IOSR Journal of Matematcs (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume, Issue 5 Ver. II (Sep. - Oct. 5), PP 43-63 www.osrjournals.org Applcaton of Pontryagn s Maxmum Prncples and Runge- Kutta Metods n Optmal Control Problems Oru, B. I. And Agwu, E. U., Department of Matematcs, Mcael Okpara Unversty of Agrculture, Umudke, Aba State, Ngera Abstract: In ts paper, we examne te applcaton of Pontryagn s maxmum prncples and Runge-Kutta metods n fndng solutons to optmal control problems. We formulated optmal control problems from Geometry, Economcs and pyscs. We employed te Pontryagn s maxmum prncples n obtanng te analytcal solutons to te optmal control problems. We furter tested te numercal approac to tese optmal control problems usng Runge-Kutta metods. Te results sow tat te Runge-Kutta metod produced results tat are comparable to analytc solutons. Terefore, we concluded tat Runge-Kutta metod gves error tat s neglgble. I. Introducton Tere are coces avalable for decson makng and te ablty to pck te best, perfect and desrable way out of te possble alternatves or varables gves us te optmal control. [3], before commencng a searc for suc an optmal soluton, te job must be well-defned; and must possessed te followng features () Te nature of te system to be controlled, () Te nature of te system constrants and possble alternatves, () Te task to be accomplsed, (v) Te crtera for judgng optmal performance. Te optmal control teory s very useful n te followng felds, geometry, economcs and pyscs. In geometry, t s nterestng to see tat by optmal control teory, te geometrcal problems suc as te problem of fndng te sortest pat from a gven pont A to anoter pont B wll be solved. Contnuous optmal control models provde a powerful tool for understandng te beavor of producton/ nventory system were dynamc aspect plays an mportant role. Some optmal control problems tat are non-lnear do not ave solutons analytcally. In fact, mostly all problems arse from real lfe are non- lnear. As a result, t s necessary to employ numercal metods to solve optmal control problems. Tere are so many numercal tecnques to optmal control problems lke [3], [7], [], recently contrbuted to te teory of optmal control. [3] used modfed gradent metod wle [7] compared te Forward Backward Sweep, te Sooter Metod, and an Optmzaton Metod usng te MATLAB Optmzaton Tool Box wereas [] compared Euler, Trapezodal and Runge-Kutta usng Forward Backward Sweep metod (FBSM). Lookng at te work done by [7] and [], we fnd out tat bot of tem were not nterested to compare ter numercal approxmatons wt analytc solutons and were only nterested on problems wt fnal value of te adjont varable. We ntend to gve attenton to problems wt ntal value of te state and adjont varables and ten generate te numercal approxmatons of bot te state and adjont varables forward nstead of forward- backward usng Runge-Kutta approac. Ts work s concerned wt te soluton of te followng tree types of problems: (a) Sortest dstance between two ponts. Tese type of problems are typcally of te form: Mnmze T t subject to x ( u ) u d t and x ( t ) x, ( T ) fr e e [5] (b) Te cocoa problem. Ts partcular problem s of te form: Maxmze DOI:.979/578-54363 www.osrjournals.org 43 Page. ( ). P u xu x d t
