Discretized Scheme Examining Convergence and Geometric Ratio Profiles for Control Problem Constrained by Evolution Equation
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1 Joural of Mathematics ad Statistics 7 (: 6-, ISSN Sciece Publicatios Discretized Scheme Examiig Covergece ad Geometric Ratio Profiles for Cotrol Problem Costraied by Evolutio Equatio O. Olotu Departmet of Mathematical Scieces, he Federal Uiversity of echology, P.M.B 74, Aure, Nigeria Abstract: Problem statemet: Here, we develop a discretized scheme usig oly the pealty method without ivolvig the multiplier parameter to examie the covergece ad geometric ratio profiles. Approach: his approach reduces computatioal time arisig from less data maipulatio. Objectively, we wish to obtai a umerical solutio comparig favourably with the aalytic solutio.. Methodologically, we discretize the give problem, obtai a ucostraied formulatio ad costruct a operator which sets the stage for the applicatio of the discretized exteded cojugate gradiet method. Results: We aalyse the efficiecy of the developed scheme by cosiderig a example ad examiig the geerated sequetial approximate solutios ad the covergece ratio profile computed quadratically per cycle usig the discretized cojugate gradiet method. Coclusio/Recommedatios: Both results, as show i the table, loo comparably ad this suggests that the developed scheme may very well approximate a aalytic solutio of a give problem to a appreciable level of tolerace without its prior owledge. Key words: Discretized scheme, operator v, evolutio equatio, examiig covergece, Cojugate Gradiet Method (CGM, cotrol problem costraied, umerical solutio, square itegrable, pealty parameter, cojugate gradiet algorithm INRODUCION Such that: Here, we examie a discretized scheme via the pealty method to examie the covergece aalysis of optimal cotrol problem costraied by evolutio equatio with real coefficiets.. he costraied problem is coverted to a ucostraied problem via the pealty method (Fletcher ad Reeves, 964. Discretizatio of its time iterval ad fiite differece method for its differetial costrait were employed to obtai the discretized formulatio of the problem. With this formulatio, a associated operator was obtaied usig the modified exteded Cojugate Gradiet Method (CGM (Hestees, 969; Ibiejugba ad Oumayi, 984. Usig the modified cojugate gradiet method, a example was cosidered to examie the umerical solutio ad the covergece aalysis as it compares favorably with each other. o this ed, a geeralized quadratic problem costraied by evolutio equatio with real coefficiets is cosidered i the ext paragraph for our developed scheme. MAERIALS AND MEHODS Geeralized problem: Miimize L,x,u (px qu dt ( 6 ax r bx cx du, x( x,x h,t [ r,] o ( where, p,q>,a,b,c are real umbers ad K deotes the Cartesia product of the followig two spaces q K H [. σ ] L [, σ, ] where H [,σ] stads for the Sobolev space of the absolutely cotiuous fuctios x( such that both x( ad x( are square itegrable over [,σ] ad L[, q σ ] stads for the Hilbert space cosistig of equivalece classes of square itegrable fuctios from [,σ]-r q. Remar: h(s,s [ r,] x r x x(s,s [, r] Whe h, the x r x Discretizatio: By discretizig ( ad ( usig (Fletcher ad Reeves, 964; Glad, 979 subdivide [,] ito equal itervals [t,t ] at meshpoits x <x <x,< <x where - is the umber of partitio
