Global Conference on Power Control and Optimization, Gold Coast, Australia, 2-4, February 2010

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1 Global Coferece o Power Cotrol ad Otiizatio, Gold Coast, Astralia, -4, Febrary 00 ASYMPTOTICAL STABILITY OF TRAJECTORIES IN OPTIMAL CONTROL PROBLEMS WITH TIME DELAY Msa A Maadov ad Aatoli F Ivaov Uiversity of Ballarat, Victoria 3353, Astralia, Pesylvaia State Uiversity, Leha, PA 867, USA, aadov@ballarateda afi@sed Abstract A otial cotrol roble with tie delay is cosidered i this aer The asytotic stability of otial trajectories i ters of a terial tye fctioal is ivestigated Uder soe coditios it is roved that there is a iqe statioary oit attractig all otial trajectories ot deedig o iitial fctios Keywords: stability, tie delay, otial cotrol, differetial eqatio Itrodctio This aer deals with the qalitative aalysis of otial trajectories for otial cotrol robles described by the followig differetial delay eqatio: xt &) = t ) F xt )) Gxt )), ae t 0, ) x s) = φ s), s [,0], [0,] ; ) where F ad G are cotios real-valed fctios, is a easrable cotrol fctio with vales i the iterval [ 0,], φ is a iitial fctio This kid of eqatios has attracted a sigificat iterest i recet years de to their freqet aearace i a wide rage of alicatios They serve as atheatical odels describig varios real life heoea i atheatical biology, olatio dyaics ad hysiology, electrical circits ad laser otics, ecooics, life scieces ad others See [, 4, 7, 9, 0,, 3, 5] for a artial list of alicatios ad frther details May ecooic odels lead to differetial delay eqatios of the for ) We refer the reader to a artial list of ecooic alicatios give i [3, 6, 8,, 6, 9] I these alicatios, x stads for the caital ad τ > 0 is the legth of the rodctio iveste cycle The cooet F t )) describes a geeral coodity beig rodced at tie t ad G ) stads for the "aortizatio" of the caital After each cycle of rodctio a certai art of the coodity caital) is sed for the ivestet while the reaiig art is cosed We shall asse that, at ay tie t 0, the art F t )) is assiged for the rodctio roses iveste while the art C = [ ] F t )) 3) is cosed The otiality is defied by the followig fctioal: J =liif C ax 4) This fctioal ais to axiize the lower level of costio whe t goes to ifiity It ca be cosidered as a aaloge of the terial fctioal defied for a ifiite tie horizo We refer to [7] for ore iforatio abot the reslts o stability of otial trajectories i ters of siilar fctioals for systes withot tie-delay There is a sigificat body of theoretical research o differetial delay eqatio ) i the ast 0-30 years They address varios asects of the dyaics i sch eqatios icldig colicated behavior ad chaos aog others However, ost of it deals with the case of liear fctiog ; that is, whe G = bx, b > 0 Paers [7, 4] rereset a artial list of related refereces This idicates that, eve for a costat cotrol fctio, de to the delay factor τ > 0, trajectories of eqatio ) ay have qite colicated strctre, deedig o iitial fctios However, the trajectories that rovide the axial vale to the fctioal 4) ight be well strctred To ivestigate the behavior of otial trajectories i roble )-4) is the ai rose of this aer Uder soe coditios it is established that, ot deedig o iitial fctioφ, all otial trajectories of roble )-4) coverge to a iqe statioary oit as tie goes to ifiity Sch a behavior of otial trajectories is called the Trike Proerty To the best of or kowledge

