Optimal control of Goursat-Darboux systems in domains with curvilinear boundaries

Transcription

1 Opmal conol of Goua-Daboux yem n doman wh cuvlnea boundae S. A. Belba Mahemac Depamen Unvey of Alabama Tucalooa, AL USA. e-mal: SBELBAS@G.AS.UA.EDU Abac. We deve neceay condon fo opmaly n conol poblem govened by hypebolc paal dffeenal equaon n Goua-Daboux fom. The condon con of a e of Hamlonan equaon n Goua fom, de condon fo he Hamlonan equaon, and an exemum pncple an o onyagn' maxmum pncple. The novel ngeden ha he doman ove whch he opmal conol poblem poed no ecangula. The non-ecangula naue of he doman affec he opmaly condon n a ubanal way. Keywod: Goua-Daboux conolled yem, Hamlonan equaon, exemum pncple. AMS Mahemac Subec Clafcaon: 49J0, 49K0, 35L70, 35R30.

2 . Inoducon. The heoy of opmal conol fo Goua-Daboux yem an mpoan exenon of he coepondng heoy fo conolled odnay dffeenal equaon. Th exenon ha he ame place n mahemacal opmal conol heoy ha he exenon fom ngle negal o mulple negal ha n he calculu of vaaon. Geneally, a conolled Goua equaon ha he fom x(, f (,,x(,,x (,,x (,,u(, --- (. Addonally, we need o pecfy a doman, ay G, fo he wo-dmenonal vaable (,, and appopae bounday condon on G. The mple fom of a doman G a ecangle, G {(, :0 a,0 b}. In ha cae, we have he andad Goua- Daboux poblem wh daa on chaacec,.e. (. accompaned by he e of bounday condon x(,0 x(, fo 0 a ; x(0, x(, fo 0 b; x(0,0 x0 x(0 x(0 --- (. I well nown (e.g. [CC] ha, unde naual Lpchz condon on he funcon f, connuy of he bounday daa x and x, and he conency condon x (0 x(0, he poblem {(., (.} ha a unque oluon. Thee ae alo moe geneal exence and unquene eul, bu hoe ae no elevan o he opmal conol ue ha we udy n he peen pape. An opmal conol poblem fo he yem {(., (.} concen he mnmzaon of a funconal J gven by J : (,,x(,,x(,,x (,,u(, da(, G a b (,x(,b,x(,bd (,x(a,,x (a,d 0 (x(a,b (.3 Neceay condon fo opmaly, n a fom analogou o onyagn maxmum pncple, have be obaned fo he poblem {(., (., (.3} n [BDMO, E, E, E3,

3 3 S, S, VST]; elaed wo, fom he pon of vew of dynamc pogammng wh wodmenonal "me" vaable, ha been done n [B, B, B3]. We defne he Hamlonan H(,,x,p,q, ψ,u : (,,x,p,q, ψ,u ψf (,,x,p,q,u --- (.4 (The vaable p, q and fo he lo of x, x, epecvely. Unde condon of uffcen dffeenably of all funcon nvolved, he Hamlonan equaon fo he coae ψ(, ae ψ(, H(,,x(,,x(,,x (,, ψ(,,u(, x D H(,,... D H(,,..., fo (, n(g; D p D q ψ(,b H(,b,... q D D (,... p (,... 0, fo 0 < a; x ψ(a, H(a,,... D (,... (,... 0, fo 0 < b; p D q x (a,... (b,... (y(a,b (a,b ψ 0 0 p q y --- (.5 D D The opeao, ae he opeao of ang oal devave wh epec o,, D D epecvely. Th mean ha fo any funcon Φ (,,x(,,x(,,x (,, ψ(,,u(,, we have DΦ D DΦ D Φ Φ x Φ x Φ x x p q Φ Φ x Φ x Φ x x p q Φ ψ Φ ψ ψ Φ u ; u ψ Φ u u --- (.6

4 4 D H D H I woh nocng ha he em and ae pecula o Goua D p D q poblem, and hee ae no analogou em n he Hamlonan equaon fo opmal conol poblem fo yem govened by odnay dffeenal equaon. D H D H The condon of he exence of he devave and a ong D p D q aumpon, nce, accodng o (.6, eque, among ohe hng, dffeenably of he conol u wh epec o and. One vaan of he andad model of conolled Goua-Daboux yem can be obaned by ang a conol funcon u n he fom u(, 0 0 v( dτ dσ --- (.7 o ha, n ealy, v become he conol funcon; h appoach ha been noduced n [B]. In cae n whch no aumed ha he funcon appeang n he Hamlonan equaon fo Goua yem ae uffcenly many me dffeenable, poble o ue an negal fomulaon of he Hamlonan equaon. The negal equaon fo he co-ae hen become a b H( ψ(, ψ(a, ψ(, b ψ(a, b d d τ σ x a H(,... H( b,... b H(, H(a, dσ dτ q q p p ψ(, b ψ(a, b a H( b,... (... (,... (a,... d σ 0 q x p p b H(a, (,... (,... (a,... (a, (a, b τ d ψ ψ 0 τ p x q q (a,... (b,... (x(a, b (a, b ψ 0 0 p q x --- (.8 ; ; ; We have menoned hee wo modfcaon,.e. ae dynamc wh nonlocal opeao acng on he conol and negal fomulaon of he Hamlonan equaon, fo he ae of compleene. We hall no ue hoe modfcaon fo he poblem we analyze n he peen pape, nce he man feaue of ou wo he effec of he non-ecangula

5 5 bounday on he opmaly condon. I poble, howeve, o nclude he wo modfcaon menoned above n he poblem ove non-ecangula doman. Thee ae many eaon fo udyng opmal conol poblem fo Goua-Daboux equaon ove non-ecangula doman: (. A hypebolc conol poblem may have been ognally fomulaed n nonchaacec coodnae, ove a doman ha ecangula n he ognal coodnae of he poblem. When we change o chaacec coodnae, n ode o ae advanage of he echnque ha have been developed fo he opmal conol of Goua-Daboux yem, he doman wll no geneally be ecangula n he new coodnae. (. The echnque we develop, n h pape, fo Goua-Daboux poblem ove nonecangula doman, can alo be ued fo poblem ha nvolve cean negal em, n he co funconal, wh negal uppoed on neo cuve of he doman, a well a em concenaed on neo pon of he doman. (3. Thee ae alo ohe eaon, n addon o elevance o applcaon, fo udyng Goua-Daboux conol poblem ove non-ecangula doman; hee ae "nenal" eaon, hey ae nnc o he dcplne of Mahemac. The udy of Goua- Daboux conol poblem ove ecangula doman no a mahemacally afacoy exenon of he coepondng conol heoy fo odnay dffeenal equaon, becaue doe no mae full ue of he wo-dmenonal paamee (,, whch play he ole of genealzed "me" fo Goua-Daboux yem. A cucal feaue of wo-dmenonal pace, compaed wh he one-dmenonal lne of eal numbe, he geae vaey of he hape of conneced open e ha ae poble on he plane. Th feaue no exploed when aenon eced o ecangula doman. Cean apec of he mahemacal ucue ha emege fom he nnc naue of vaaonal calculu fo conolled yem govened by Goua-Daboux equaon ae obcued when aenon eced o ecangula doman, and ae evealed only when we examne moe geneal non-ecangula doman.

