Optimal Control and Hamiltonian System

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1 Pure ad Appled Maheacs Joural 206; 5(3: 77-8 hp:// do: 0.648/.pa ISSN: (Pr; ISSN: (Ole Opal Corol ad Haloa Syse Esoh Shedrack Massawe Depare of Maheacs, College of Naural Sceces, Uversy of Dar es Salaa, Dar es Salaa, Tazaa Eal address: To ce hs arcle: Esoh Shedrack Massawe. Opal Corol ad Haloa Syse. Pure ad Appled Maheacs Joural. Vol. 5, No. 3, 206, pp do: 0.648/.pa Receved: Aprl 6, 206; Acceped: Aprl 28, 206; Publshed: May 0, 206 Absrac: I hs paper, a opal corol for Haloa corol syses wh exeral varables wll be forulaed ad aalysed. Necessary ad suffce codos whch lead o Paryag s prcple are saed ad elaboraed. Fally s show how he Poryag s prcple fs very well o he heory of Haloa syses. The case of Poryag s axu prcple wll be cosdered deal sce s capable of dealg wh boh ubouded couous corols ad bouded corols whch are possbly dscouous. Keywords: Opal Corol, Haloa Syses, Codos for Opaly. Iroduco I has bee esseal for ay physcal syses whch are govered by dffereal equaos o be corolled such a way ha a gve perforace dex would be opzed. Large savgs cos have bee obaed by a sall provee perforace. The opal corol proble whch wll be forulaed wll be he so called Bolza proble [] wh he added codo ha he corol varables le a closed se. [2] hs paper of Opal corol of sochasc dyacal syses developed exsece of sochasc opal corols for a large class of sochasc dffereal syses wh fe eory s cosdered. [3] esablshed a feedback corol law s developed for dyacal syses descrbed by cosraed geeralzed coordaes. They revealed ha for cera coplex dyacal syses, s ore desrable o develop he aheacal odel usg ore geeral coordaes he degrees of freedo whch leads o dffereal-algebrac equaos of oo. [4] developed a copuaoal approach o oor corol ha offers a ufyg odellg fraework for boh dyac syses ad opal corol approaches. I dscussos of several behavoural experes ad soe heorecal ad robocs sudes, hey deosraed how he copuaoal deas allow boh he represeao of selforgazg processes ad he opzao of ovee based o reward crera. [5] proposed a ew aheacal forulao for he proble of opal raffc assge dyac eworks wh ulple orgs ad desaos. Several researchers have suded opal corol ad dyacal syses. [6] suded Dyacal Syses based opal corol of copressble fluds. They proposed a cos fucoal based o a local dyacal syses characerzao of vorces. Coecos of opal corol ad Haloa syses especally he ecessary codos of opaly has o bee suded ye. I hs paper, s eded o focus o he lk bewee opal corol ad Haloa syses. The case of Poryag s axu prcple wll be cosdered deal sce s capable of dealg wh boh ubouded couous corols ad bouded corols whch are possbly dscouous. 2. Forulao of Opal Corol Proble We cosder he sae of a corol syse descrbed by a -vecor ( = ( x (,..., x ( x whose evoluo s govered by a syse of dffereal equaos ( x, xɺ = g ( where u s a corol fuco fro a closed subse of ad X =R. I =,, ope ses, Gve a copac erval [ ] R X R,

