Hndaw Mathematca Probems n Engneerng Voume 7 Artce ID 85368 8 pages https://do.org/.55/7/85368 Research Artce Traectory Optmzaton Based on Mut-Interva Mesh Refnement Method Nngbo L Humn Le Le Shao Tao Lu and Bn Wang Ar and Msse Defense Coege Ar Force Engneerng Unversty X an 75 Chna Schoo of Eectronc and Informaton Engneerng X an Jao Tong Unversty X an 749 Chna Correspondence shoud be addressed to Nngbo L; nb_as@63.com Receved February 7; Accepted 8 Apr 7; Pubshed 3 August 7 Academc Edtor: Wam MacKuns Copyrght 7 Nngbo L et a. Ths s an open access artce dstrbuted under the Creatve Commons Attrbuton Lcense whch permts unrestrcted use dstrbuton and reproducton n any medum provded the orgna wor s propery cted. In order to mprove the optmzaton accuracy and convergence rate for traectory optmzaton of the ar-to-ar msse a mutnterva mesh refnement Radau pseudospectra method was ntroduced. Ths method made the mesh endponts converge to the practca nonsmooth ponts and decreased the overa coocaton ponts to mprove convergence rate and computatona effcency. The traectory was dvded nto four phases accordng to the worng tme of engne and handover of mdcourse and termna gudance and then the optmzaton mode was but. The mut-nterva mesh refnement Radau pseudospectra method wth dfferent coocaton ponts n each mesh nterva was used to sove the traectory optmzaton mode. Moreover ths method wascomparedwthtradtonah method. Smuaton resuts show that ths method can decrease the dmensonaty of nonnear programmng (NLP) probem and therefore mprove the effcency of pseudospectra methods for sovng traectory optmzaton probems.. Introducton Beyond vsua range (BVR) combat s beng the man form of future ar combat [ 3]. Deveopng the msse wth BVRcombatcapabtybymprovngtheexstngmedum range msse has great practca sgnfcance. Consderng the exstng propuson capabty we can effectvey ncrease the range by traectory optmzaton. Traectory optmzaton has been a hot research topc n the fed of arcraft gudance and contro. Many schoars have studedtnvewofthedfferentbacground. Consderng the ong range of the fght probems such as structura strength norma wor of the engne and contro stabtygversetostrctrequrementsonthedynamc pressure overoad contro and other characterstcs of the process. Therefore traectory optmzaton of the ar-to-ar msse can be regarded as a compex nonnear optmzaton probem. Wth the deveopment of computer technoogy the drect method based on parameterzaton has deveoped remaraby[4 6].Thsmethodconvertstheoptmacontro probem of contnuous tme nto NLP probem and then soves t by NLP sover such as the sequenta quadratc programmng (SQP) [7] whch s recognzed as the best sovng method for NLP probem. Durng ast decades pseudospectra methods have been a research hotspot n the doman of traectory optmzaton [8 9]. In ths method the contro and state are frsty dscretzed at certan coocaton ponts n the tme nterva of nterest whch are then approxmated by poynoma nterpoaton. Fnay the optma contro probem s transformed nto NLP probem whch can be soved by NLP sover. Dfferent pseudospectra methods seect dfferent coocaton ponts and base functons of nterpoaton. In Huntngton et a. [] three nds of commony used pseudospectra methods Legendre Radau and Gauss were compared. The computatona effcency of the three methods s smar whe the Gauss and Radau pseudospectra methods (RPM) are superor to the Legendre pseudospectra method n the aspects of convergencerateandestmatonaccuracyofthecostate.in addton the RPM produces the most accurate propagated souton among three methods and the dscretzaton of RPM