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control subject to x u and x( ), u( ) (c) Pendulum problem: ts type of problem requres te descrpton of te mecancal system of a pont mass m constraned by a lgt wre of lengt l to swng n an arc. It s of te form: Mnmze t ( c o s ) t.3 S m l u m g l d t subject to u and θ t = θ θ(t) free. II... Dervaton Of Euler- Langrage Equaton Consder te functonal t Prelmnares ' S [ y ] F ( t, y, y ) d t.. t t W [ y ] F ( t, y, u ) d t..a t Were n equaton.. F s a dfferentable functon of te tree varables F = t, y, y and F = F(t, y, u) n equaton(..a). Suppose y t s any curve passng troug te two ponts P t, y and Q t, y. Let te orgnal sape of te curve be y = y(t). Suppose tere s a small varaton due to certan dsturbance, te curve canges sape to y * t = y(t) +aq(t), were a s a small parameter,q(t) s an arbtrary functon, y * (t) s te new curve, q(t ) = and q(t ) =, mples no varaton n t. Ts means tat y (t) = y (t) + aq (t) Let te new functonal denoted as S*[ y] t ] dt.. => S [y] = F[ t, y + aq, y + aq t Also let te varaton on S denoted as δs, t => δs = S S = F t, y + aq, y + aq F t, y, y dt..3 t Snce t s fxed, t mples tat F vared n two varables y and y. We recall tat Taylor seres of two varables s gven by: DOI:.979/578-54363 www.osrjournals.org 44 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control F[t + k, y + l] = F(t, y) + (k + l t y )(F(t, y) + (k t + l F(t, y) ) + y! Takng te frst order varaton, equaton (..3) becomes t δs = a (q F F + q y y )dt..4 t Were, k = aq, l = aq [4].For S[y] to be statonary (maxmum or mnmum), ds da a== We recall tat, s lm a a = ds da a== t => q F t t + q F y dt = => q F y dt F + q y dt = t t Integratng te second ntegral by part we ave, Tus, t t F y d dt t t t q F y dt F + q y F y q(t)dt t t t q d dt Snce q(t) s an arbtrary functon, te equaton can only be satsfed f F y dt = =..5 t t F y d dt F y dt = Dfferentatng bot sdes equaton (..6) becomes..6 F d F y d t If F = F t, y, u, ten (..7) becomes y F d F u d t u Equatons..7 and (..8) are known as Euler-Lagrange equatons...8..7.. Pontryagn s Maxmum Prncple (Pmp) Ts s a powerful metod for te computaton of optmal controls. It gves te fundamental necessary condtons for a controlled trajectory ( x, u ) to be optmal. For te soluton of optmal control problems, te prncpal metod resolves a set of necessary condtons tat an optmal control and te consstent state equaton must satsfy. Te necessary condtons are derved from Hamltonan, H, wc s defne as DOI:.979/578-54363 www.osrjournals.org 45 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control H ( t, x, u,, ) f ( t, x, u ) g ( t, x, u ) Pontryagn s maxmum prncple states tat: let [ t, T ] wt * u pecewse contnuous. If ( t ) suc tat te followng condtons are satsfed * * * * ( x, u ) be a controlled trajectory defned over te nterval * * ( x, u ) s optmal, ten tere exst a constant and te adjont H ( t, x, u,, ) H ( t, x, u,, ) for all t [, T] H u (Optmalty condton) x H H x (State equaton) (Adjont equaton) (..) ( T ) free (Transversalty condton). [5]..3. Dervaton Of Pontryagn s Maxmum Prncple Consder te basc optmal control problem of te form T J ( u ) f ( x ( t ), u ( t ), t ) d t.3. subject to t x ( ) g ( ( ), ( ), ),,,..., n.3. were we ws to fnd te optmal control vector u tat mnmzes equaton (.3.). In (.3.), tere are tree varables: tme t, te state varable, x and te control varable u. We now ntroduce a new varable, known as adjont varable and denoted by (t). Lke te Lagrange multpler, te adjont varable s te sadow prce of te state varable. Te adjont varable s ntroduced nto te optmal control problem by a Hamltonan functon, H ( t, x, u,, ) f ( t, x, u ) ( t ) g ( t, x, u ). Were H denotes te Hamltonan and s a functon of fve varables t, x, u,,. [], for te t constrant equaton n (.3.) we form an augmented functonal were te ntegrand T * J f ( g x ) d t.3.3 n Te Hamltonan functonal, H s defned as n F f ( g x ).3.3a J * as suc tat * n.3.4 H f g T n.3.5 J H x d t Now te new ntegrand F F ( x, u, t ) becomes DOI:.979/578-54363 www.osrjournals.org 46 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control