2 Global Coferece o Power Cotrol ad Otiizatio, Gold Coast, Astralia, -4, Febrary 00 this roerty has ot bee cosidered i the literatre for cotios tie systes with delay We refer to [6, 7, 8, 0] ad refereces therei for ore iforatio abot the Trike Theory for cotios tie systes Preliiaries We asse that fctios F,G : R R are defied ad cotios o the ositive seiaxis R ={ x : x > 0}, G is iforly Lischitz cotios ad F 0) 0, G0) = 0 Iitial fctio φ is assed to be a cotios fctio with vales i R : φ C[,0], R ) The followig are the ai asstios abot the fctios F ad G Asstio A ) F ad G are strictly icreasig i R ; ) For each 0,] there exists a iqe oit x > 0 satisfyig the followig coditios: F x ) = G x ) ad [ F ][ x x] > 0 for all x > 0, x x These asstios are jstified by ecooic iterretatios of the ivolved oliearities [3, 9] Note that x < xv if < v The oit x that corresods to = will be deoted by M Clearly M > 0 ad F M ) = G M ) ad F < G 5) for all x > M It is also qite atral to defie x = 0 if = 0 Iterval [ 0, M ] will be refereed to as a set of statioary oits Itrodce the otatios x * = arg ax{ F : x 0} ; 6) c* = F 7) x * will be called a otial statioary oit Asstio A ilies c * > 0 The followig asstio will also be sed Asstio A Poit x * is iqe Fro this asstio it follows that F > G 0 ad F < c * for all x x * 8) Deote * = c * / F 9) The, fro 7) ad 9) we have * F = G Therefore, Asstio A leads to * [0, ) ad x * = x * Note that this relatio also ilies M > x * as it ca be see fro 5) Defiitio Give a iitial fctio φ C[,0], R ) ad a cotrol =, a absoltely cotios fctio x = is called a trajectory of syste ), ), if eqatio ) is satisfied for alost allt 0 x, will be called a rocess We asse that, for every iitial fctio φ C[,0], R ) ad cotrol =, t 0, there exists a iqe trajectory that will be deoted by x =, t 0 We do ot address i detail the qestio of global existece of trajectories of eqatio ) We oly ote that the reslts are well-kow ad readily available i the literatre see eg [, 5] ad frther refereces therei) Oe of sch coditios of global existece ca be the asstio that G is iforly Lischitz cotios Give rocess x, fctioal 4) defies the vale ~ J x, φ ) = li if C ~ where C = [ ] F t ) Defiitio Process x *, *, φ *) is called otial i the roble )-4) if the ieqality J x, J x*, *, φ*) holds for all rocesses x, I this case, x * ad * will be called otial trajectory ad otial cotrol, resectively Deote by J * the axial vale of the fctioal 4) over all the rocesses x, : J* = s J x, x, φ Give trajectory x, we defie the set of liit oits as follows P )) = { y : t, t ) y} k k

3 Global Coferece o Power Cotrol ad Otiizatio, Gold Coast, Astralia, -4, Febrary 00 Clearly, if this set is ot ety, the it is a closed iterval that will be deoted by P )) = [, ] 3 The existece of otial trajectories This sectio resets soe reslts abot the strctre of trajectories of syste ) I articlar, give ay iitial fctio, the existece of otial trajectories is established Theore Asse that A holds Let x = be a trajectory of syste ) The P )) [0, M ] 9) Proof Take ay iitial fctio φ ad cotrol Let x = be the corresodig trajectory Sice φ is a cotios fctio, we have M = ax{ φ : t [,0]} < τ Let M = ax{ M, M} ε where ε >0 is a arbitrary ber ad M is the axial statioary oit defied above First we show that x M, t 0 O the cotrary asse that x t ) = M for soe t > 0 ad x < M for t < t The, sice M > M > x *, there exists t < t sch that x & t ) 0, t ) > M ad x t ) ) t Fro Asstio A it follows 0 x& t) = t) F t )) t)) F t τ )) t)) F t )) t )) 0 < This is a cotradictio Therefore, x M, t 0 ; that is, trajectory is boded above The the set of liit oits P )) is ot ety With P )) = [, ] we eed to show that M O the cotrary asse that > M Clearly, there is a seqece sch that t x& t ) 0, t ) as Sice x is boded, we ca asse that the seqece x ) also coverges to soe oit t x : t ) x otherwise we ca choose a sbseqece of { t }) We have 0 x& t ) = t ) F t )) t )) F t )) t )) By takig the liit as we obtai 0 F ) F ) ) This cotradicts 4) as it is assed > M Therefore, M, which eas 9) is tre Theore is roved Theore Asse that A holds The, for ay trajectory x = of syste ) the followig ieqality holds: J x, φ ) c * Proof Take ay iitial fctio φ ad cotrol Let x = be the corresodig trajectory ad C = [ ] F t )) Fro Theore it follows that P )) = [, ] [0, M ] ) The, there is a seqece t sch that x& t ) 0, t ) ad x t ) x Fro ) ad 3) we have 0 x& t ) = t ) F t )) t )) F t )) C t ) t )) or C t ) F t )) t )) ) Takig liit as ad sig Asstios A we obtai liif C t ) F ) F ) ) c* Therefore, J x, φ ) = liif C liif C t ) c* Theore is roved Now, we cosider costat cotrol fctios [0,] Theore 3 Asse that A holds ad x = is a trajectory corresodig to

4 Global Coferece o Power Cotrol ad Otiizatio, Gold Coast, Astralia, -4, Febrary 00 a costat cotrol [0,] ad a iitial fctio φ The x as t Here x is defied i Asstio A Proof Fro Theore it follows that trajectory x is boded ad P )) = [, ] [0, M ] First we show that x * Clearly, there is a seqece t sch that x& t ) < 0, t ) ad x t ) x Sice F is icreasig, F F ) The, takig liit i x& t ) = F t )) t )) we obtai F ) 0 or F ) ) 0 Ths, fro Asstio A it follows x I a siilar way, we ca show that x Therefore, P )) = { x } which eas that x Theore is roved Corollary If Asstio A holds, the give ay iitial fctio φ there is a rocess ~ x, ~, sch that J ~ x, ~, φ ) = c * Proof Take ay iitial fctio φ Cosider cotrol ~ * ad let ~ x t ) be the corresodig trajectory ~ Fro Theore 3 it follows that x * as t, where x * = x* The, fro 3),4) ad 9) it follows J ~ x, ~, φ ) = [ *] F = c * Fro Theore ad Corollary, we obtai the followig theore abot the existece of otial trajectories Theore 4 Asse that A holds For ay give iitial fctio φ there exists a otial rocess ~ x, ~, i the roble )-4) with the fctioal vale J ~ x, ~, φ ) = J* = c * Sarizig the reslts i Theores -4 it follows that, if we take a costat cotrol fctio *, the ot deedig o iitial fctio φ, corresodig trajectory x is otial i the roble )-4) ad x * as t I the followig we ai to stdy the strctre of all otial trajectories Or ai target is to rove the Trike Proerty for the roble )-4); that is, we ai to show that for ay otial rocess ~ x, ~, the covergece ~ x * as t is satisfied 4 Asytotical stability of otial trajectories The followig theore describes the strctre of otial trajectories i roble )-4) A stroger reslt below will be roved for a secial class of fctios G Theore 5 Asse that A ad A hold The, for ay otial trajectory x = the roble )-4), we have P x )) = [, x *] 3) Proof Cosider the otial trajectory x = corresodig to a iitial fctio φ ad a cotrol By Theore 4 J x, φ ) = li if C = c * 4) Let P )) = [, ] We eed to show that = x * As is a liit oit of x, there is a seqece t sch that t x& ) 0, t ), t ) x as The, siilar to ), the followig ieqality holds C t ) F t )) t )) Sice F F ), takig also 4) ito accot we obtai c* F ) ) Therefore, fro Asstio A it follows that = x * Theore is roved The followig theore resets a sefl roerty of otial trajectories related to the dratio of cycles whe otial trajectory ay stay otside the otial statioary oit x * It shows that sch cycles ca ot last loger tha the tie delay τ