Mathematca Probems n Engneerng spans the entre nterva wthout overap. So t s desrabe to mpement for mut-nterva optmzaton probems. And n ths paper the RPM method s seected for ar-to-ar msse traectory optmzaton. Many researchers have studed pseudospectra methods for traectory optmzaton n dfferent bacground such as Mars atmospherc entry traectory optmzaton [] onboard traectory generaton [] and UAV path pannng [3]. Though some achevements have been made there are st some ntractabe probems to sove. For exampe to mprove the accuracy of the souton we can add the number of the coocaton ponts. However wth ncrease of the ponts the NLP probem becomes much more compex and the computaton effcency decreases. Moreover the hgh effcency and accuracy of pseudospectra method depend on thesmoothnessoftheprobem.ifthecontroandstatehave dscontnuous ponts the convergence rate and accuracy of the souton w decrease. In auson to above probems some mproved optmzaton agorthms were ntroduced [4 5].ZhaoandTsotras[4]proposedasmpebuteffectve method whch empoys a probabty densty functon to sove the traectory optmzaton probem by generatng a fxedorder mesh. Gong et a. [5] desgned a spectra agorthm for pseudospectra method whch uses the pseudospectra dfferentaton matrx to ocate swtches ns corners and other dscontnutes. In ths method the tme nterva was dvded nto smaer subntervas and the dstrbuton of the nodes was controed by nottng method. Ths paper advances pseudospectra methods n computatona optma contro. In auson to the probems such as too many coocaton ponts n mesh optmzaton hgh dmensonaty of NLP probem and the ow computatona effcency a mutnterva mesh refnement method wth unfxed coocaton ponts was ntroduced n ths paper. The mesh needs to be further refned f the reatve error s arger than the gven vaue and f nonsmooth pont exsts n the mesh the number of mesh ntervas w be ncreased; otherwse the number of coocaton ponts w be ncreased.. Ar-to-Ar Msse Traectory Optmzaton Probem The range of ar-to-ar msse s beyond m. Accordng to the worng tme of the engne and handover of mdcourse and termna gudance the msse traectory can be dvded nto four phases as shown n Fgure. The frst phase s from the msse separated from the carrer to the engne startng worng where the msse s n a state of ow speed wth no thrust. The second phase s the worng process ofengnewherethemssesspeedngupandthefght attude veocty overoad and dynamc pressure of msse reach the pea vaue. The thrd secton s from shutdown of the engne to the seeer capturng the target where the veocty and attude of msse change sowy. The ast phase s the termna gudance. It s necessary to adopt mut-nterva traectory optmzaton for the prevous three phases because of the sharp change of thrust and the dscontnuous contro caused by the swtch of the engne. Traectory Phase and phase Phase connected and phase 3 connected Phase Phase Phase 3 Phase 3 and phase 4 connected Phase 4 O A B C D Tme Fgure : Ar-to-ar msse traectory. Target.. Moton Mode. When we desgn the nomna optma traectory the ar-to-ar msse shoud be drecty guded towards the target so the dynamc mode s consdered n the ongtudna pane. The mass pont moton equatons are gven as foows: V = P cos α C xqs gsn θ m θ= P sn α+c yqs mv x=v cos θ y=v sn θ g cos θ V where V θ xandy are veocty fght path ange and ocaton n the nerta coordnate respectvey; m s the msse mass; g s gravty acceeraton; α s ange of attac; P s engne thrust; q s the dynamc pressure; and C x C y denote the drag and ft coeffcents respectvey whch are expressed as a functon of Ma the Mach number and α the ange of attac: C y =C α y (Ma) α C x =C x (Ma) +K(Ma) C y where C x K denote zero-ft drag coeffcent and nduced drag coeffcent respectvey and C α y denotes the parta dervatve of C y wth respect to α. q s the dynamc pressure: () () q =.5ρV. (3) And ρ s the atmosphere densty expressed as ρ=ρ exp ( y y ) (4) where ρ =.5 g/m 3 y = 754.3 m... Constrant Condtons... Boundary Condtons. As the msse s aunched from thecarrerthefoowngntacondtonsneedtobesatsfed: X (t )=[V θ x h ]. (5)