6 6. Saemen of he poblem. We conde a doman G n he f quadan ( 0, 0 of he -plane, bounded by he egmen [0, a] on he -ax, he egmen [0, b] on he -ax, and a connuou and pecewe C cuve (γ wh a mo a fne numbe of pon a whch he nomal veco feld o (γ ha ump dconnue. The cuve (γ aumed o be a non-nceang gaph, and evey mooh egmen of (γ can be epeened n a lea one of wo fom: ϑ (, ϑ(. Clealy, boh epeenaon ae poble on evey mooh pa of (γ ha ha nowhee a angen paallel o any of he coodnae axe. We denoe by µ he paamee of ac lengh on (γ, meaued fom he pon 0 (a,0 on he -ax. The cuve (γ alo paamezed by ac lengh, a L( µ, L( µ, 0 µ L, whee L he oal lengh of (γ. We conde a e of pon on (γ, : {(a,0 0,,,..., N, N (0, b}, ha conan all he pon of ump dconnue of he nomal veco feld on (γ bu may alo conan addonal pon. The eaon fo allowng he pobly of ncludng addonal pon n wll become clea fahe down n h econ. We hall call he pon n vece. We e ( γ R : ( γ \, and we hall call ( γ R he egula pa of (γ. o lae ue, we alo defne he egula pa of G, denoed by G R, a he e obaned by emovng fom G all agh lne, paallel o ehe he -ax o he -ax, and pang hough a veex. If (, ( ae he - and - coodnae of each veex, hen G : {(, G: ( and ( 0,,..., N. R The fac ha he cuve (γ a non-nceang cuve mple he well-poedne, unde andad condon of Lpchz connuy of f elave o x, x, x, and connuy of f wh epec o u, of he Goua-Daboux poblem x(, f (,, x(,, x(, x(,0 x( fo 0 a ;, x x(0, (,, u(,, fo (, (n G ( γ; x( fo 0 b --- (. whee he funcon x (., x(. ae connuou on [ 0,a], [0, b], epecvely, and afy he conency condon x (0 x(0, and we e x0 : x(0 x(0. Th follow fom he fac ha, unde he aed condon, he e W(,, defned by W(, : {( : 0 σ, 0 τ } --- (.

7 7 afy W(, (n G ( γ (, (n G ( γ, and he Goua-Daboux poblem (. can be wen n negal fom a x(, x( x( x0 f ( τ, x(, xσ(, xτ(, u( W(, fo (, (n G ( γ --- (.3 and well-poedne can be hown by andad echnque fo uch wo-dmenonal Volea equaon. We noe ha moe geneal eul of exence and unquene, unde condon ha do no eque he Lpchz connuy of f wh epec o all he vaable we led pevouly, ae nown n he eeach leaue, bu hoe eul ae no elevan o he conol poblem we udy n h pape. Thu, we conde a conolled dynamcal yem of he fom (., n whch he ae x (, ae value n n-dmenonal Eucldean pace, and he conol funcon u (, ae value n m- dmenonal Eucldean pace. Coodnae wll be denoed by upecp, and he convenon of eno algeba abou ummaon wh epec o epeaed ubcp and upecp wll be cly ued, excep n cae n whch we explcly ae ha no ummaon aen. Befoe we defne he co funconal ha o be mnmzed, we need ome noaon. We n(, denoe by n(, he un nomal veco feld on he mooh pa of n(, (γ oened owad he exeo of G. o any dffeenable funcon ϕ, we denoe by ϕ ϕ µ o he angenal devave of ϕ on (γ,.e. µ ϕ(, ϕ(, ϕ µ (, n(, n(,, fo (, ( γr. The ouwad nomal ϕ devave of ϕ wll be denoed by ϕ n o, hu n ϕ(, ϕ(, ϕ, n (, n (,. n ( o lae ue, we defne a hd devave, denoed by ϕ o n ϕ n : ϕ n ϕ(, ϕ(, (, : n(, n(, --- (.4

8 8 The funconal ha o be mnmzed J : G Φ(,,x(,,x (,,x (,,u(, da(, ( γ Φ (,,x(,,x µ (,dµ(, Φ 0 (,x( --- (.5 whee d µ (, d d he dffeenal of ac lengh on (γ. I clea ha he co funconal (.4 nclude (. a a pacula cae. m The conol funcon u ae value n IR, aumed o be pecewe connuou n G (.e. hee a fne collecon of open e, uch ha he unon of he cloue equal G, and u connuou on each of hoe open e, and fo mplcy we aume ha he componen of u afy u u u, m. The e of admble value of he conol funcon wll be denoed by U. Naually, moe geneal conol can be condeed, fo example bounded meauable conol ang value n a compac ube m of IR ; howeve, uch genealy would no conbue o he ue of h pape, whch concen he effec of he cuvlnea bounday on he Hamlonan equaon.

9 9 3. Hamlonan equaon. We deve he Hamlonan equaon ang ou of he f vaaon of he funconal (.4 ubec o he conan of he ae dynamc (.. By andad vaaonal mehod, he vaaon of he ae, δ x(,, nduced by an admble vaaon δ u(, of he conol funcon, afe δx (, f (,,... δx (, f (,,... δx f (,,... δx (, x p q f (,,... δu (,; δx (,0 δx (0, 0 u --- (3. The vaaon of he co funconal J δj(u, δu Φ ( γ N u (,,... δu { Φ Φ,x 0,x G { Φ (,,... δx (, Φ (, }da(, (,,... δx (, Φ (,x( δx ( x, η p (,,... δx (,,... δx (, Φ µ (,}dµ(, q (,,... δx (, --- (3. The vaaon δ x can be expeed n em of he max-valued Remann funcon R (,, σ, aocaed wh he lnea Goua-Daboux poblem (3.. The heoy of he cala Remann funcon explaned n deal n [CC], and alo can be found n many andad efeence on paal dffeenal equaon. The poof of he popee of he max-valued Remann funcon ae analogou o he poof of [CC], and fo h eaon we hall only menon, whou poof, hoe of he popee of [R ] ha ae needed n h pape. The max-valued Remann funcon afe

10 0 R (,, f (,,...R (,, x f (,,... R (,, f (,,... R (,,, fo > σ and > τ; p q R (, τ, f (, R (, τ, ; q R (, f (,...R (, ; p R ( τ, δ ( Konece' dela --- (3.3 o lae ue, we alo we down he adon poblem o (3.3 ha he max-valued Remann funcon alo olve: f (,,... R ( τ,, R ( τ,, x D f (,,... D f (,,... R ( τ,, R ( τ,,, D p D q fo < σ and < τ; f (,... R ( τ, R ( τ, ; p f (, R ( τ,, R ( τ,, ; q R ( τ, δ --- (3.4 The oluon δ x(, of (3. can be expeed a x (, R (,, f u W(, δ ( δu ( da( --- (3.5

