S c g T U 2 ds c dt, 49 S c U 1T HU 2 U 2T HU 1 ds c dt, 50 S c S c U 2T HU 2 ds c dt. 51 J 2 1 = 0, 52 J 1 1 J 2 2 = 0, 53 then J J

8 ths condton s derved by the followng condtons, Thus, from Eqs.52 and 53, the followng necessary condtons can be derved. J 1 2 0, 55 J g T U 2 = 0, 57 g T U U 1T HU U 2T HU 1 = from Eq.57, followng condton can be derved. g =0, 59 whch s the basc dea of the weghted gradent method. From Eq.58 t s obtaned that, g HU 2 =0, 60 usng the fact that H s symmetrc. Eq.60 s the basc concept of the Newton based method, the algorthm of whch s conventonally expressed as follows: U n1 = U n H n 1 g n 61 the control varable U s updated by ths formulaton. where n s the teraton number Frst Order Adjont Equaton J 1 1 s expressed as follows: J 1 1 = { ċ 0 c 0, u 0 κc 0, Q c 0 c obj c 1 ddt { u 0 u 0.j u0 j u 0 j u 0 j, ν u 0,jj u 0 j,j u 0 f η 0, η 0 H { η 0 u 0, g η 0, u 0 η 1 ddt u 0 t f u 1 t f u 0 u 1 d η 0 t f η 1 t f η 0 η 1 d c 0 t f c 1 t f c 0 c 1 d { u 1 u 0 u j u 0,j η g 0 H ξ, ν u 0 j, u0,j,j fu0 ddt Γ C η 1{ η 0 η 0 Hu 0 c 1 {ċ 0 u 0 c 0, κc 0, ddt { u 0 u 0 j ν u 0,j, ddt c 0 c 0, u 1 ddt u 0 j, η 0 η 0 H δ j c 0 c 0 δ j n j U 1 dγ C dt 62 8

9 consderng each term equals zero, statonary condtons are obtaned as follows: u 0 u 0,j u0 j u 0 j u 0 j, ν u 0,jj u 0 j,j u 0 f η 0, η 0 H c 0 c 0, = 0 n, 63 η 0 u 0, g η 0, u 0 = 0 n, 64 ċ 0 c 0, u 0 c 0, κ Q c 0 c obj = 0 n, 65 u 0 t f = 0 n, 66 η 0 t f = 0 n, 67 c 0 t f = 0 n, 68 u 0 u j u 0,j η g 0 H ξ, ν u 0 j, u0,j,j fu0 = 0 on, 69 η 0 η 0 Hu 0 = 0 on, 70 Eqs are called the frst order adjont equaton. rearranged as follows: { J 1 1 = u 0 u 0 j ν u 0,j Γ C, ċ 0 u 0 c 0, κc 0, = 0 on, 71 b = c 0, n = 0 on Γ N,72 Introducng Eqs nto Eq.62, J 1 1 can be u 0 j, η 0 η 0 H δ j c 0 c 0 δ j n j U 1 dγ C dt. 73 then, the gradent of the performance functon can be computed as follows: gradj = u 0 u 0 j ν u 0,j u 0 j, η 0 η 0 H δ j c 0 c 0 δ j n j on Γ C Second Order Adjont Equaton J 2 1 s expressed as follows: { J 2 1 = ċ 0 c 0, u 0 κc 0, Q c 0 c obj c 2 ddt { u 0 u 0.j u0 j u 0 j u 0 j, ν u 0,jj u 0 j,j u 0 f η 0, η 0 H { η 0 u 0, g η 0, u 0 η 2 ddt u 0 t f u 2 t f u 0 u 2 d η 0 t f η 2 t f η 0 η 2 d c 0 t f c 2 t f c 0 c 2 d { u 2 u 0 u j u 0,j η g 0 H ξ, ν u 0 j, u0,j,j fu0 ddt Γ C η 2{ η 0 η 0 Hu 0 c 2 {ċ 0 u 0 c 0, κc 0, ddt { u 0 u 0 j ν u 0,j, ddt c 0 c 0, u 2 ddt u 0 j, η 0 η 0 H δ j c 0 c 0 δ j n j U 2 dγ C dt 75 9