We recall Euler-Lagrange equatons n.3.6 F H x F d F,,,..., n x d t x F d F, j,,..., m u d t j u j If we relate equatons (.3.3 a ), (.3.7 ), (.3.8), we ave f x f u n g.3.7 a x n g.3.8a u If we relate equatons (.3.4 ), (.3.7 a ), (.3.8 a ), we ave H,,,..., n x H u, j,,..., m (.3.9 ) (.3. ).3.7.3.8 were equaton (.3.9) s known as adjont equaton. Te optmum solutons for x, u, can be obtan by equaton (.3. ), (.3.9 ), (.3. ). We can now state te varous components of te maxmum prncple for problem (.3.) as follows: * * H ( t, x, u,, ) H ( t, x, u,, ) for all t [, T] H u (Optmalty condton) x H H x (State equaton) (Adjont equaton) (.3.) ( T ) free (Transversalty condton). Condton one and two n (.3.) state tat at every tme t te value of u( t ) cosen so as to maxmze te value of te Hamltonan over all admssble values of u( t ), te optmal control, must be. Condton tree and H H four of te maxmum prncple, x a n d, gve us two equatons of moton, referred to as te y Hamltonan systems for te gven problem. Condton fve, ( T ) free s te transversalty condton approprate for te free termnal state problem only. DOI:.979/578-54363 www.osrjournals.org 47 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control.4 Hamlton s Prncple In Mecancs Te evoluton of many pyscal systems nvolved te mnmzaton of certan pyscal quanttes. Te mnmzaton approac to pyscal systems was formalzed n detal by Hamlton, and resulted n Hamlton s prncple wc states tat. Of all te possble pats along wc a dynamcal systems may move from one pont to anoter wtn a specfed tme nterval, te actual pat s tat wc mnmzes te tme ntegral of te dfference between te knetc and potental energes [9]. Expressng ts prncple n terms of te calculus of varatons, we ave t S = mn T V dt t Were S s te acton to be mnmzed, T s te knetc energy, V s te potental energy and te quantty (T V) s called te Lagrangan L. [3]. Applyng te necessary condton to mnmze te acton, te Euler- Lagrange equatons become L d L x d t x L d L u d t u Hence, n any dynamcal system, we wll frst nvestgate te mecancal energy of te system and set up te Hamltonan for te system. Ten applyng Hamltonan equatons, we wll obtan an equaton descrbng te moton of te system nstead of usng Newton s approac wc wll be more dffcult to andle because t requres te total force on te system []..5 Procedures To Analytcal Soluton Form te Hamltonan for te problem. Wrte te adjont dfferental equaton, transversalty boundary condton, and te optmalty condton n terms of tree unknowns, u, x, and λ. H Use te optmalty equaton to solve for u n terms of x and λ. u Solve te two dfferental equatons for x and λ wt two boundary condtons. After fndng te optmal state and adjont, solve for te optmal control usng te formula derved by trd procedure..6. Runge- Kutta metod of Order 4 (RK4) Ts metod s developed for solvng ODE numercally and to avod computaton of dervatves. Snce optmal control problems are descrbed by a set of ODE, we sall use ts tecnque to obtan te numercal approxmatons to optmal control problems..6.. Dervaton Of Runge-Kutta Metod Of Order 4, Consder an ODE of te form In general form, RK4 s stated as X f ( t, x ) x x t, x, Were 3 3 4 4.6.a.6.b t, x, a k a k a k a k... a k, wle a,,,..., n are constants n n and k,,,..., n are functonal relaton gven by DOI:.979/578-54363 www.osrjournals.org 48 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control k f ( t, x ) k f t, x k k f t, x k k... 3 k f t, x k k... k n n n, n, n, n n Note tat s called ncrement functon. Let 3 F f ff, F f ff f f, F f 3 ff 3 f f f f t x tt tx x x 3 ttt ttx tx x x x x Now dfferentatng te equaton (.6.a), we ave x f f x f f f F t x t x x f ff f f f ( f ff ) F f F tt tx x x x