Mathematca Probems n Engneerng 3 The nta pont of the traectory optmzaton s the startng contro pont after aunchng from the carrer and the end pont s the nta pont of termna gudance. In order to successfuy capture the target the seeer must satsfy certan constrantswhchcanbeexpressedas V (t f ) V fmn θ fmn θ(t f ) θ fmax r(t f ) r fmax where V fmn denotes the mnmum veocty that termna gudance requres to ht the target. θ fmn θ fmax denote the mnmum and maxmum of termna fght path ange. r(t f ) denotes the dstance between msse and target at the begnnng of termna gudance. r fmax denotes detecton range of the seeer.... Path Constrants Dynamc pressure constrant: q mn q q max where q mn q max denote aowabe mnmum and maxmum dynamc pressure. In order to mantan the basc aerodynamc fght the dynamc pressure cannot be too sma or too arge. Overoad constrant: n n max wheren max denotes the maxmum aowabe overoad. Due to the stabty mtaton of the msse body structure t s necessary to mt overoad durng the fght. Attude constrant: h h max whereh max denotes the maxmum aowabe attude. To ensure the norma wor of engne the attude cannot be too hgh...3. Connecton Pont Constrants. In order to ensure smooth transton between two phases the state and contro shoudbethesameateachphaseconnectonpont: t (+) =t f () (6) 3. Mut-Interva Radau Pseudospectra Method For concson and wthout oss of generaty et us begn wth dscussng the standard form of nonnear optma contro probem. In the mesh refnement Radau pseudospectra method [6] τ [ +] s composed of mesh ntervas S = [T T ] =...Kwhere K = = T <T < <T =+ S = [ +]. X () (τ) and U () (τ) denote state and contro n nterva S. They can be dscretzed at Legendre-Gauss-Radau (LGR) ponts expressed as foows: X () (τ) = U () (τ) = () (τ) = = X () N U () = == (τ) () (τ) (τ) () (τ) τ τ () τ () τ () (9) () where () (τ) =... denotes the bass of Lagrange poynomas. (τ ()...τ() N ) are LGR coocaton ponts of mesh nterva [T T ) and τ () = T s a noncoocated node pont. We can get state equatons as foows: dx () (τ) dτ = = X () d () (τ) dτ U (+) = U f () X (+) = X f () (7) D () X() = () where denotes the th fght phase and subscrpts and fdenote the nta and termna pont respectvey...4. Contro Constrants. α α max whereα max denotes the maxmum ange of attac. Due to the stabty mtaton of thebodystructureandatttudecontrosystemtheangeof attac shoud not be too arge..3. Obectve Functon. The obectve s to ncrease the range wth predetermned thrust profe. The performance ndex canbeexpressedas mn J= x(t f ). (8) where D () = t f t = d() (τ () dτ f(x () ) (τ) U () (τ) t(τ () t t f )) (=...N =...). () D () s the LGR dfferenta matrx of dmenson N ()of mesh nterva S. After LGR dscretzaton the nonnear optma contro probem s transformed nto the foowng NLP probem.
4 Mathematca Probems n Engneerng Obectve functon: J=M(X () t X (K) N K + t f) + t f t K N ω () = = L(X () Dscretzed dfferenta equatons: D () X() = =. t f t f(x () Dscretzed path constrants: c mn c(x () Dscretzed boundary condtons: U () t(τ () t t f )). U () t(τ () t t f )) (3) (4) U () t(τ () t t f )) c max. (5) b mn b(x () t X (K) t f) b max. (6) The state s contnuous at ponts (T...T K )sothe condton X () = X(+) must be satsfed. 