10 ntroducng Eqs nto Eq.75, J 2 1 can be rearranged as follows: { J 2 1 = u 0 u 0 j ν u 0,j u 0 j, η 0 η 0 H δ j c 0 c 0 δ j n j U 2 dγ C dt. 76 Γ C whle J 2 2 s expressed as follows: J 2 2 = { ċ 1 c 1, u 1 κc 1, Qc 1 c 2 ddt { u 1 u 1.j u1 j u 1 j u 1 j, ν u 1,jj u 1 j,j u 1 f η 1, η 1 H { η 1 u 1, g η 1, u 1 η 2 ddt u 1 t f u 2 t f u 1 u 2 d η 1 t f η 2 t f η 1 η 2 d c 1 t f c 2 t f c 1 c 2 d { u 2 u 1 u j u 1,j η g 1 H ξ, ν u 1 j, u1,j,j fu1 ddt Γ C η 2{ η 1 η 1 Hu 1, ddt c 2{ ċ 1 u 1 c 1, κc 1, ddt { u 1 u 1 j ν u 1,j c 1 c 1, u 2 ddt u 1 j, η 1 η 1 H δ j c 1 c 1 δ j n j U 2 dγ C dt 77 consderng each term equals zero, statonary condtons are obtaned as follows: u 1 u 1,j u1 j u 1 j u 1 j, ν u 1,jj u 1 j,j u 1 f η 1, η 1 H c 1 c 1, = 0 n, 78 η 1 u 1, g η 1, u 1 = 0 n, 79 ċ 1 c 1, u 1 c 1, κ Qc 1 = 0 n, 80 u 1 t f = 0 n, 81 η 1 t f = 0 n, 82 c 1 t f = 0 n, 83 u 1 u j u 1,j η g 1 H ξ, ν u 1 j, u1,j,j fu1 = 0 on, 84 η 1 η 1 Hu 1 = 0 on, 85, ċ 1 u 1 c 1, κc 1, = 0 on, 86 b = c 1, n = 0 on Γ N,87 Eqs are called the second order adjont equaton. Introducng Eqs nto Eq.77, J 2 2 can be rearranged as follows: J 1 1 = Γ C { u 1 u 1 j ν u 1,j u 1 j, η 1 η 1 H δ j c 1 c 1 δ j n j U 2 dγ C dt. 88 therefore, comparng Eq.88 wth Eq.50, h whch s the product of Hessan matrx H and control varable U 2 canbeexpressedasfollows: h = u 1 u 1 j ν u 1,j u 1 j, η 1 η 1 H δ j c 1 c 1 δ j n j U 2 on Γ C

11 J 2 3 s expressed as follows: J 2 3 = 1 2 c 1T Qc 1 ddt u 1 { u 1 u j u 1,j η g 1 H ξ, ν u 1 j, u1,j,j fu1 ddt η 1{ η 1 η 1 Hu 1, ddt c 1 {ċ 1 u 1 c 1, κc 1, ddt, 90 J 2 3 s transformed by Eqs as follows: J 2 3 = 1 c 1T Qc 1 ddt The term J 2 3 s calculated by the square sum of c 1, whch means that J 2 3 s non negatve. 6 Mnmzaton Technque 6.1 Weghted Gradent Method In ths study, the weghted gradent method s appled as mnmzaton technque. In the weghted gradent method, a modfed performance functon to whch a penalty term s added K s used and expressed as follows: K = J 0l 1 2 U l1 U l T W l U l1 U l dt, 92 where l, W l and U l are number of teraton, the weghtng dagonal matrx and the control velocty, respectvely. In case that the modfed performance functon s converged to the mnmum value, the penalty term wll be zero. To mnmze the modfed performance functon s equal to mnmze the extended performance functon. Let U be the control velocty, then the followng equaton holds: The control velocty s updated by followng equaton: δk =0, 93 grad J = W l U l1 U l Algorthm of Weghted Gradent Method The algorthm of the weghted gradent method s shown as follows: Step 1. Chose the ntal control velocty U l, and set the number of teraton l to 0. Step 2. Solve the state varables u 0l, η 0l and c 0l usng Eq.1, Eq.2 and Eq.10. Step 3. Compute the ntal performance functon J l. Step 4. Solve u 0l, η 0l and c 0l usng Eq.63, Eq.64 and Eq.65. Step 5. Solve the gradent g l usng Eq.74 Step 6. Update the control velocty U l usng Eq.94 Step 7. Solve the state varables u 0l1, η 0l1 and c 0l1 wth the control velocty usng Eq.1, Eq.2 and Eq.10. Step 8. Compute the performance functon J l1. 11