t x x ( v ) 3 x f 3 ff 3 f f f f f ( f ff f f ) 3( f ff )( f ff ) f ( f ff ) ttt ttx tx x x x x x tt tx x x t x tx x x x t x F f F 3 F ( f ff ) f F 3 x tx x x x Now te Taylor seres con be wrtten as 3 4 x x f F ( F f F ) F f F 3 F ( f ff ) f F... x 3 x tx x x x 6 4.6.c And te functonal at n=4 s gven by k f ( t, x ) k f t, x k k f t, x k k 3, If we substtute nto (.6.b), we ave k f t x k k k 4 3 3, 3, 3,3 3 x x a f ( t, x ) a f t, x k a f t, x k k a f t, x k k k 3 4 3 3, 3, 3,3 3 If we compare (.6.c) wt (.6.d) and te classcal Runge-Kutta vales for te constants,, 3,,,,, 3 3 3 3 a, a, a, a 3 4 6 3 3 6 [6]. Terefore, te RK4 becomes k f ( t, x ).6.d k f, t x k k f t, x k 3 k f t, x k 4 3 x x k k k k 3 4,,,, 3... N 6 DOI:.979/578-54363 www.osrjournals.org 49 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control Te algortm for Classcal Runge-Kutta metod of order four for X f ( t, x, u ) s gven by k f ( t, x, u ) k f,, t x k u k f,, 3 t x k u k f t, x k, u 4 3 x x k k k k 3 4,,,, 3... N.6.e 6 Te algortm for Classcal Runge-Kutta metod of order four for f ( t,, x, u ) s gven by k f ( t, x, u ), k f,,, t k u x k f,,, 3 t k u x.6.f k f t, k, u, x 4 3 k k k k,,, 3... N 6 3 4 x x k k k k N 6,,,, 3... s te teratve metod for generatng te next value for 3 4 x ; t s calculated usng te current value of x plus te wegted average of four values of K, j,...4. Were K, j,...4 are functonal relatons. Note tat s te step sze. j j III. Results 3.. Analytcal Solutons 3.a Geometrcal Problems; Sortest Dstance Between Two Ponts PROBLEM: Wat curve jonng two dfferent ponts P and Q as te sortest lengt? [8]. SOLUTION: Let te curve be y = y(t) and te two ponts be P[t, y ] and Q[t, y ] DOI:.979/578-54363 www.osrjournals.org 5 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control were, r t y j, r ( s ) t ( s ) y ( s ) j, d ( r ) d t d y j, d s d r ( d t ) ( d y ) ( d t ) ( d y ) ( dt) ( dt) ( y ) d t Were s s arc lengt parameter and ds small element of arc lengt from P Te dstance between te two ponts s L t t ' ( ) 3.a. L d s y d t If we let y = u be te control varable, (3.a. ) can be expressed as }dt 3.a. t L = { + u t Tus, te sortest-pat problem s Mnmze T t ( u ) d t 3. a.3 subject to y u and y t = y, were, F = + u y(t) free Te Hamltonan for te problem s, H ( u ) u 3. a.4 Recall tat te necessary condton for optmal control s gven by H y H u 3. a.5 3. a.6 DOI:.979/578-54363 www.osrjournals.org 5 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control (3. a.5) & (3. a.6 ) ( u ) u ( u ) u ( u ) u u u u ( u ) u ( ) From, But, u( t) y u, ( t) c y y ( t ) t k If we substtute te value of c, we ave te control varable as u( t) and te correspondng state varable s c c y ( t ) c t k c 3.b. Te Economc Problems; Cocoa Producton Problem Problem: a farmer wo owns a cocoa plantaton and as a problem to decde at wat rate to produce cocoa from s plantaton. He s meant to manage te plantaton from date to a perod of years. At date, tere s x cocoa n te farm, and te nstantaneous stock of cocoa x( t ) declnes at te rate te farmer produces u( t ). Te plantaton owner produces cocoa at cost x and sells u + xu of cocoa at constant prce $. He does not value te cocoa remanng n te farm at te end of te perod (tere s no scrap value). At wat rate of producton n tme u t wll maxmze s profts over te perod of ownersp wt no dscount tme. SOLUTION: Let P represents te farmer s profts, f e produces cocoa at cost x and sells u + xu at constant prce $ ten, P s gven by P = $ (u + xu) x 3.b. Te total proft over te perod of ownersp s DOI:.979/578-54363 www.osrjournals.org 5 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control Ten te cost functonal s gven by Maxmzes ( ) 