5 Global Coferece o Power Cotrol ad Otiizatio, Gold Coast, Astralia, -4, Febrary 00 Theore 6 Asse that A ad A hold ad x = is a otial trajectory If t ) x * for soe seqece t, the the covergece t ) x * is also tre x Proof Let t ) x * for soe seqece t We will show that for ay sbseqece tk ) of t sch that the seqece x t k ) ) is coverget, the liit x t ) * is tre k ) x Take ay sch a sbeqece t k ) For the sake of silicity, we deote it by t Let t ) x x We eed to show x = x * Fro Theore 5 we kow that x x * 5) Cosider a seqece of ositive bers δ 0 Take ay We will show that for ay ε δ there is a sfficietly large ber = ) sch that s{& s) : s [ t ε, t ε ]} > δ 6) By the cotrary, asse that this is ot tre; that is, there is a ositive ber ε δ ad a seqece t k ) sch that x& s) δ for all s [ t k ) ε, t k ) ε ], k =,, The x t ) t ) = k ) k ) ε t k ) = x& s) ds δ ε t k ) ε As t k ) ) x *, we have li if t k ) ε ) x * δ ε > x * k ) This eas that the set of liit oits P )) cotais a oit grater tha x * This cotradicts Theore 5 Therefore 6) is tre The, there are seqeces ε 0, t ad a seqece of oits s ) [ t ) ε, t ) ε ], =,, sch that x& s ) δ 7) I this case, sice x satisfies the Lischitz coditio ad t ) ) x *, t ) τ ) x, the followig is also satisfied: s ) x*, s ) x as 8) Fro 7) it follows that F s τ )) C s) s)) or F s )) s )) δ C s ) Takig ito accot 8) ad the otiality of x we obtai F liif C s) c * O the other had c* = F ad therefore F F Sice F is icreasig, fro 5) we coclde the reqired eqality x = x * Theore is roved The followig theore establishes the Trike Proerty for the roble )-4) i the case whe fctio G is liear Theore 7 Asse that A ad A hold ad G = bx, b > 0 The, all otial trajectories x = of the roble )-4) coverge to x * : x * as t Proof Let x = be a otial trajectory ad C = [ ] F t )) By Theore 4 we kow that J x, φ ) = liif C = c * Fro Theore 5 we have P x )) = [, x *] We eed to show = x * By the cotrary, asse that < x * 9) Take ay ositive berδ < x * ) / 4 Give this berδ, we defie a seqece of itervals [ t, ] ad a seqece of oits t s [ t, t ], =,,, sch that δ

6 Global Coferece o Power Cotrol ad Otiizatio, Gold Coast, Astralia, -4, Febrary 00 t ) = t ) = x * δ ; x < x * δ, t t, t ); s ) = δ Fro the choice of δ it is clear that δ < x * δ Deote α t ) = F t )) Clearly, α is a easrable fctio ad α = F t )) C The x satisfies the followig x& = α b, t s 0) We fid the soltio to 0) as follows t = ex b ex bs) α s) ds) s ex bs ) s )] By the defiitio of t it is clear that at the oit t = t the followig eqality holds t ex bt ) ex bs) α s) ds) s ex bs ) s )] = x * δ ) Clearly, 0 α for all t > 0 Moreover, give δ, there is a ber T δ sch that the followig two ieqalities hold for all t Tδ ; F t )) F δ / ; ad C c * δ / The first ieqality follows fro P )) = [, x*], ad the secod oe follows fro liif C = c * The, α t ) F c * δ = = bx * δ, t T δ Let N δ sch that t Tδ for all Nδ Note that t > s > t T δ for all Nδ Take ay Nδ ad cosider the followig fctio k = ex b t ex bs) bx * δ ) ds s ex bs ) s )], t [ s, t ] Clearly, k s ) = s) < x * δ Moreover, sice α s ) bx * δ, fro ) we have k t ) x * δ Ths, as k is cotios, there exists a oit t [ s, t ] satisfyig k = x * δ Deote by t the iial soltio to this eqatio; that is, k t ) = x * δ, k < x * δ, t [ s, t ); ) ad t t 3) We fid t i the for t = s lξ δ ) ; 4) b where b x * ) δ b) ξ δ ) = δ b) Clearly, ξ δ ) as δ 0 The, it is clear that, the ber δ ca be chose so sall that lξ δ ) > τ b I this case, fro 3) ad 4) we obtai t s τ for all Nδ 5) Now, we show that 5) leads to a cotradictio First we ote that t ) = x * δ ad x < x * δ if t [ s, t ) The, we ca fid a seqece v < t, t v 0 sch that x& v ) 0 for all Clearly Nδ x v ) x * δ Moreover, there is a sbseqece, that for the sake of silicity will be deoted agai by v, x& v ) ξ 0, ad v τ ) x x * δ The, x& v ) = v ) F v )) b v ) or C v ) = v ) F v )) b v & ) Therefore, we have liif C liif C v ) = ξ F b x * δ ) F x * δ ) b x * δ )