11 Thu, he vaaon of he co funconal J δj G R (,, f ( δu ( da( da(, u Φ (,,... δu (, da(, u G {n(, Φ p ( γ W(, D Φ (,,...}R (,, f Dµ, η u N { Φ 0,x { Φ x W(, (,,... D D Φ (,,... p (,,... n(, Φ q D D (,,... Φ,x ( δu ( da( dµ (, (,x( V ( Φ } R (, f, η u W( Φ (,,...} q (,,... ( δu ( da( --- (3.6 By ung he negaon fomulae of appendx A, we fnd δj R ( τ,, f (,,... δu (, da( da(, u Φ (,,... δu (,da(, u G G G {n( Φ p γ(, D Φ Dµ, η N { Φ 0,x { Φ x E(, ( D D ( n( Φ q ( }R ( τ,, f u (,x( V ( Φ, η Φ ( p (,,... δu (, dµ ( da(, } G D D ( Φ,x R (,, f u Φ ( } q ( (,,... δu (, da(, --- (3.7

12 The eplacemen of R (, f ( δu ( da( by u W( R (,, f (,,... δu (, da(,, n he la em n (3.7 above, ufed by u G he fac ha R (,, 0 wheneve (, W(. o addonal empha, we may we h em a R (,, f (,,... χ((, W( δu (, da(, whee χ u G, f S ue and fo he uh funcon of a logcal aemen, χ( S :. 0, f S fale We defne he co-ae ψ(, by ψ (, : { Φ x E(, R ( τ,, da( {n( Φ p γ(, D Φ ( }R ( τ,, dµ ( Dµ, η N { Φ 0,x ( D D ( n( Φ q Φ ( p D D ( Φ,x Φ ( } q ( (,x( V ( Φ }R (,, χ((, W(, η --- (3.8 Alo, we defne he Hamlonan H by H(,,x,p,q, ψ,u : Φ(,,x,p,q,u ψ f (,,x,p,q,u --- (3.9 Then he vaaon of J ae he fom δj G H(,,x(,,x(,,x (,, ψ(,,u(, δu (, da(, u --- (3.0

13 3 We hall pove Theoem 3.. Aumng adequae connuou dffeenably of all funcon nvolved, he co-ae ψ defned n (3.8 afe he followng hypebolc equaon, n he egula pa of he doman G: ψ(, H(,,x(,,x(,,x (,, ψ(,,u(, x D H(,,x(,,x(,,x (,, ψ(,,u(, D p D D H(,,x(,,x (,,x (,, ψ(,,u(, q --- (3. oof: We hall ue he adon equaon (3.4 fo he Remann funcon and he dffeenaon fomulae of appendx A. In ode o mplfy he noaon, we e D D ( : Φ ( Φ ( Φ ( ; x Dσ p Dτ q, ( : n( Φ ( n( Φ ( Φ p q,x D Φ Dµ, η ( ; 0, (,x( : Φ 0,x (,x( V ( Φ, η ( --- (3. Then (3.8 can be wen a ψ (, ( R ( τ,, da( E(, N, ( R ( τ,, dµ ( 0, (,x( R (,, γ(, --- (3.3

14 4 Accodng o he dffeenaon fomulae ha ae poved n appendx A, we have ψ (, B (, R (, τ,, dτ ( R ( τ,, da( E(, ( γ N, ( R ( τ,, dµ ( 0, (,x( χ((, W( R (,,, (B,...R (B,, n (B --- (3.4 ψ (, A (, R (, τ,, dσ ( R ( τ,, da( E(, ( γ N, ( R ( τ,, dµ ( 0, (,x( χ((, W( R (,,, (A,...R (A,, n (A --- (3.5

15 5 ψ (, E(,, ( γ N (A,...R (A,,, (B,...R (B,, n (A n (B ( R ( τ,, da(, (, R (, τ,,dσ ( R ( τ,, dµ ( 0, A (,x( χ((, W( R (,, B (, R (, τ,, dτ --- (3.6 We noe ha, n he neo of G R, he funcon χ ((, W( eman conan. o hoe em of (3.6 ha conan R ( τ,,, we ue he expanded fom of he f equaon n (3.4: f R ( τ,, f (,,... R ( τ,, R q (,,... R x f ( τ,, ( τ,, D D f (,,... R ( τ,, p (,,... R p ( τ,, D D f --- (3.7 (,,... q The expanded fom of he waned equaon (3. ψ (, Φ(,,... x D D Φ(,,... p Φ(,,... q f (,,... D f (,,... D f (,,... ψ(, D x D p q ψ(, f (,,... ψ(, f (,,... p q D D --- (3.8

16 6 We examne he em ha ae ou of he gh-hand de of (3.6, and, ung (3.7 and he emanng equaon ou of (3.4, we how ha hee em mach he em on he gh-hand de of he waned equaon (3.8. R Each em conanng on he gh-hand de of (3.6, by nvong (3.7, R R mached wh he coepondng em conanng, f f mulpled by,, p q ψ ang fom he em on he gh-hand de of (3.8 ha conan ψ,, afe hoe em have been expeed by ulzng he econd and hd equaon ou of he e D f D f (3.4, and wh mla em conanng R mulpled by, ang fom D D p q he em conanng ψ on he gh-hand de of (3.6. Th machng can be ealy een, fo example, fo he em conng of double negal ove E(, ; he poof R mla fo he ohe em ha conan on he gh-hand de of (3.6. Now, by f (,,... he hd equaon n (3.4, he em R ( τ,, equal R ( τ,,, q ψ and h mached wh he coepondng em n he expanon of f. Th q A compaon ypfed by he em conanng he negal n (3.6 afe ψ and f devave (fom (3.4 and (3.5 have been ubued no (3.6; he compaon fo he emanng em mla. Of coue, he compaon fo he em ha conan conan R R. /// paallel o he compaon, explaned above, fo he em ha

17 7 4. Sde condon fo he Hamlonan equaon. The oluon of he Hamlonan equaon, whch wee deved n he pevou econ, eque de condon ha mae he coepondng Goua poblem (Hamlonan equaon plu de condon well-poed poblem. Thee de condon ae no meely bounday condon bu ahe hey ae moe complcaed condon, and hey conue he man dffeence beween he cae of ecangula doman and he cae of geneal doman wh cuvlnea bounday of he ype decbed n h pape. We hall need ome defnon and noaon peanng o he geomey of (γ. Defnon 4.. A conneced pa of (γ wll be called fla f a agh lne egmen paallel o ehe he -ax o he -ax; a conneced pa of (γ wll be called oblque f doe no conan any agh lne egmen ha paallel o ehe he -ax o he - ax. /// Theoem 4.. On he elave neo (n he elave opology of (γ of a fla pa of (γ, a pon (, ha do no belong o, he co-ae ψ afe [(n (, (n (, ] Φ(,,... D Φ (,,... 0 x Dµ η ψ (, H(,,... n(, n µ p H(,,... (, q --- (4. If a pon n he elave neo of he oblque pa of (γ and, hen lm (, lm (, ψ (, 0; n ψ (, ( lm (, ψ (, n(, (, (4. oof: o a pon (, n he elave neo of a fla pa of (γ wh (,, we have, n ome elave neghbouhood of (,, ehe con. o con., and each of he funcon χ (( W( eman conan fo ( n a uffcenly mall elave neghbouhood of (,. Clealy, uffce o pove (4. n one of he cae con. o con., fo example n he cae con. In ha cae, we have (,, fo B