2 J. Math. & Stat., 7 (: 6-, poits chose arbitrarily, thus havig ( partitio poits, with x j j* j, j,,,.. ad j is the fixed legth of each subiterval for j or ot. By j* j, it meas j multiplied by j. Let t ad t j,,,,...,,t j : x( x,u u,,,,., By Euler's scheme or fiite differece method: c μ ab p abμ ac m ad μ m μ a m μ ab μ ab μac w μad w μb x( (x( x( /,,,,... the geeralized problem ( becomes. Geeralized problem.: Miimize j,x,u (x ( uqu px Subject to Eq. ad 4: a(x - -γ -x -γ / b(x -x / cx du ( (4 Applicatio of the pealty parameter: Miimize j,x,u Such that: (x (5 uqu px µ[ax -γ -ax bx - cx - du ] (6 Now (5 ad (6 become the followig usig (Di Pillo ad Grippo, 979; Glad, 979 Eq. 7: MiimizeJ,x,u, xa u μ b x r c x -r -r cx -r c x -r -r px -r u m x -r mx -r ( t r u( t w x ( t wx ( t v x u q x u d (7 Where: a p b μ bcμc μ b q u d c μ a c μ a 7 v μ b q μ bd d μbd μcd Settig y x i Eq. 7 we have Eq. 8: (x a u b y r MiimizeJ,x,u, μ c y r r c y t r y t c y t r x ( ( ( ( t p y ( t r u ( t m K rm ry m r r u w y w y v yuq xud (8 Costructio of operator v: We ow costruct the operator for the cojugate gradiet method. he biliear form associated with (7 usig (Ibiejugba ad Oumayi, 984 is give as Eq. 9 ad : z,z {x a u u β y r y r c } y ( t r x ( t r c y ( t r x ( t r c yk ( t r y ( t c y ( t r y ( t c y ( t r x ( t p y( t r x( t p y ( t r u ( t m y ( t r u( t m ( t r x( t r m ( t r y ( t m ( t r y ( t m ( t r x ( t ( t r x ( t ( t r u ( t w ( t r u( t w y ( t y ( t w y ( t x ( t v y ( t x( t v y( t u( t q y ( t u ( t q ( t u( t d ( t u ( t d (9
3 x a u u z,z b x r r c } r r c r r c K r r c r r c K K r r c r r c K r r c r c r c K r c r c r c K r c r c r c K r p r p r p K r p r u m r u m x K r u m r u m r r m K r r m r m K r m r r K r u w r u w w K r w w w K v x v v K v u q u q u q Ku q ( u d x u d Now, we state the heorem establishig operator ad provide a proof; heorem 5. Let the iitial guess of the cojugate gradiet algorithm be z so that z (x,u,h-, the the cotrol operator associated with the geeralized problem is give by Eq. : x z u h J. Math. & Stat., 7 (: 6-, ( 8 where, z (x, u h Proof: Fid x by settig u h i (5. ad otig the fact that if h the x γ x. So Eq. becomes: z,z x [x (a,c p w w v] ( c p w v r (c c c p m m r ( (c c c m] [x ( (c w [ ( c m v w ( r ( (c c r ( (c c c p u [x (m w q d] ( (m q h γr ( (c c c p r ( (c c For further compactess, ( becomes (: x [x N x z,z N x r N x r (N4] [x N5 N6 r N7 r N8] u [x N9 N] h r [x r N x r N] Where: N a c p w w v N c p w v N c c c p m m N4 (c c c m N5 (c w N6 (c m v w N7 (c c N8 (c c c p N9 (m w q d N (m q N (c c c p N (c c (
4 J. Math. & Stat., 7 (: 6-, Now, we represet (4 by the followig Eq. 5-8: x u h Where: (4 [x r N r N] (5 [x N9 N] (6 o solve for, set: his is a secod order ordiary differetial equatio that eeds to be solved. So we impose the followig iitial coditios Eq. 9-: ( p ad (r where, p ad γ are to be determied. Let Q(s L(q ad (s L( deote the Laplace trasform of q ad respectively. aig the Laplace trasform of (, we have Eq. 4: s (s-p s-γ- (s Q(s (4 Ω x N N r N r N4 (7 Q(s ps γ (s (5 s s s ad: f x N5 N6 x r N7 r N8 (8 Now, both Ω ad f are cotiuous fuctios o [,]. is cotiuous ad at least twice differetiable o [,]. Hece Ω - ad f - are cotiuous o [,]. x(. D [,] such that x( x( ad by (Pola, 97, we have: We tae the iverse Laplace trasform of (5 ad usig covolutio theorem for the first term o the right to obtai Eq. 6 ad 7: q(ssih s dt p cosh sih f (s cosh sih(f ( s ds Ω sih s ds (6 pcosh γ sih (7 {x [ Ω ] [f ]}dt (9 where, qs f (s -Ω (s ad Eq. 8: Hece: d (f Ω ( dt So: f Ω, t ( Let: f Ω q ( By: {f (s cosh( s ds τ sih(f ( Ω(s sih( s ds sih( Ω (cosh( Ω ( (8 Now, solve for u by settig x h x i EQ. 5., followig the same sequece as from the begiig of the proof to equatio(8, we the have Eq. 9-4: u m (9 u b ( x N5 x N6 x r N7 r N8 [x N N r i N r (N4] ( 9 f u q ( Ω u (q z u γ (m w m (