3. b. P u xu x d t ( ) 3. b.3 P u xu x d t subject to x u and x( ), ( ) Takng te Hamltonan equaton, we ave x H f g u x u x u 3. b.4 Recall tat te necessary condton for optmal control s gven by H y H u (3. b.5) & (3. b.6 ) 3. b.5 3. b.6 u x 3. b.7 u x 3. b.8 If we dfferentate equaton (4.a.8), we ave u x But, x u u x u x u x u x 3. b.9, equaton (3.b.9) becomes x x x x x x If we take te auxlary equaton, we ave r r r x ( t ) A c o s t B sn t Were A, B are constants applyng te ntal condtons, we ave DOI:.979/578-54363 www.osrjournals.org 53 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control x ( ) A c o s B s n A,.. x ( t ) A s n t B c o s t x ( ) A s n B c o s B We recall tat, from equaton (3.b.3) we ave. x u u ( t ) s n t u ( t ) A sn t Substtutng te values our constants, te control becomes u ( t ) sn t Te correspondng state trajectory s gven by x ( t ) co s t 3.c. Applcatons To Pyscal Prncples 3.c. Te Pendulum Problem. PROBLEM: Descrbe te mecancal system of a pont mass m constraned by a lgt wre of lengt l to swng n an arc. SOLUTION: Let fgure (3.c. ) be te dagrammatcal llustraton of te system. Let s be te dsplacement of te bob from ts mean poston. Let OA = l be te lengt of te lgt wre before swng. Let OB be te lengt after swng. => OA = OB = l Let te angle θ (n radan) be te angle made by OB and te vertcal OA. From te dagram above, at pont A te potental energy V = and te knectc energy T = T, but at pont B, knetc energy T = and potental energy V = V In order to set up te Lagrangan for te system, we frst defne te work done by te system as were r s te poston and F s te force W = F. dr, => m dv dr. dr = m dt v. dv dt DOI:.979/578-54363 www.osrjournals.org 54 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control v = m v. dv = mv Snce te energy of te partcle s ts capacty to do work, we ave Knetc energy T = mv 3.c. a Smlarly potental energy V = mg 3.c. b Te crcular measure of an angle n radan s equal to te rato of te arc wc te angle subtends wen at te centre of te crcle to te radus of tat crcle, []. Tat s θ = arcab OA = S l Te velocty s = > S = lθ ds ds = d(lθ) dt Ts mples tat te knetc energy becomes Also from te dagram above = ldθ dt = lθ T = m(lθ ) = ml θ But, OA OC = AC = OA = l, cos θ = OC OB = OC l => OC = l cos θ OA OC = = l l cos θ = l( cos θ) Substtute nto equaton 3.c. b, we ave potental energy V = mg(l l cos θ) V = mgl( cos θ) Now tat we ave bot knetc energy and potental energy of te system, ten we can defne te Lagrangan L of system as L = knetc energy potental energy []. L = ml ( ) mgl( cos θ) [], usng Hamlton prncple, te acton S becomes t S = ( ml ( ) t mgl( cos θ))dt 3.c. c If we let θ = u be te control varable, (4.c..c) can be expressed as Tus, te pendulum problem s t S = ( ml u mgl( cos θ))dt t Mnmze t S = ( ml u mgl( cos θ))dt t DOI:.979/578-54363 www.osrjournals.org 55 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control were,. subject to u and θ t = θ θ(t) free F = ml u mgl( cos θ) Te Hamltonan for te problem s, H = ml u mgl( cos θ) + u Recall tat te necessary condton for optmalty s gven by H x H u m g l s n m l u m l u, From, But, u m l u m l u m g l s n u m l m g l s n m l m g l s n g s n...(3. c. d ) l Te equaton (3.c..d) above consttutes te Newton s second law of moton wc descrbes te moton of pendulum oscllaton to arbtrary angles. Now solvng for equaton (3.c..d) above, we ave and θ t = A cos g l t + B sn g l t u t = θ = g l { A sn g l t + B cos g l t} were A and B are constants. If we let w = g, we ave l θ t = A cos w t + B sn wt 3.c. e u t = w{ A cos w t + B sn wt } Terefore te equaton (3.c. e) above sows tat te system s a Harmonc oscllator (SHM). 