18 8 ome value uch ha {(, : 0 } a maxmal agh lne egmen conanng he pon (,, and B a membe of, accodng o ou defnon and aumpon n econ. Thee can be ohe elemen, ay, of n he egmen of (γ ha connec (, and B. On ha egmen, we have µ (, 0. A, (,, wh ' d d, n(,, n ( (, n he neo of he egula pa of G, he double negal ove E(, goe o 0, and we have, accodng o he defnon of ψ, ψ (, (, R (, τ,,dτ 0, ((,,x(, R (,,, ' 0, ((,(,x(,( R (,(,, ' ' ' --- (4.3 fom whch ψ (, (,,...R (,,, (,,... R ( 0, ((,,x(, R (,,, ' 0, ((,( ',x(,( ' R (,(',, By ung (3.4, we can we (4.4 a, τ,,dτ --- (4.4 ψ (, (,,... 0, ' f (,,... ((,,x(, R p 0, f ((,(,x(,( ' (,,... p f (,,... [ ψ (, f (,,...] ' (,,, (,,... R p (,,... R p (, τ,,dτ (,(,, ' --- (4.5

19 9 Now, (4.5 anamoun o (4.; h la aeon a dec conequence of he defnon of H and n (3.9 and (3.., If (, a pon on he oblque pa of (γ and no on, hen, a ( ',' (, wh ( ',' n(gr, boh he double negal ove E(', ' and he lne negal ove γ(', ', n he defnon of ψ(', ', wll go o 0; he dcee em conanng 0, wll alo be zeo f (', ' uffcenly cloe o (, o ha hee no pon n uch ha W( conan (', '. Th pove he f equaon n (4.. To pove he econd equaon n (4., we ue (3.4 and (3.5; a (, ((,(, he only em, ou of he gh-hand de of (3.4 and (3.5, ha have (pobly nonzeo lm ae he em, (B,...R (B,,,, (A,...R (A,, ; hu n (B n (A lm (, n ψ (, ( lm (,, n ( (B,...R (B,,, (,... n(b --- (4.6 and analogouly lm (, ψ (, n(, (, (4.7 Th conclude he poof of (4.. /// The examnaon of he lmng value of ψ a a pon of eque he followng Defnon 4.. o evey pon of G (ncludng pon on ( γ R and pon on, we defne he e W W W III II IV ( : {(, n(g: < (, < (}; ( : {(, n(g: < (, > (}; ( : {(, n(g: > (, < (}

20 0 --- (4.8 (The upecp n (4.8 efe o he quadan of he - plane aound he pon. o evey pon on (γ, we defne he followng e: T( he collecon of all vece, excludng elf n cae a veex, ubequen o elave o he couneclocwe oenaon of G, ha le on a agh lne egmen paallel o he -ax, ha pa of (γ and pae hough ; S( he collecon of all vece, excludng elf n cae a veex, pecedng elave o he couneclocwe oenaon of G, ha le on a agh lne egmen paallel o he -ax, ha pa of (γ and pae hough ; V( : T( S( {} ; L ( he maxmal agh lne egmen conaned n (γ, excludng elf, conng of pon ubequen o elave o he couneclocwe oenaon of G, ha paallel o he -ax and pae hough ; L ( he maxmal agh lne egmen conaned n (γ, excludng elf, conng of pon pecedng elave o he couneclocwe oenaon of G, ha paallel o he -ax and pae hough ; L( : L ( L ( {}. Clealy, n ome cae, ome of he e defned hee may be empy. /// We have: Theoem 4.. The lmng value of ψ a (, appoache a veex ae deemned by

21 lm ψ (,, ( R ( τ, dµ( (, L( III (, W ( 0, ( ',...R (', ; ' V( lm ψ (,, ( R ( τ, dµ( (, L ( II (, W ( 0, ( ',...R (', ; ' T( lm ψ (,, ( R ( τ, dµ( (, L ( IV (, W ( 0, ( ',...R (', ' S( --- (4.9 III oof: o (, n W ( and uffcenly cloe o, we have E(, L(. III A (, wh (, W (, he ac A B convege o L( n he ene ha he wo pon A, B convege o he wo exeme of L(. To how h, we obeve ha, f L ( ' ' ' (.e. he e L( equal o he conneced ac of (γ ha ha exeme ' and ' ', wh ' pecedng ' ', hen he egmen '' a maxmal agh-lne egmen hough ha pa of (γ and paallel o he -ax, and ' a maxmal agh-lne egmen hough ha pa of (γ and paallel o he -ax; h uaon nclude he pobly ha one o boh of he agh-lne egmen '', ' mgh degeneae no he ngleon { }. By defnon, fo III (, W (, we have < ( and < (, hu alo < ( ' ( and < ( '' (, and conequenly E(, L(. By he maxmaly of he egmen '', ', he ac A ' ' no paallel o he -ax, and he ac ' B no paallel o he -ax. Theefoe, he pon B ha he epeenaon B (, ϑ( and conequenly, a ( (' ', we have B ' '. By he ame oen, a ( ( ', we have A ( ϑ(, '. By ang lm n (3.8, whch he he defnon of ψ, ung he nfomaon above, we oban he f equaon n (4.9. The poof of he emanng pa of (4.9 ae mla. ///

22 Defnon 4.3. We denoe by ( S he agh lne egmen hough and paallel o he -ax, by ( T he agh lne egmen hough and paallel o he -ax, and by V, he neecon of ( S and ( T fo >. We denoe by D, 0 N, he (geneally cuvlnea-angula doman bounded by he ub-ac ( γ and he agh lne egmen V,, V,. (Clealy, D degeneae no he agh-lne egmen when ha egmen a fla pa of (γ, and heefoe, n ha cae, n D. We denoe by Q,, 0 < N, he ecangula doman wh vece V,, V,, V,, V,. The collecon of doman { D, Q, } paoned no zone a follow: zone Z 0 compe he angula doman p N, zone Z p compe all ecangula doman Defnon 4.4. The ump Ω D ; fo each nege p, Q, wh p. /// ( T ϕ of a funcon ϕ aco a agh lne egmen (T ha paallel o he -ax defned a Ω (T ϕ(, : ϕ(t (, ϕ(t (, fo (, (T, whee ϕ( T (, : lm ϕ(, ϕ(t (, : lm ϕ(. Smlaly, he ump Ω ( S ϕ of a σ σ funcon ϕ aco a agh lne egmen (S ha paallel o he -ax defned a Ω (S ϕ(, : ϕ(s (, ϕ(s (,, whee ϕ( S (, : lm ϕ(,, ϕ(s (, : lm ϕ(,, fo (, (S. Of coue, hee τ τ defnon ae condonal on he exence of he ndcaed lm. /// Theoem 4.3. The ump of he co-ae ψ aco he lne ( S, (T ae gven by Ω(S ψ (, L fo (, n G wh ( ; Ω(T ψ (, L, ( R ( τ,, dµ ( ( ':, ( R ( τ,, dµ ( ( '': 0, ( ',...R (',, S( 0, ( '',...R ('',, S( fo (, n G wh ( --- (4.0 ' '' oof: A n he poof of heoem 4., he conbuon of he double negal (negal ove E(, o he calculaon of he ump of ψ zeo. We pove he f equaon n