2 78 Esoh Shedrack Massawe: Opal Corol ad Haloa Syse V R, a se U V ha ad fucos, g, L ad φ such L: X V R, : X V R, φ: X R, g: X V R. he opal corol proble ca be saed as follows: Mze (, = φ ( ( + (, J x u x L x u d (2 over all couous fucos x ad easurable fucos u sasfyg (, xɺ = g x u u ( U, I., I, L s called he rug cos ad φ he eral cos. [7]. The Poryag s prcple requres he roduco of he Haloa fuco H: X R U U gve by (,, = (, + T (, H x p u L x u p g x u (3 I aalogy wh he correspodg quay classcal echacs. (,..., L p = p p = s he geeralzed qɺ oea. Slar o he forulao of he Haloa syses, he followg se of equaos hold [8] x ɺ ( = ( x(, p(, u( = g ( x(, u( p pɺ ( = ( x(, p(, u( (4 x L T g = ( x(, u( p ( ( x(, u( x x wh boudary codos p x = x 0 0 = φ x ( ( x( Necessary Codos for Opaly I hs seco we shall sae he ecessary codos for opaly whch he lead o Poryag s axu prcple. Theore The ecessary codo s for x, 0 u ( o be a opal al codo ad opal corol for he opal corol proble saed above are he exsece of a ozero k - desoal vecor λ wh λ 0 ad a = desoal vecor fuco P ( such ha for [, ] : [] ( Pɺ ( = P ( f, x (, u ( for [, ] ad u U, ( x ( P f,, f,, x u x u 0, ( P = ( e wh = (,, x(, x( λ φ x (v ( = λ φ P e, 0 x 0 (v e, (,,, = P f x x u λ φ e (,,, = P f x x u λ φ e (v If f (, x, u has a couous paral dervave (,, f x u he he codo (,, = λ φ ( 0,, ( 0, ( P f x u x x (v s + P f s x s u s ds,, holds for each [, ]. Codo ( above ca be wre as { H ( ( } = H ( ( 0 ax, x, u, x, u. (5 Ths s called Poryag s axu prcple. The erpreao of hs prcple s ha o he opal corol, H s zed wh respec o he corol varables u, =,...,. For splcy we shall rea proble. Ths s a specal case of he proble of opal corol whch he al e ad fal e are fxed ad here are o codos o he fal sae. We shall resae he Poryag s prcple so ha fs aurally o our fraework of free eral po proble. Theore: (Poryag s prcple for free eral po proble [] A ecessary codo for opaly of a corol u for he free eral po proble s ha ( f ( ( f ( ( ( P, x, v, x, u 0 (6 For each U soluo of v ad [, ], where 0 P s he

3 Pure ad Appled Maheacs Joural 206; 5(3: wh boudary codo ( = ( fx (, (, ( Pɺ P x u ( = φ x ( ( P x. The Poryag s prcple gves oly ecessary codos for opaly bu hese codos eed o be suffce. Sce each opal corol us be exeral, here us be exeral corols whch are o opal. However s aural o ask for codos whch are o opal. However s aural o ask for codos whch are boh suffce ad ecessary for opaly. Cosder a space ν of corol fucos u defed o, wh values o R. Le he subse κ V be he se [ ] of corol fucos u such ha [, ] ad 0, u U for each x u s a feasble par for he fxed al sae x. 0 The ecessary ad suffce codos ha a corol u be opal for free eral po proble s ha, we have for each fxed [ ] ( ( + (, = φ ( P fu, x, u v 0 for each v U such ha u v 0 K s covex ad he appg J u x s a fuco o K. To f hs o he perforace dex = φ ( + (,, J u x L x u d (7 I s assued ha ha L s a real couously x, u ad φ s dffereable fuco ad covex couously dffereable covex fuco of x. For splcy we shall cosder a lear syse. Theore A ecessary ad suffce codo for opaly of a corol u for free eral po proble wh syse []. ( = ( ( + ( ( xɺ A x B u, B R, (8 ad perforace dex s ha for [ ] for each U soluo of = φ ( x( + (, x, u J u L d, : ( ɶ Lu, x, u v + P B v 0 v such ha u( + v U where ( P ɶ s he P ɺɶ Pɶ A x u, ( = ( ( + L x (, (, ( ( = φ x (, x( P ɶ. Moreover f L(, x, u s srcly covex (, each fxed, he opal corol u ( s uque []. x u for Proof Sce he correspodg dffereal equaos s a lear u U ad syse, he se κ of corols such ha (, ( 0 couous fucos such ha u ( U se. Le u 0 ( ad u ( be corols κ ad 0 ( ( equaos wh x( 0 = x 0. If 0 α L(, x, u ad φ ( x ples x u s a feasble par cosss of all pecewse. Ths s a covex x ad x he correspodg soluos of he dffereal < <, he covexy of αj α J α L d φ ( u0 + ( ( u = (, x0, u0 + ( x 0 ( + + ( α L(, x (, u ( d φ ( x ( (, α 0 ( α, α 0 ( α L x + x u + u d ( x0 ( ( α x ( + φ α + ( α = Jαu + u (9 sce αx( + ( α x ( s he soluo of he dffereal equao correspodg o αu0 ( + ( α u (. Therefore J ( u s a covex fuco o κ. I ca be show ha for each v ( sasfyg u( + v ( U for each [, ] ha u ( u, v = (, x, v Pɶ B v 0. (0 δ J L d Hece J ( u has a u a u = u ( []. If L(, x, u s srcly covex, he above equaly s src. Thus J ( u s a srcly covex fuco o κ ad he u s uque. 3. Coecos o Haloa Syses To f he Poryag s prcple o heory of o he heory of Haloa syses, a corol syse