3.. Te Numercal Results. DOI:.979/578-54363 www.osrjournals.org 56 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control Here, we want to apply te results we ave n (.6.e) and (.6.f) to obtan te numercal approxmatons to problems (3.a), (3.b) and (3.c).. te sortest-pat problem: Mnmze 5 ( u ) subject to x u and x =, x(t) free were, F = + u Taken te frst four procedures n (.5), we ave u d t, x u, x Employng Runge-Kutta metod of order 4, wt ntal guess of u we ave, k f ( t, x, u ) u k f,, t x k u u k f t, x k, u u 3 k f t, x k, u u 4 3 x x k k k k x u u u u 3 4 6 6 x x 6 u 3,,,, 3... N, (3.. a ) 6 Also, for teratve formula k f ( t, x, u ), k f,,, t k u x k f t, k, u, x 3 k f t, k, u, x 4 3 k k k 3 k 4 6,,, 3... N, (3.. b ) DOI:.979/578-54363 www.osrjournals.org 57 Page
X...4.8 X.exact..4.8 U..99.. U.exact.6.8...4 Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control Te table below sows te results of problem =. S/N Tme u. x. u.exact x.exact.777..5.777. 3...777. 4.3.35.777.3 5.4.4.777.4 6.5.55.777.5 7.6.63.777.6 8.7.735.777.7 9.8.84.777.8.9.945.777.9 3.5.777 3 Table...4.6.8. tme...4.6.8. tme...4.6.8. tme...4.6.8. tme Fg. 3..a, optmal state and control values at =. (). Te cocoa problem: Maxmzes P ( u xu x ) d t subject to x u and x( ), u( ) Taken te frst four procedures n (.5), we ave DOI:.979/578-54363 www.osrjournals.org 58 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control u x u x u x Employng Runge-Kutta metod of order 4, wt ntal guess of we ave, u k f ( t, x, u ) u k f,, t x k u u k f,, 3 t x k u u k f t, x k, u u 4 3 x x k k k k x u u u u 3 4 6 6 x x 6 u 3,,,, 3... N, (3.. c ) 6 Also, te teratve formula for k f ( t, x, u ) u x, k f,,, t k u x u x k f,,, 3 t k u x u x k f t, k, u, x u x 4 3 k k k k u x 6 3 4 u x,,, 3... N, (3.. d ) Te table below sows te numercal results to problem =. S/N Tme u x u. exact x. exact...9998.4.9999.9998 3.4.4396.99996.7984.39989.999 4.6.658.99888.7536.59964.998 5.8.8744.996776.554.7995.9968 6..879.99496.968.99833.9954 7..978.99744.7445.97.9989 8.4.433.994.636.39543.996 9.6.636.9874.9679.5938.9877.8.8978.983683.393.793.983844..85.979864.3643.98669.9867 DOI:.979/578-54363 www.osrjournals.org 59 Page
X. -.5..5. X.exact -.4..4.8 U...4.8 U.exact..4.8 Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control..649.975646.3959.83.975897 3.4.436.9734.479.3773.97338 4.6.5998.9666.457.578.96639 5.8.7953.9667.4859.76356.9655 6.3.9895.954837.5754.955.955336 7.3.38.948658.53374.34567.94935 8.34.337385.9494.558959.333487.94755 9.36.35647.93546.58347.3574.935897.38.37533.9788.6696.379.98665.4.39487.9.68395.38948.96 Table..5..5. tme..5..5. tme..5..5. tme..5..5. tme (3) Te pendulum problem: Mnmze subject to u and θ t = θ θ(t) free were, F = ml u mgl( cos θ) Taken te frst four procedures n (.5), we ave u Fg. 3..b. Optmal state and control values at =.. t S = ( ml u mgl( cos θ))dt t m g l s n x u m l g Employng Runge-Kutta metod of order 4, wt ntal guess of DOI:.979/578-54363 www.osrjournals.org 6 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control we ave, k f ( t,, u ) u u k f,, t k u u k f,, 3 t k u u k f t, k, u u 4 3 3 4 6 6 k k k k u u u u 6 u 3,,,, 3... N, (3.. e ) 6 Also, for teratve formula k f ( t, x, u ) sn, k f,,, sn ( ) t k u x k f,,, sn ( ) 3 t k u x k f t, k, u, x s n ( ) 4 3 k k k k 3 4 s n s n ( ) s n ( ) s n ( ) 6 6 s n 4 s n ( ) s n ( ),,, 3... N, (3.. f ) 6 Te table below sows te numercal approxmatons to problem 3. =. S/N Tme u.. u.exact.exact. -..99563.5 -.9956.9899.9949667 3..983.46 -.983.9796.3979734 4.3.974375.59896 -.97437.9695.595434 5.4.96563.7969 -.96563.9584.7978775 6.5.956794.99396 -.95679.9475.9878599 7.6.947866.94 -.94787.936437.88553 8.7.93885.3854 -.93885.9559.37436695 9.8.99753.5798 -.9975.93689.566394.9.9577.7739 -.958.97.757983..935.96584 -.93.8975.9467999..9.5748 -.9.87834.357 3..896.3488 -.896.86595.3335 4.3.88357.53794 -.8836.85349.58379 5.4.87364.7675 -.87364.84889.6936 6.5.8647.946 -.8647.884.87647343 7.6.854447.35 -.85445.8536.35863697 8.7.844774.38747 -.84477.8987.33949465 9.8.83556.34745 -.8356.788658.34984.9.8596.365645 -.853.7755.35975..85497.383948 -.855.76464.3774539 DOI:.979/578-54363 www.osrjournals.org 6 Page