7 Global Coferece o Power Cotrol ad Otiizatio, Gold Coast, Astralia, -4, Febrary 00 Fro 8) it follows that F x * δ ) b x * δ ) < F bx* = c * The we obtai liif C < c * This cotradicts Theore 4 de to the otiality of x Ths, we have show that 5) ad therefore 9) leads to a cotradictio Theore is roved 5 Coclsios Qalitative aalysis of otial trajectories for otial cotrol robles described by differetial delay eqatios is cosidered i this aer Otiality is cosidered i ters of a terial tye fctioal defied for a ifiite tie horizo Sch differetial delay eqatios aear i a wide rage of alicatios i atheatical biology, olatio dyaics ad hysiology, electrical circits ad laser otics, ecooics, life scieces ad others The dyaics of trajectories of sch systes eve withot a cotrollable araeter ay have a qite colicated behavior I this aer, the asytotic behavior of otial trajectories is stdied It is show that otial trajectories have a well defied strctre ideedet of iitial coditios I articlar, it is roved that der soe coditios, all otial trajectories are attracted by a iqe statioary oit 6 Refereces [] M Adiy, F Craste ad S Ra A atheatical stdy of the heatooiesis rocess with alicatios to chroic yelogeos lekeia SIAM J Al Math, 65 4), ) [] ODieka, S va Gils, SVerdy Lel, ad HOWalther, Delay Eqatios: Colex, Fctioal ad Noliear Aalysis Sriger- Verlag, New York, 995 [3] G Gadolfo, Ecooic Dyaics Sriger-Verlag, 996, 60 [4] KP Hadeler ad J Toik, Periodic soltios of differece differetial eqatios Arch Rat Meck Aal ), [5] JKHale ad SMVerdy Lel, Itrodctio to Fctioal Differetial Eqatios Sriger Alied Matheatical Scieces, vol 99, 993 [6] AF Ivaov ad MA Maadov, Global dyaics of soltios i a class of differetial eqatios with delay Research Reort 009/0, Uiversity of Ballarat, Astralia [7] AF Ivaov, AN Sharkovsky, Oscillatios i siglarly ertrbed delay eqatios I: Dyaics Reorted, New Series 99), 64-4 Editors: CKRT Joes, U Kirchghaber & H-O Walther) [8] AF Ivaov ad AV Swishchk, Otial cotrol of stochastic differetial delay eqatios with alicatio i ecooics Iteratioal Joral of Qalitative Theory of Differetial Eqatios ad Alicatios 008), 0-3 [9] Y Kag, Delay Differetial Eqatios with Alicatios i Polatio Dyaics Acadeic Press Ic 003, 398 Series: Matheatics i Sciece ad Egieerig, Vol 9 [0] RN Mada ed), Chas s Circits: A Paradig for Chaos World Scietific Series o Noliear Sciece, Series B, vol 993), 043 [] MC Mackey, Coodity rice flctatios: rice deedet delays ad oliearities as exlaatory factors J Ecooic Theory ), [] MC Mackey, Matheatical odels of heatooietic cell relicatio ad cotrol I: "The Art of Matheatical Modelig: Case Stdies i Ecology, Ohysiology, ad Bioflids", HG Other, FR Adler, MA Lewis, ad JP Dalto, eds), Pretice-Hall, Uer Saddle River, NJ, 997, [3] MC Mackey, C O L Pjo-Mejoet, ad J W Periodic oscillatios i chroic yelogeos lekeia SIAM J Math Aal ), o, [4] J Mallet-Paret ad RD Nssba Global cotiatio ad asytotic behavior for eriodic soltios of a differetial-delay eqatio A Mat Pra Al ), 33-8 [5] JMallet-Paret ad RDNssba A differetial delay eqatio arisig i otics ad hysiology SIAM J Math Aal 0 989), 49 9 [6] MA Maedov Maadov), Trike theores i cotios systes with itegral fctioals Dokl Akad Nak 33 99) o 5, ; Eglish trasl I: Rssia Acad Sci Dokl Math 45 99), No, [7] MA Maadov, Trike theory: Stability of otial trajectories Ecycloedia of Otiizatio, d ed 009, XXXIV, 466, Flodas, CA; Pardalos, PM Eds), ISBN: [8] MA Maedov Maadov), Asytotical Stability of Otial Paths i Nocovex Probles C Pearce ad E Ht Eds) Otiizatio: Strctre ad Alicatios, Sriger, series Otiizatio ad Its Alicatios, Vol 3, 009, 95-34

8 Global Coferece o Power Cotrol ad Otiizatio, Gold Coast, Astralia, -4, Febrary 00 [9] FP Rasey, A atheatical theory of savigs Ecooic J 38 98), [0] A Zaslavski, Trike Proerties i the Calcls of Variatios ad Otial Cotrol Sriger: Nocovex Otiizatio ad its Alicatios, 80), 005, 396