23 3 (4.0, nce he wo equaon n (4.0 ae ymmec wh epec o an nechange of he vaable and. We conde wo agh lne S, ε, S,ε conng of pon of he fom {( (,': ' ε, ((, S}, {((,'': '' ε, ((, S}, epecvely. o ε>0 and uffcenly mall, he pon of ha ae ncluded n γ ( (, ε bu no n γ( (, ε ae pecely he pon n L (. If he vece ', '' ae defned n he ame way a he pon ', ' ', epecvely, n he poof of heoem 4., hen he e-heoec dffeence γ ( (, ε \ γ((, ε convege, a ε 0, o he ac ' whch concde wh he e S(. Theefoe, we have Ω (S lm ε 0 E((,ε ': S(, L ( E((,ε ( R ( τ,(, εda(, γ((,ε, γ((,ε 0, : W((,ε ψ ((, ' ( R ( τ,(, εdµ ( ( R ( τ,(, εdµ ( (,...R (,(, ε ' ( R ( τ,(, εda( 0, ( ',... ( R ( τ,(,dµ( ( R (,(, ε R (,(, ε ': S( ' 0, (,...R (,(, ' ' --- (4. Theoem 4.3 poved. /// The de condon of heoem 4. and 4.3 can be wen n Hamlonan fom. The de condon depend on he Remann funcon hough em of he fom R (, τ,,, R (,,. We e

24 4, (,, : R (, τ,,,,(,, σ : R (,, --- (4. The quane α,, α,, wll play he ole of addonal new co-ae. Accodng o (3., we have,, f (,,... (,, p, f (,,... (,, σ q (,, ;,, (,, σ; (, τ, δ, ; (, σ δ --- (4.3 We defne wo new famle of Hamlonan h(,, τ,x,p,q,,u : f (,,x,p,q,u, ; h (,, τ,x,p,q,,u : f (,,x,p,q,u, ; h, h, n a follow: --- (4.4 Then (4.3 can be wen n Hamlonan fom a h(,, τ,x(,,x(,,x (,, (,,,u(,, (,, ; p h (,, τ,x(,,x(,,x (,, (,,,u(,, (,, σ ; p, (, τ,, (, σ δ --- (4.5 The condon of heoem 4. and 4.3 become

25 5,(,( ((,...,( (( d,,( ((,..., (( (, lm ;,(,( ((,...,( (( d,,( ((,...,( ( (, lm,...;,( ((,(,( ((,...,( ((,(,( ((,...,( (( d,,( ((,..., (( d,,( ((,...,( ( (, lm, 0, '',, ( ( ( W (, (,, 0, ',, ( ( ( W (, (, 0,, 0, '', 0, ',, ( (,, ( ( ( W (, (, '' IV ' II '' ' III τ τ τ ψ σ σ σ ψ τ τ τ σ σ σ ψ < < < < --- (4.6,(,( ((,...,( (( d,,( ((,..., (( (, ;,(,( ((,...,( (( d,,( ((,...,( ( (,, 0, '',, ( ( (T, 0, ',, ( ( (S ' ' τ τ τ ψ Ω σ σ σ ψ Ω < < --- (4.7 The gnfcance of he de condon n heoem 4., 4., and 4.3 hown n he followng:

26 6 Theoem 4.4. We aume ha he gh-hand de of he Hamlonan equaon (3. ψ ψ Lpchz connuou wh epec o ψ,,, unfomly n all he ohe vaable. Then he de condon fo ψ, eablhed n heoem 4., 4., and 4.3, ae uffcen fo he unque olvably of he Hamlonan equaon (3.. oof: o each non-degeneae angula doman D, he Hamlonan equaon (3. ae wen, n negal fom (afe epeaed applcaon of he Gau-Geen heoem, a ψ(, B A ψ ( ψ (B ψ (A n ( dµ ( ψ( ψ( g( τ, ψ(,, da( τ σ E(, --- (4.8 whee g denoe he gh-hand de of (3.,.e. H(,,x(,,x(,,x (,, ψ(,,u(, g(,, ψ, ψ, ψ : x D H(,,... D H(,,... D D p q --- (4.9 o mplcy, we do no explcly how he dependence of g on x,x,x,x,x, u n n n ψ (4.9. The ymbol ψ ha he meanng ψ n ψ ψ n. Snce D n aumed o be non-degeneae, he ac D ( γ an oblque pa of (γ, and he value of n ψ ae gven by (4., and we have n ψ (, (, (, n (, ψ ψ n (,, (,, (4.0 Conequenly, (4.8 an negal equaon of he fom

27 7 ψ( ψ( (, ψ0,(, g( τ, ψ(,, da( σ τ E(, --- (4. ψ and exence and unquene of oluon of negal equaon of h ype can be hown by povng convegence of a equence of cad eaon; h poof gven n appendx B of he peen pape. Thu he Hamlonan ae unquely olvable n he neo of each doman n zone Z 0 (cf. defnon 4.3. Now, he value ψ (S (, ae nown fo (, eln(v, V, fom he oluon ψ n n D, and he value ψ (T (, ae nown fo (, eln(v, V, fom he oluon ψ n n D. The ump condon of heoem 4.3 hen allow he deemnaon of ψ (S (, fo (,,, eln(v V and ψ fo, eln(v V. (T (, (,, Conequenly, he Hamlonan equaon fo ψ ae unquely olvable a odnay Goua-Daboux poblem n n Q, fo all,.e. ψ now nown n he neo of each doman n zone Z. Inducvely, f ψ nown n he neo of each doman n zone Z p, hen he value ψ (S (, ae nown fo (, eln(v, V, fom he oluon ψ n n Q, p, and he value ψ ae nown fo, ( (T (,, eln(v, V, fom he oluon ψ n n Q, (vally, when Q, n zone Z p, Q, alo n zone Z p ; he ump condon hen yeld he value of ψ (S (, fo (, eln(v, V, and ψ (T (, fo (, eln(v, V,, and hen ψ deemned a he unque oluon of an odnay Goua-Daboux poblem n n Q,, hu n he neo of evey doman n zone Z p. The nducon complee and he heoem poved. /// Rema 4.. The eamen peened n econ 3 and 4 can be exended o he moe geneal cae n whch G a doman, on he - plane, conanng he ogn O n neo and bounded by a connuou cuve (γ wh he popee aed below. Le G,,,3, 4 denoe he pa of G ha le n he coepondng quadan of he - plane, and le ( γ,,,3, 4 be he pa of (γ n he ame quadan. (The quadan of he - plane ae labelled n he andad way, namely he f quadan { 0, 0}, he econd quadan { 0, 0}, he hd quadan { 0, 0}, and he fouh