12 Step 9. Solve u 0l1, η 0l1 and c 0l1 usng Eq.63, Eq.64 and Eq.65. Step 10. Solve the gradent g l1 usng Eq.74 Step 11. Update the control velocty U l1 usng Eq.94 Step 12. Check the convergence; f U l1 U l < 10 6 then stop, else go to step 13. Step 13. Update a weghtng parameter W l ; f J l1 <J l,thensetw l1 =0.9W l andgotostep4 else W l1 =2.0W l andgotostep BFGS method The BFGS method s one of the Newton type mnmzaton technque. In ths method, Hessan matrx s approxmated by followng equaton: H l1 = H l g k g kt g kt U k cont Hl U k cont U kt cont Hl U cont kt Hl Ucont k, 95 where l and H are number of teraton and Hessan matrx, respectvely. H l, U cont and g are denoted as follows: H l = 2 J U 2 cont, 96 U k cont = U l1 cont U l cont, 97 g k = J l1 J l = g l1 g l, 98 U cont U cont where s whch means the tracton gven by the perturbaton for the Lagrange multpler can be sown as follows: s l = H l U k cont = 2 J U 2 cont Consequently, the update equaton of Hessan matrx can be represented as follows: H l1 = H l g k g kt g kt U k cont U k cont. 99 s l s lt U cont kt s l. 100 In generally, the product between the Hessan matrx H and the U cont can not be obtaned drectly. Conventonally, ths product s calculated by usng an approxmated Hessan matrx. On the other hand, the tracton gven by the perturbaton for the Lagrange multpler expressed by exact Hessan matrx can be obtaned n ths approach. Therefore, t s consdered that the more exact Newton drecton can be obtaned n comparson wth conventonal method. 6.4 DFP method The DFP method s one of the Newton type mnmzaton technque. Ths method whch are formulated by the secant equaton are the methodologes used to obtan a Hessan matrx. Ths method s ensured that the Hessan matrx s postve defnte and symmetrc. Therefore, ths method s well known to calculate the Hessan matrx. The DFP method s appled to obtan the Hessan matrx. The update equaton of Hessan matrx by usng the DFPmethodswrttenasfollows: H l1 = H l g k H l U cont kt g k T g k g k H l U cont kt T g kt U cont k g k H l U cont kt T U contg k g kt g kt U cont k

13 where H l, U cont, g l and s are denoted as well as the BFGS method. Consequently, the update equaton of Hessan matrx can be represented as follows: H l1 = H l g k s lt g k T g k g k s lt T g kt U cont k g k s lt T U contg k g kt g kt U cont k Therefore, the product between the Hessan matrx H and the U cont s calculated by usng approxmated the Hessan matrx. The tracton gven by the perturbaton for the Lagrange multpler expressed by exact the Hessan matrx can be obtaned n ths approach. 6.5 Algorthm of Newton Based Method The Newton equaton can be wrtten as follows: H l d l = g l, 103 where l s number of teraton. Hessan matrx, Newton lne search drecton and gradent are denoted by H, d and g, respectvely. The algorthm of the Newton method usng the BFGS method s shown as follows: Step 1. Chose the ntal control velocty U 0, an ntal Hessan matrx H 0 and set the number of teraton l to 0. Step 2. Solve the state varables u l, ηl and c l usng Eq.1, Eq.2 and Eq.10. Step 3. Compute the ntal performance functon J l. Step 4. Solve u 0l, η 0l and c 0l usng Eq.63, Eq.64 and Eq.65. Step 5. Compute the steepest decent drecton g l usng Eq.74. Step 6. Solve the Newton Equaton H l d l = g l by the conjugate gradent method. Step 7. Generate a new control varable Ucont l1 by the Newton drecton d l1. Step 8. Solve the state value u 0l1, η 0l and c 0l usng Eq.1, Eq.2 and Eq.10. Step 9. Compute the performance functon J l1. Step 10. Solve u 0l1, η 0l1 and c 0l1 usng Eq.63, Eq.64 and Eq.65. Step 11. Compute the steepest decent drecton g l1 usng Eq.74. Step 12. Check the convergence; f U l1 cont U cont l < 10 6 then stop, else go to step 13., η 1l1 and c 1l1 usng Eq.84, Eq85 and Eq.86., η 1l1 and c 1l1 usng Eq.78, Eq.79 and Eq.80. Step 13. Solve the perturbaton vector u 1l1 Step 14. Solve u 1l1 Step 15. Compute the tracton gven by the perturbaton for the Lagrange multpler s l. Step 16. Compute the Hessan matrx H l1 by the U l1 cont U l cont, g l1 g l and s l. Step 17. Solve the Newton Equaton H l1 d l1 = g l1 by the conjugate gradent method and go to step 7. The ntal Hessan matrx s generally set an unt matrx. 7 Numercal Study In ths study, the optmal control of the concentraton of DO n Teganuma rver s carred out. Teganuma rver s located n Chba prefecture n Japan. And the total length of ths rver about 4km. Fg.5 shows the fnte element mesh of Teganuma rver. Ths mesh has 2171 nodes and 3991 elements. Fg.6 shows the dffusng of the concentraton of DO. The tme ncrement t s 2.0sec. As an ntal condton, velocty, water elevaton, and the concentraton of DO whch s analyzed by the fnte element method usng observed data are gven at the whole doman. In nverse analyss, the upstream boundary becomes the control boundary. As a boundary condton, water velocty set to 0.0 on the control boundary. The target concentraton of DO whch set to 10.0 s gven on the objectve pont. Then, the varable whch s calculated on the control boundary becomes the control varable. 13