5 J. Math. & Stat., 7 (: 6-, sih(f ( f (s cosh s ds ( Ω (s sih s ds p cosh r sih τ {sih(f ( f (s cosh( s sih( ds Ω (s sih ( s ds Ω (4 (cosh( Ω (} Fially, we solve for h by settig x u x, i Eq. ad followig the same sequece as from the begiig of the proof to (8, we have Eq. 5-4: h [ ( c c ] (5 h [ m ] (6 f ( h ( (7 Ω h [ (c p c c c ] (8 Covergece aalysis of the discretized scheme: his sectio shows that the discretized pealty exteded cojugate gradiet method coverges geometrically by(shao, 978; Aruchua ad Sulaima,. Defiitio 6. he sequece {z } geerated by the discretized cojugate gradiet method of K coverges geometrically to z such that: z z lim p < z z where, p is the covergece ratio. Now we sate the cojugate gradiet method algorithm usig our developed scheme for quadratic objective fuctio: f(x x Ax bx c Step : Pic the iitial solutio z arbitrarily, wherez (x,u o,h K Step : Compute the iitial directio; p -(Axa - g Step : Update the iitial guess as follows: x x a x p x, u u a p u, h h a p h, sih(f ( f (s cosh s ds Ω sih s ds p cosh γ sih (9 (g,g where, a p,ap Step 4: Update the gradiet ad stepsize at the ( th step as follows: τ {sih(f ( f (s cosh( s sih( ds Ω (s sih( s ds (4 Ω (cosh( Ω (} Havig costructed operator, writte as: (4 g x, g x, a Ap x, g u, g u, a Ap u, g x, p x, g g a Ap whereg g ad p p h, h, h, u, u, g h, p h, ad: P x, -g x, β p x, P u, -g u, β p u, (g,g P h, -g x, β p h, where g (g g Settig Eq. equal to Eq. 4-45:, with the geeralized scheme ad associated operator, a z x z x z x program is writte usig the Cojugate Gradiet z z x z x z x Method Algorithm (Pola, 97 to execute ad z x z x z x examie the covergece profile ad the geometric z ratio profile.next we give the theoretical bacig for z the geometric covergece ratio. z (4
6 J. Math. & Stat., 7 (: 6-, Where: q (s sih s dt z sih γ pcosh q (s sih s dt γ sih p cosh q (s sih s dt sih γ pcosh (4 ad: Hece: N j (z,p, j j (z,z (z,z (48 is a bouded operator the there are positive umbers m ad M such that for every z i K: z X K (m w q d ( (m q u b h ( m (44 z X K γ(( (c c c p ( (c c u b h (45 ( (m c c Now, we state ad prove the geometric covergece property of the developed scheme. heorem 6.: he sequece of estimated solutios {z } geerated by the discretized pealty method coverges to z* with ratio ω give by: mz z (z z Mz z (49 Hece: (z,z z z (5 Substitute iequality (5 i Eq. 47 to obtai: (z,z f (z f(z f(z (5 (p,p z z Usig (6.: ω τ z z where, τ max{ } ad z is the iitial guess. p,p Proof: Defie: f(z m z z ad Eq. 5 becomes Eq. 5 ad 5: f(z z f(z z (p, p z z z z (5 f(z (z-z*,z-z* ad z t (x,u,h (46 With optimality coditio z*. From (46 f(z (z -z*,z -z* (z,z -(z*,z f(z (z a p -z*,(z a p -z*, (z,z a (z,p a (p,z a (p,p -(z*,z -a (z*,p herefore: (z,z f (z f(z f(z. (47 (p,p (z,z sice is self-adjoit ad z*.agai Eq. 46-5: N j j (z,z (z,z (z,p herefore: z z (p, p z z z z z, where τ sup { } (p,p his completes the proof. subject to the liear restorig dyamic equatio: (5 Numerical example: his example validates heorem (9 ad heorem(9 by miimizig the followig quadratic fuctioal with real coefficiets Eq. 54 ad 55: Miimize L,x,u (x u dt (54