28 8 quadan { 0, 0}. We defne he anfomed doman G ~ ( ~ γ, fo,, 3, 4, by and anfomed cuve G ~ ~ G ~ ~ G ~ : G, ( γ : ( γ; {(, : (, G}, ( γ : {(, : (, ( γ} ; {(, : (, G }, ( ~ : {(, : (, ( }; G ~ 3 3 γ3 γ3 {(, : (, G }, ( ~ 4 4 γ4 : {(, : (, ( γ4} --- (4. We poulae ha each pa ( G,( γ,,,3, 4 hould have he ame popee a hoe of he pa (G, (γ ha wee aed n econ of h pape. We conde he poblem wh ae equaon x(, f (,, x(,, x(, x(,0 x( fo (,0 G;, x x(0, (,, u(,, fo (, G ; x( fo (0, G --- (4.3 and co funconal J, o be mnmzed, gven by J : G Φ(,,x(,,x (,,x (,,u(, da(, ( γ Φ (,,x(,,x µ (,dµ(, Φ 0 (,x( --- (4.4 whee now he e can conan a mo a fne numbe of pon n each pa ( γ,,,3,4 of (γ. The e can be wen a a don unon U. The 4 4 funconal J can be expeed a a um J J wh each J gven by

29 9 J : G Φ(,,x(,,x(,,x (,,u(, da(, Φ(,,x(,,xµ (,dµ(, Φ0(,x( ( γ --- (4.5 The vaaon δj, unde a vaaon δu of he conol, can analogouly compued a 4 δj δj --- (4.6 and he vaaon of each J can be evaluaed by he ame echnque a fo he andad cae of econ. The co-ae ψ can be evaluaed n each ub-doman G by he ame mehod a fo he andad poblem. ///

30 30 5. An exemum pncple. Maxmum pncple of onyagn' ype have been poved n [BDMO, E, E, E3, S, S, VST] fo conolled Goua-Daboux yem ove ecangula doman. A maxmum pncple fo an opmal conol poblem ha wo man componen: he Hamlonan pa,.e. a poof ha he co-ae afe a e of Hamlonan equaon, and he exemum pa,.e. a poof ha an opmal conol mu mnmze he Hamlonan ha coepond o an opmal aecoy. We have aleady poved he appopae Hamlonan equaon fo he poblem of he peen pape. We hall now fomulae and pove he exemum pa of a maxmum pncple, an o onyagn' maxmum pncple. The dea of he vaaonal agumen below baed on he appoach of Gabaov and Kllova [GK]. The pncpal effec of he non-ecangula doman manfeed n he de condon fo he Hamlonan equaon; he poof of he exemum popey doe no ubanally devae fom he poof fo ecangula doman, and ncluded hee pmaly fo he ae of compleene. We have: Theoem 5.. Aumng he exence of a pecewe connuou opmal conol funcon u (. wh coepondng ae aecoy x (.,. and co-ae aecoy ψ (.,., and aumng ha H unfomly dffeenable wh epec o he vaable u wh connuou H(,,x (,,x bounded devave (,,x (,, ψ (,,u (, fo (, G R, we u have, fo evey pon (, of G R ha alo a pon of connuy of u, H(,,x (,,x H(,,x (,,x (,,x (,, ψ (,,u (, (,,x (,, ψ (,,u u U --- (5. oof: o evey admble vaaon δ u(.,. of he conol funcon, we denoe he fne ncemen of J by J, hu J(u (.,., δu(.,. : J(u (.,. δu(.,. J(u (.,. --- (5.

31 3 The ncemen of he Hamlonan, wh epec o a vaaon n he eal vaable u(, alone, denoed by u H, hu uh(,,x (,,x (,,x (,, ψ (,,u (,; δu(, : H(,,x (,,x (,,x (,, ψ (,,u (, δu(, H(,,x (,,x (,,x (,, ψ (,,u (, --- (5.3 Becaue of he defnon of J, we have J(u (.,., δu(.,. δj(u, δu o ( δu L (G --- (5.4 whee δ J(u, δ u gnfe δ J a defned n econ 3, evaluaed fo conol funcon u (.,. and admble vaaon δ u(.,.. Alo, by he dffeenably aumpon on H, we have uh(,,x (,,x (,,x (,, ψ (,,u (,; δu(, H(,,x (,,x (,,x (,, ψ (,,u (, δu o( δu(, u --- (5.5 Accodng o he eul we have hown n econ 3, we have δj(u, δu G H(,,x (,,x (,,x (,, ψ (,,u (, δu da(, u --- (5.6

32 3 Conequenly, J(u (.,., δu(.,. uh(,,x (,,x (,,x (,, ψ (,,u (,; δu(, da(, G o( δu L (G --- (5.7 We denoe by B(,; ε he open d n IR wh cene a (, and adu ε,.e. B(, ; ε : {( IR : ( σ ( τ < ε } --- (5.8 We ae ε>0 o mall ha B(, ;ε GR and u connuou n B(, ;ε. We ae a an admble vaaon a funcon v ε wh he followng popee: v ε connuou n B(, ;ε ; v ε vanhe oude B(,; ε ; vε (, u u (,, whee u an abay bu fxed elemen of U. Then we have J(u (.,.,v (.,. G o( ε u ε H(,,x (,,x (,,x (,, ψ (,,u (,;v (, da(, ε --- (5.9 and alo

33 33 G u B(,; ε u H(,,x (,,x (,,x (,, ψ (,,u (,;v (, da(, H(,,x (,,x (,,x (,, ψ (,,u (,;v (, da(, πε uh(,,x (,,x (,,x (,, ψ (,,u (,;u u (, o( ε --- (5.0 ε ε Theefoe J(u (.,.,vε(.,. u πε H(,,x (,,x (,,x (,, ψ (,,u (,;u u (, o( ε --- (5. We ue he opmaly condon J(u (.,.,vε(.,. 0, we dvde (5. by we hen we le ε 0, and we oban πε, and uh(,,x (,,x (,,x (,, ψ (,,u (,;u u (, (5. I plan ha (5. anamoun o (5.. /// Rema 5.. The ame exemum pncple hold fo he moe geneal poblem decbed n ema 4.. ///

34 34 6. Applcaon o an nvee poblem n he modellng of unam. The echnque of econ 3, 4, and 5 can be appled o a vaey of phycal model ha nvolve lnea o nonlnea econd-ode hypebolc equaon n wo vaable. Afe changng o chaacec coodnae, we end up wh an equaon o yem of equaon n Goua-Daboux fom. Unde cean condon, uch poblem can fall no he famewo of h pape. A an example, we peen a poblem ha ae n he mahemacal modellng of unam wave, nduced by emc moon of he boom of he ea, and alo ang no accoun he eah' oaon. Th only an example, ou of many poble example, of poblem o whch he mehod of he peen pape can be appled. An nvee poblem n he modellng of unam wave ha been eaed by opmal conol mehod n [VST]. Hee, we ue a model ha ncopoae addonal feaue and paally baed on he model ued n [VS]. Ou mehod of oluon alo ele on opmal conol mehod, bu ung a dffeen pocedue han ha of [VST]. Ou pocedue baed on educng he poblem o a Goua-Daboux equaon ove a nonecangula doman, and applyng he eul of he peen pape. We ue he followng model: v(, y(, γ(, ωw(, g w(, ωw(, 0 ; y(, v(, c h( 0 ; --- (6. Hee, ω he angula velocy of he ban aound a vecal ax z; he equaon ae wen n cylndcal coodnae (, ϑ, z unde he aumpon of adal ymmey; ζη( he equaon of he boom of he ea; v(,, w(, ae, epecvely, he adal and angenal componen of he ma velocy of flud pacle; y(, he elevaon of he uface of he wae fom elave equlbum poon; zγ(, decbe he vecal moon of he boom of he ea; c a popoonaly conan; g he acceleaon of gavy. By ang of he f equaon n (6., we oban