Teta...5..5 Teta.exact..4.8. U...6. U.exact.5..5. Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control Table 3....4.6.8. tme...4.6.8. tme...4.6.8. tme...4.6.8. tme Fg. 3..c, optmal state and control values at =. IV. Dscusson We use Pontryagn s maxmum prncples to obtan analytc solutons to problems formulated from tree dfferent felds. We derved Runge-Kutta algortm tat generates te numercal approxmatons to te problems consdered. Bot te state and adjont varables are solved wt forward Runge-Kutta teratve formulas. It s observed tat te Runge-Kutta sceme produces results tat are comparable wt analytc results as t s sown n above tables. Te errors occur are gly nfntesmal. V. Concluson We ave formulated optmal control problems from tree dfferent felds. Te Pontryagn s maxmum prncples were employed for obtanng analytc solutons to optmal control problems. A Runge-Kutta metod of order four for numercal approxmatons to optmal control problems as been developed. From te numercal experments, te results sow tat te Runge-Kutta metod of order four produced results tat are comparable to analytc solutons to te problems consdered. Terefore, we conclude tat Runge-Kutta metod gves error tat s neglgble. Acknowledgements Te successful completon of ts paper could ave not been possble wtout te contrbutons made by varous autors, persons and groups. We owe a lot to all autors of varous lteratures we consulted and tey are dully acknowledged n te relevant sectons of ts wrte up. We express our most grattude to Dr. Eze, E. O. and Dr. Nkem Ogbonna for pontng out errors and approprate correctons made n ts paper. References [] Borcardt, W.G. and Perrott, A.O, New Trgonometry for Scools, G.Bell and Sons Lmted, London, 959, 9-9. [] Byron, F. and Fuller, R, Matematcs of Classcal and Quantum Pyscs, Addson-Wesly, 969. [3] Ejej, C.N, A modfed conjugate gradent metod for solvng dscrete optmal control problems: PD tess, Unversty Ilorn, Ilorn, Ngera, 5. [4] Elsogolts, L, Dfferental Equatons and Calculus of Varatons, Mr Publser, Moscow, 973. [5] Eugene, S. and Wng, S, Te structure of Economcs, Irwn McGraw Hll, New York,. [6] Francs, S., Numercal Analyss, Irwn McGraw Hll, New York,, 968 DOI:.979/578-54363 www.osrjournals.org 6 Page
Applcaton of Pontryagn s Maxmum Prncples and Runge-Kutta Metods n Optmal Control [7] Garret, R. R, Numercal metods for solvng optmal control problems: M.sc tess, Te Unversty of Tennessee, Knoxvlle, 5. [8] Lyusternk, L.A, Te Sortest Lnes, Mr Publser, Moscow, 976. [9] Rley, K.F., Hobson, M.P. and Bence, S.J, Matematcal Metods for Pyscs and Engneerng, Press Syndcate of te Unversty of Cambrdge, U.K,, 74-746. [] Saleem, R, Habb, M. and Manaf, A, Revew of forward backward Sweep metod for bounded and unbounded control problem wt payoff term, Scence Int. Laore, 7(). 69-7. 4 [] Sngresu, S. R, Engneerng optmzaton: teory and practces, New age nternatonal lmted, West Lafayette, Indana, 8. [] Spegel, M.R, Teoretcal Mecancs wt Introducton to Lagrange s Equaton and Hamltonan Teory, McGraw Hll, New York, 967. [3] Yourgrav, W. and Mandelstexolowatam, S, Varatonal Prncple n Dynamcs and Quantum Teory, Ptman and Sons, London, 968. DOI:.979/578-54363 www.osrjournals.org 63 Page