4. Mesh Refnement Method 4.. Reatve Error Estmaton. It s assumed that there are N LGR coocaton ponts at mesh nterva S =[T T ]. When estmatng whether to add the coocaton ponts we suppose that there are M =LGR coocaton ponts... τ() M )where τ () =τ () =T τ () M + =T.Thestate s approxmated as (X )...X( τ() M )) at LGR coocaton ponts... τ() M ). The bass of Lagrange nterpoaton of contro s expressed as foows: U () (τ) = () (τ) = N U () = N = = (τ) () (τ) τ τ () τ () τ () And X () ) canbeexpressedasfoows: X () )=X () (τ )+ t f t M ( I () = f(x () )U () )t t t f ))). =...M + (7) (8) where I () =...M are ntegra weght matrces of dmenson M M of LGR ponts... τ() M ).The absoute error and reatve error between X ) and X ) are expressed respectvey as foows: E () e () )= )= X () ) X () ) (9) E () ) +max [...N +] [...K] X() (τ () ) =...n x =...M +. () The maxmum reatve error estmaton at mesh nterva S s expressed as foows: e () max = max 4.. Hp-Adaptve Mesh Update [...n x ] [...M +] e() ). () 4... Nonsmooth Pont Locaton. When the mesh nterva needs to be further refned t s necessary to decde whether to add the number of coocaton ponts or the number of mesh ntervas. In ths paper f the mesh nterva s nonsmooth we ncrease the number of mesh ntervas; otherwse we ncrease the number of coocaton ponts. For smpcty n the mesh nterva S P (M) s the oca maxmum of X (M) (τ). SmaryP (M ) s the oca maxmum of X (M ) (τ) τ S ( =...n x =...M ). M denotes the current number of teratons. If R = P(M) P (M ) R () where R denotes the gven rato mesh nterva S s nonsmooth [7]. 4... Mesh Interva Dvson. If e () max >εand R R at mesh nterva S whereε denotes maxmum aowabe error the nonsmooth ponts exst n ths mesh nterva. It s necessary to dvde ths mesh nterva nto S submesh ntervas. And S can be cacuated as foows: S=[og N ( e() )]. (3) ε The upper bound H max on the number of submesh ntervas can be cacuated as foows. If e () ε(such as 6 ) H max can be 5 5 [8] whe f e () ε H max s cose to S=mn (H +H max ). (4) 4..3. Coocaton Ponts Increasng. If e () max > ε and R < R at mesh nterva S themeshntervas s smooth. In order to mantan the reatve error estmaton under ε the error must be reduced e () max /ε tmes as ts current vaue. If the requrements are satsfed by ncreasng the number of
Mathematca Probems n Engneerng 5 Intaze meshes 4 Sove NLP for nta meshes 3 Goba error < 휀 Yes Ext h (m) No Loca error < 휀 Yes Yes Add ponts No R R Sove NLP No Add meshes Fgure : Mut-nterva mesh refnement method. coocaton ponts the coocaton ponts of mesh nterva S can be N (M) at Mth teraton and N (M+) at (M + )th teraton. And N (M+) canbeexpressedasfoows: (e(m) N (M+) =N M ε ) /(q.5) (5) where q s reevant to N (M).BecauseN (M+) s an nteger (9) canbetransformedntothefoowngform: N (M+) = [ [ N M (e(m) ε ) /(q.5) ]. (6) ] 4.3. Agorthm Fow. The cacuaton steps of mut-nterva mesh refnement pseudospectra method are as foows and the fow chart s shown n Fgure. Step. Intaze the mesh and dscretze t by the method proposednthspaper.transformtheoptmacontroprobem ntoanlpprobemwhchcanbesovedbysnoptthen. Step. If e () to Step 3. max <εat a mesh ntervas then termnate or go Step 3. If e () max <εat mesh nterva S gotostep5orgoto Step 4. Step 4. If R R at grd nterva S ncreasethenumber of coocaton ponts or ncrease the number of the mesh ntervas; then go to Step 5. Step 5. Accordng to the resut of Step to Step 4 bud new mesh ntervas and coocaton ponts and sove the current NLP probem; then return to Step. 3 4 5 x (m) Mesh ntazaton Mesh teraton Mesh teraton Mesh teraton 3 Mesh teraton 4 Fgure 3: Traectory curves of teratve process. 