7 J. Math. & Stat., 7 (: 6-, x. x 5u x( x h,t [.,] (55 For better uderstadig of the results, the followig abbreviatios ad otatios shall be ivoed: J(x,x -γ,u the objective fuctioal F(x,x -γ,u,µ the pealized fuctioal he aalytic solutio is give as.8 by (Olotu ad Adeule,. RESULS AND DISCUSSION For every fixed pealty costat per cycle, it is see that covergece is attaied as evideced by the values of the objective fuctioal. For istace, for pealty costat µ., the objective fuctioal values iitiatig at.6 ad edig at.57889, a value attaied at the sixth iteratio compares more favorably with the aalytic solutio.8. he same applies to every cycle with at most six iteratios. As the pealty parameter icreases, the better the approximatio to the aalytic solutio. Also, the geometric covergece ratio profile is attaied favorably, sice the geometric ratio is greater tha zero ad less tha oe, as exhibited uder the geometric ratio colum i able. For istace, for µ., we see that the geometric covergece ratios;.78594,.98449, , ad ad are all greater zero ad less tha oe. he same iformatio is see for other cycles. able : Numerical results comparable to the aalytical solutio (.8 Costr Pealized Parameters Iteratio No Obj. fuctioal satisfactio fuctioal Geometric ratio µ µ µ µ µ
8 J. Math. & Stat., 7 (: 6-, CONCLUSION Cosiderig the miimum of the objective fuctioal values for each cycle, it is readily see that the umerical solutio approximates the aalytic solutio.8 withi a error tolerace of.. he covergece aalysis is equally favorable. his shows that the developed scheme for the solutio of optimal problem costraied by evolutio equatio is adequately effective ad efficiet. Recommedatio: he results obtaied ad the iformatio gathered from the geometric ratio profile imply that a solutio of a hypothetical costraied problem ca be approximated with this scheme, eve without a prior owledge of the aalytical solutio. REFERENCES Aruchua, E. ad J. Sulaima,. Numerical solutio of secod-order liear fredholm itegrodifferetial equatio usig geeralized miimal residual method. Am. J. Applied Sci., 7: DOI:.844/ajassp Di Pillo, G. ad L. Grippo, 979. A ew class of augmeted lagragias i oliear programmig. SIAM, J. Cot. Optimiz., 7: Fletcher, R. ad C.M. Reeves, 964. Fuctio Miimizatio by Cojugate Gradiets. Comput. J., 7: DOI:.9/comjl/7..49 Glad, S.., 979. A combiatio of pealty fuctio ad multiplier methods for solvig optimal cotrol problems. J. Optimiz. heory Appl., 8: -9. DOI:.7/BF977 Hestees, M.R., 969. Multiplier ad gradiet methods. J. Optimiz. heory Appl., 4: -. DOI:.7/BF9767 Ibiejugba, M.A. ad P. Oumayi, 984. A cotrol operator ad some of its applicatios. J. Math. Aal. Appl., : -47. DOI:.6/- 47X( Olotu, O. ad A.I. Adeule,. Aalytic ad umeric solutios of discretized costraied optimal cotrol problem with vector ad matrix coefficiets. It. J. Adv. Modell. Optimiz., : 9-. Pola, E., 97. Computatioal Methods i Optimizatio: A Uified Approach. st Ed., Academic press, Lodo, ISBN-: 5595, pp: 9. Shao, D.F., 978. O the covergece of a ew cojugate gradiet algorithm. SIAM, J. Numerical Aal., 5:
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