35 35 v(, w(, y(, ω g (6. In vew of he econd equaon n (6., we can ewe (6. a v(, y(, 4ω v(, g (6.3 Dffeenaon of he hd equaon n (6. wh epec o gve y(, c ( h(v(, γ(, --- (6.4 Elmnaon of he em y(, beween (6.3 and (6.4 yeld v(, 4ω v(, gc ( h(v(, γ(, g --- (6.5 We e γ(, u(, : --- (6.6 We denoe by, ' he chaacec coodnae fo (6.5. Thee ae found a 0 β( ; ' 0' β( ; β( : d gc gc 0 h( --- (6.7

36 36 (Naually, we aume ha he ogn of coodnae choen o ha h ( > 0 fo all value of. The obevaon doman G aen a G : {(,:, } --- (6.8 whee,,, afy he condon β ( β( gc ( --- (6.9 Then, fo a uable choce of coodnae, by 0, 0', he doman G decbed, n chaacec G : {(,': A ' A, A ' A} --- (6.0 The value of A, 0, 0' ae deemned a oluon of he yem A 0 0' 0 0' (0 0' (0 0' β( gc gc β( --- (6. Clealy, he f 3 ou of he 4 equaon n (6. uffce fo he deemnaon of A, 0, 0' ; he fouh equaon conen wh he ohe 3 becaue of he condon (6.9. The paal dffeenal equaon fo v become

37 37 v(,' v(,' v(,' gc h'( gch''( gc ω v(,' u(,' ' ' h( (6. In (6., expeed a a funcon of he chaacec coodnae a gc (,' β ( ' 0 0' --- (6.3 whee dβ( d β he nvee funcon of β, and well defned nce 0. h( > We examne he poblem of emang u fom obevaon v~ (, ' of he funcon v ove he doman G. We fomulae h poblem a a poblem of opmal conol, and we ee o mnmze a funconal J gven by J G v~ (,' (,', v(,',u(,'dd' --- (6.4 o example, we may ae (,', v(,' v~ (,',u(,' v(,' v~ (,' λ u(,' --- (6.5 whee λ a egulazaon paamee. We noe ha h poblem dffeen fom he elaed poblem n [VST]. Hee, we ue only obevaon of v, bu no obevaon of y, whle y wa equed n [VST]; alo, γ(, γ hee we emae, wheea he emaed quany n [VST] wa. The doman G made up of 4 pa G, 4, whee each G he pa of G ha le n he -h quadan of he ' - plane. Each of hoe 4 pa afe he condon ued n

38 38 he above econ of he peen pape, and neceay opmaly condon have he fom of he Hamlonan equaon, he exemum pncple of econ 5, and he de condon of econ 4 on each pa ( γ of he bounday of G n each of he 4 quadan. The numecal ealzaon of he opmaly condon of econ 3-5, fo h pecfc poblem of unam modellng, beyond he cope of he peen pape.

39 39 Appendx A: negaon and dffeenaon fomulae. In h econ, we peen a numbe of eul ha ae needed fo he manpulaon of negal and devave ang n he devaon of he Hamlonan equaon. We hall ue he noaon and emnology of econ of h pape. In addon, we defne E(, : {( G: (, W( } --- (A. Conequenly, fo funcon ϕ (,, ha ae negable ove G G, we have, by ubn' heoem, G W(, ϕ(,, da( da(, G E(, ϕ( τ,, da( da(, --- (A. o evey pon (, n G, we denoe by A, B he pon of neecon of (γ wh he agh lne hough (, paallel o he -ax and o he -ax, epecvely; hu A ( ϑ(,, B (, ϑ( --- (A.3 Theefoe, ( σ, γ(, (, W( --- (A.4 and ubn' heoem fo funcon ϕ (,, ha ae negable ove ( γ G ae he fom ( γ W(, ϕ(,, da( dµ (, G γ(, ϕ( τ,, dµ ( da(, --- (A.5

40 40 o he ae of noaonal conency, he pon (, wll alo be denoed by. When he funcon ϕ aboluely connuou (n he ene of abolue connuy fo funcon of wo vaable, we have, fo almo all (, n G, B ϕ(,, da( E(, ϕ(,,, dτ ϕ(,, da( E(, --- (A.6 A ϕ(,, da( E(, ϕ(,, dσ ϕ (,, da( E(, --- (A.7 A ϕ(,, da( ϕ(,,, E(, ϕ (,, da( E(, ϕ(,, dσ B ϕ (,,, dτ --- (A.8 ϕ(,,b (,,, (,,, d (, ϕ σ τ ϕ σ τ µ σ τ dµ ( n(b γ(, γ(, --- (A.9 ϕ(,,a (,,, (,,, d (, ϕ σ τ ϕ σ τ µ σ τ dµ ( n(a γ(, γ(, --- (A.0

41 4 ϕ (,,A (,,B (,,, d (, ϕ ϕ σ τ µ σ τ n(a n(b γ(, ϕ(,, dµ ( γ(, --- (A. We noe ha, fo (, GR, he quane n (A, n(b ae boh nonzeo, becaue of he geomec meanng of he pon A and B. Nex, we pove one fomula ou of he goup (A.6, (A.7, and (A.8. A a epeenave cae, we pove (A.6, and he poof of he emanng fomulae wll be mla. o mplcy of he expoon, and whou lo of genealy, we peen he poof fo one of he poble cae, namely fo he cae n whch he ac A B ( γ ha he epeenaon τ ϑ ( σ. We have (A ϕ(,, da( σ E(, ϑ( σ τ ϕ(,, dτdσ --- (A. and conequenly ϑ( ϕ(,, da( τ E(, B ϕ(,,, dτ ϕ(,,, dτ ϕ(,, da( E(, (A σ ϑ( σ τ ϕ (,, dτdσ --- (A.3 Smlaly we pove fomula (A.9 ou of he goup of fomulae (A.9, (A.0, (A., he poof of he emanng fomulae beng mla. When he ac γ (, can be epeened boh n he fom σ ϑ ( and n he fom τ ϑ ( σ, he poof become omewha mplfed, and we peen ha mple poof f. We have

42 4 γ(, ϕ(,, dµ ( ϑ( τ ϕ(,, ϑ (, ( ϑ '( dτ --- (A.4 and conequenly ϕ(,, dµ ( ϑ' ( ϕ(,,, ϑ( ( ϑ' ( γ(, τϑ ( γ(, ϕ (,, dµ ( ( ϑ '( ϕ(,, B n (B ϑ '( ϕ(,,, ϑ ( ϕ(,,, ϑ γ(, ' ( ϑ ( γ(, ϕ (,, dµ ( γ(, ϕ (,, dµ ( ϕ (,, dµ ( --- (A.5 In he geneal cae, we denoe by µ, µ he value of he paamee µ ha coepond o he pon A, B, epecvely. Then µ ϕ(,, dµ ( ϕ(,, L ( µ, L ( µ dµ µ γ(, --- (A.6 and, conequenly, µ ϕ σ τ µ σ τ ϕ µ µ (,,, d (, (,, L (, L ( γ(, µ µ ϕ(,, L( µ, L( µ dµ --- (A.7

43 43 I follow fom he geomec defnon of A and B ha, nea B, he cuve (γ ha he epeenaon τ ϑ ( σ. When ep conan and change o δ, he coepondng pon (, τ on (γ, wh τ ϑ (, change o ( δ, τ δ whee δτ ϑ' ( δ o( δ, and µ change o µ δµ whee δµ ( ϑ' ( δ o( δ. Theefoe, µ ( ϑ' ( n(b --- (A.8 and hen (A.9 follow fom (A.7 and (A.8.