14 Ths optmal control problem s to fnd an optmal control velocty on the control boundary so as to mnmze the performance functon. The weghted gradent method and the Newton based method usng BFGS method are appled as mnmzaton technque. 8 Numercal Results Fg.8 shows the performance functon. The weghted gradent method and the Newton based method usng BFGS method are converged at 114 and 52 teraton, respectvely. Therefore, optmal control of dssolved oxygen n Teganuma rver can be obtaned. Fg.9 s control velocty on control boundary. Fg.10 s the concentraton of DO at objectve pont. 9 Concluson In ths paper, the analyzng of Teganuma rver and the optmal control of concentraton of DO n the shallow water flow and presented. The water flow behavor and dffusng of DO are calculated usng the fnte element method n Teganuma rver. As a numercal study, the optmal control of DO s shown n Teganuma rver. The control velocty can be derved so as to mnmze the performance functon usng the weghted gradent method. The target concentraton of DO can be obtaned by control velocty. Newton based method usng BFGS method and DFP method are appled to ths control problem. The performance functons converged. However, the results of BFGS method and DFP method are same convergence. Moreover, Computatonal tmes are lttle hgher than weghted gradent method. Accuracy s also not good varable than weghted gradent method. Future work s to solve these problem. Reference 1. J.Matsumoto, T.Umetsu and M.Kawahara, Shallow Water and Sedment Transport Analyss by Implct FEM, Journal of Appled Mechancs, vol.1, J. Matsumoto, T. Umetsu and M. Kawahara, Stablzed Bubble Functon Method for Shallow Water Long Wave Equaton Int. J. Comp. Flud Dyn., vol.174, pp , Aleksey K.Alekseev and I.Mchael Navon Calculaton of Uncertanly Propagaton usng Second Order Adjont Equatons,Internatonal journal of Computatonal Flud Dynamcs vol.174, pp , Aleksey K.Alekseev and I.Mchael Navon On Estmaton of Temperature Uncertanly Usng the Second Order Adjont Problem,Internatonal journal of Computatonal Flud Dynamcs vol.162, pp , S.G.Nash; Precondtonng of Truncated-Newton Methods, SIAM J. SCI. STAT. COM- PUT., vol.6, no.3, pp , July M. Kawahara and K. Sasak, Adaptve and Optmal Control of The Water Gate of A Dam Usng A Fnte Element Method, Int. J. Comp. Flud Dyn., vol.7, pp ,

15 7. M. Paseck, N. D. Katopodes and Member, ASCE, Control of Contamnant Releases n Rvers. 1: Adjont Senstvty Analyss, J. Hydr. Eng., ASCE. vol.123,pp , M. Paseck, N. D. Katopodes and Member, ASCE, Control of Contamnant Releases n Rvers. 2: Optmal Desgn, J.hydr. Eng., ASCE. vol.123,pp , T.Kurahash and M.Kawahara, Development of Second-order Adjont Technque for Boundary Value Determnaton Problems,Internatonal journal for Numercal Methods n Fluds, To be publshed Fg.3 : Fnte Element Mesh Fg.4 : Dffusng of DO These 4 crcle ponts are the observed pont. The trangle pont s the objectve pont OBSERVED DATA BFGS METHOD DFP METHOD WEIGHTED GRADIENT METHOD DISSOLVED OXYGEN PERFORMANCE FUNCTION TIMEh Fg.5 : Observed Data ITERATION Fg.6 : Performance Functon 15

16 4 3.5 WITH CONTROL WITHOUT CONTROL WITH CONTROL WITHOUT CONTROL 3 9 WATER VELOCITY DISSOLVED OXYGEN TIMEh Fg.7 : Control Velocty TIMEh Fg.8 : DO at The Objectve Pont 16