4 80 Esoh Shedrack Massawe: Opal Corol ad Haloa Syse xɺ = g ( x,, x X wll be cosdered wh he pu space U a afold whou boudares H ( x, p, u ad a sooh fuco all of s varables. Uder hese codos, he Poryag s prcple ples he frs order codo H = ( x, p, u = 0 for opzao. u gve by Cosder a sple corol syse ( X, B, g wh B = ( x, u. M = T X has a aural syplecc for ω = dθ ad T ( T X has a syplecc for ωɺ. The space of exeral varables W T U ( u, y e for ω for fuco ad Haloa e = dθ. Therefore e Ω = πω + π2ω. Le : H: T X U = = have a syplecc T T X T U s a syplecc L X U R be a sooh R. These fucos defe he (,, = (, + T (, H x p u L x u p g x u. Ths s a geerag fuco of he Lagraga subafold N ( T ( T X T U, Haloa equaos Ω gve by he xɺ = ( x, p, p H pɺ = ( x, p, ; =,..., ; =,..., ( x y = ( x, p, u [8] has show ha Haloa corol syse s gve by qɺ = ( q, p, p pɺ = ( q, p, ; =,..., ; =,..., (2 q y = c ( q, p, u Coparg equaos ( ad (2 ca be cocluded ha a corol syse ( X, B, g ogeher wh a sooh fuco L: X U R defes a full Haloa syse T X, T U, T B, f f g, h wh H where H ( H H gh: T X U T ( T X such ha gh ( x p x p H H,, =,,, p x H ad hh : T X U T U such ha hh ( x, p, u, u. I s assued ha B s a rval budle [9]. Le d( X = ad d( U =. Le also L( x, u, H ( x, p, u ad he Haloa syse ( T X, T U, T B, f H be as defed above. The f 2 H rak u u =,,..., H u = he equao ( x p u,, = 0 has a local Haloa fuco H = H x, p, u ( x, p [0]. We he oba locally a Haloa vecorfeld X Hop o T X. The proeco of he soluo curves of X H o op X for a se of curves whch by Poyags prcple x.. I s oed ha coas he opal raecory 2 H rak = u u, =,..., ples ha also H rak =. If we have oly rak of he u H ap = he we oba a ersed Lagraga u subafold V of (, op T T X ωɺ. Ths s slar o he P,, ɺ, = 0. plc Haloa dffereal equao ( x p x pɺ If T ( T X s proeced oo T X he here ay be soe pos V where he proeco does o have xal rak ad hus he soluo of he dffereal equao wll o be defed. If T X s proeced oo X, sgulares ad ouqueess of he opal raecores y occur [0]. 4. Cocluso I hs paper, a opal corol for Haloa corol syses wh exeral varables has bee forulaed ad aalysed. Necessary ad suffce codos whch led o Paryag s prcple are saed. I was show how he Poryag s prcple o he heory of Haloa syses. The case of Poryag s axu prcple was ake abroad because s capable of dealg wh boh ubouded couous corols ad bouded corols whch are possbly dscouous.

5 Pure ad Appled Maheacs Joural 206; 5(3: Refereces [] W. H. Fleg ad R. W. Rshel, Deersc ad Sochasc Opal Corol, Sprger, New York, Ic., 975. [2] N. U. Ahed, Opal corol of sochasc dyacal syses, Iforao ad Corol, Volue 22, Issue, pp. 3-30, 973. [3] V. Radsavlevc ad H. Baruh, Joural of Dyac Syses, Measuree, ad Corol, 2(4, pp , 999. [4] S. Schaa, P., Mohaera ad A. Ispeer, Progress Bra Research, Vol. 65, pp , [6] M. H. Huller, K. Kusch, Y. S., S. Volkwe, Ieraoal Joural for Nuercal Mehods Fluds, 00:-6, [7] S. Bare ad R. G. Caero R. G, Iroduco o Maheacal Corol Theory, Claredo Press, 985. [8] E. S. Massawe, Haloa Corol Syses, Ieraoal Joural of Theorecal ad Maheacal Physcs 206, 6(: pp [9] V. der Schaf, Syse Theorec Descrpo of Physcal Syse, Docoral Thess, Maheacal Ceru, Aserda, 984. [0] E. S. Massawe, Haloa Corol Syses, Upublshed M.Sc Thess, Uversy of Dubl. [5] S. Laforue, Iroduco o Dscree Eve Syses, The Ieraoal Seres o Dscree Eve Dyac Syses, 993.

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