44 44 Appendx B: olvably of cean negal equaon. We examne he yem of negal equaon of he fom (4., whch we ewe hee fo he eade convenence: ψ( ψ( (, ψ0,(, g( τ, ψ(,, da( τ σ E(, --- (B. ψ The condon on E(, and he doman D ove whch a oluon of (B. ough ae a aed n econ 4. We noe ha he mehod of poof of h econ can alo be ued fo a moe geneal negal equaon, n whch he funcon g nde he negal alo depend on and,.e. ψ( ψ( he cae g g(,, τ, ψ(,,. Howeve, h exenon no τ σ needed fo he peen pape. The mple exence and unquene eul acheved hough he andad mehod of cad eaon. We aume ha each g connuou wh epec o all agumen, and afe a Lpchz condon g( τ, Ψ,,Q g( τ, Ψ,,Q L [ Ψ Ψ n n Q Q n ],IR,IR,IR --- (B. n The ymbol n denoe he Eucldean nom on IR. Snce no aumpon made,ir on he magnude of L, clea ha any ohe nom can be ued n he Lpchz condon. n n We ee a oluon of (B. n he pace C (D ;IR of IR - valued funcon ha ae connuou n D and have f paal devave n n D whch ae connuou n D. By dffeenang (B. wh epec o and, and ung he dffeenaon fomulae of appendx A, we oban

45 45 ψ(, ψ0,(, ψ(, ψ0,(, B A ψ(, ψ(, g(, τ, ψ(,,, dτ ; τ ψ( ψ( g(, ψ(,, dσ σ --- (B.3 We e ψ (, (, : (, ψ (, Q (, : ( --- (B.4 We eplace (B. by he yem ψ (, ψ Q ( ( 0, (, E(, ψ0,(, (, ψ0,(, (, B g ( τ, ψ(,,(,,q(, da(, ; A g (, τ, ψ(,,(,,q(, dτ ; g (, ψ(,(,q( dσ --- (B.5 I can be hown, by ung he dffeenaon fomulae (B., ha he poblem (B.5 equvalen o he ognal poblem (B.. o each 0, we defne he nom ψ ψ : : ( ψ, (, D (, D (,Q ( ψ,,q max max ( exp( (a b ψ(, : ψ exp( (a b : ψ ( Q ψ Q (, ( ;,IR ; n ; --- (B.6

46 46 (The nom (, Q(,, Q ae defned n he ame way a he nom ψ, ψ. We defne he opeao Ψ,, Q, S by ( Ψ ( ψ,,q ( ( ψ,,q (, : ψ0,(, g( τ, ψ(,,(,,q(, da(, ; E(, ψ0, (, (, : ψ (, B A g(, τ, ψ(,,(,,q(, dτ ; 0, ( Q( ψ,,q (, : g(, ψ(,(,q( dσ ; ( S( ψ,,q (, : (( Ψ( ψ,,q (,,( ( ψ,,q (,,( Q( ψ,,q (, ( Ψ( ψ,,q (, : (( Ψ( ψ,,q (, : n ; ( ( ψ,,q (, : (( ( ψ,,q (, : n ; ( Q( ψ,,q (, : (( Q( ψ,,q (, : n ; ( S( ψ,,q (, : (( Ψ( ψ,,q (,,( ( ψ,,q (,,( Q( ψ,,q (, ; --- (B.7 o wo e of funcon, (,,Q, ( ψ,, Q emae ψ [] ([] ([] [] ([] ([], we have he

47 47 ( ( Q Q L ( b exp( a( exp( ( Q Q L ( b ( exp( a( ( exp( ( d d b (a exp( Q Q L ( b (a exp(, da( b (a exp( Q Q L ( b (a exp(, da( Q Q L ( b (a exp( b (a exp(, da(, (,Q, (,, (,, ( g, (,Q, (,, (,, ( g b (a exp( (,,Q, ( (,,Q, ( b (a exp( [] [] [] [] [] [] [] [] [] [] [] [] b a [] [] [] [] [] [] E(, [] [] [] [] [] [] [] [] [] [] [] [] E(, [] [] [] [] [] [] E(, [] [] [] [] [] [] ψ ψ ψ ψ σ τ τ σ ψ ψ τ σ τ σ ψ ψ τ σ ψ ψ τ σ τ σ τ σ τ σ τ σ ψ τ σ τ σ τ σ τ σ ψ τ σ ψ ψ Ψ Ψ --- (B.8

48 48 exp( (a b B exp( (a b g(, τ, ψ[] (,,[] (,,Q[] (, dτ B exp( (a b [] [] Q[] Q[] dτ b exp( (b dτdσ ( exp( ( a( exp( ( b ( ( ψ,,q (, ( ( ψ,,q g(, τ, ψ[] (,,[] (,,Q[] (, exp( (b L( ψ[] ψ[] exp( (a b L( ψ[] ψ[] [] [] Q[] Q[] L( ψ[] ψ[] [] [] Q[] Q[] [] [] [] (, B exp( (b dτ exp( (a b L( ψ[] ψ[] [] [] Q[] Q[] ( exp( a( exp( b L( ψ[] ψ[] [] [] Q[] Q[] --- (B.9 [] [] [] wh a mla emae fo he opeao Q. ( exp( a( exp( b Snce lm 0, follow ha, fo uffcenly lage, he opeao S a conacon elave o he nom ( ψ,,q, and conequenly ha a fxed pon whch obaned a he unfom lm of he cad eaon coepondng o he yem (B.5. Thu, f we ae an abay ple ( ψ [ 0], [0], Q[0] n n ( C (D ;IR 3, and defne a equence of eaon nducvely by (,,Q ( ψ,, ψ [ n ] [n ] [n ] S [n] [n] Q[n] --- (B.0 hen he followng lm ex, n he nom n convegence n ( (D ;IR 3 C : ( ψ,,q, hu n he opology of unfom

49 49 lm n ( ψ,,q ( ψ,,q [n] [n] [n] [ ] [ ] [ ] --- (B. and he funcon ψ [ ] he unque oluon of (B., wheea ψ[ ] ψ[ ] [ ], Q[ ].

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