5. Smuaton and Anayss The worng tme of engne s 5 s. Smuaton nta constrants are as foows: x =m h =m V = 4 m/s and θ = ; smuaton process constrants: h 3m n g α and Pa q Pa; smuaton termna constrants: V f 4 m/s θ f = 45. The number of coocaton ponts at each mesh nterva n mut-nterva method whch s expressed as ph (N mn N max ) s unfxed. N mn N max denote the mnmum and maxmum aowabe number of coocaton ponts. h N f denotes the h method and the number of coocaton ponts at each mesh nterva s a fxed vaue N f. The smuaton resuts of ths paper are obtaned from Matab smuaton on Lenovo CPU 3.4 GHz Core 7 Inte computer and NLP probem s soved by SNOPT. M denotes the teraton tmes. And the maxmum aowabe error of mesh optmzaton accuracy s ε = 6 R =.. Durng the ntazaton the traectory s dvded nto 3 phases each wth mesh ntervas. There are nta coocaton ponts n each mesh nterva. The traectory curve s shown n Fgure 4. In the smuaton the souton converges after 4 teratons. The tme consumpton of smuaton s 9.7 s. In order to anayze tsconvergencetheocapartoffgure4senargedas shownnfgure5fromwhchwecanseethattheeffectof ntaton s good but the frst teraton error s arge then the mesh converges qucy and the fourth teraton satsfes the accuracy requrement whch means the error s under ε. We can see from Fgure 3 that the attude changes rapdy at the begnnng and the end whch ndcates that the states change rapdy at the begnnng and end of the mesh. The dstrbuton of the mesh coocaton ponts s shown n Fgure 5 from whch we can see that coocaton ponts at the begnnngandtheendareobvousymuchmorethanthat at the mdde secton whch verfes the adaptaton of the method. When the h 4method s adopted the dstrbuton of coocaton ponts s shown n Fgure 6. And when M= the coocaton ponts n h method are smar to those n
6 Mathematca Probems n Engneerng 3 5 4 9.5 h (m) 9 M 3 8.5 3 4 5 6 7 8 x (m) Mesh ntazaton Mesh teraton Mesh teraton Mesh teraton 3 Mesh teraton 4 3 4 5 6 Fgure 6: Coocaton ponts dstrbuton of h 4. Fgure 4: Loca traectory curves of teratve process. Tabe:Comparsonoftwomethods. M 4 3 3 4 5 6 Fgure 5: Coocaton ponts dstrbuton of ph (3 ). N mn N max Tmes/s N M h 34.5 346 6 h 3 3 4.5 5 h 4 4 3. 8 5 h 5 5 4.5 9 5 ph 3 4 5. 83 5 ph 3 5 6.7 77 5 ph 3 6 3.9 75 5 ph 3 8 9.69 75 4 ph 3 4. 77 5 ph 4 5.4 58 5 ph 4 8.6 58 5 ph 4.7 58 5 ph 5 6. 8 5 ph 5 8.7 8 5 ph 5.5 8 5 mut-nterva method. But when M = 3 4thecoocaton ponts of h method are more eveny dstrbuted n each mesh nterva and the tota coocaton ponts are sgnfcanty more than those n mut-nterva method because the mutnterva method has fewer coocaton ponts n the mdde secton wth sma change of state and contro. Then the mut-nterva method and h method are compared n terms of computatona effcency. The runnng tme the tota number of coocaton ponts N and the mesh teraton tmes M are shown n Tabe. Tabe shows that n the aspect of tme consumpton ph (3 8) sthesmaestofthemut-ntervamethodandh 4 s the smaest of h method.thetotanumberofcoocaton ponts of ph (3 8) method (N = 75) sessthanh 4 method (N = 8). In fact as shown n Tabe for h N f and ph (N mn N max )fn f and N mn are the same the coocaton ponts of mut-nterva method are fewer than those of h method whch ndcates that the mut-nterva method s more effcent than h method. In addton the performance ndex of these two methods s consstent. The smuaton resuts of ph (3 ) method are shown n Fgures 7. In Fgure 7 the traectory s smooth and the maxmum attude s 3 m. In order to maxmze the range the traectory s optmzed as hgh attude traectory. In practca appcaton the boost-gde traectory has the characterstcs of ncreasng range. In ths paper the maxmzed range s consdered as the performance ndex and the traectory has the boost-gde process whch s consstent wth the practca stuaton and verfes the effectveness of the method. From Fgures 8 we can now that the change of ange of attac s smooth and dynamc pressure and overoad are satsfyng the path constrants whch ensures the stabty of fght. 6. Concuson In order to ncrease the range of exstng ar-to-ar msse the traectory was dvded nto four phases accordng to the worng tme of the engne and handover of mdcourse and termna gudance and then the optmzaton mode was
Mathematca Probems n Engneerng 7 35 h (m) 5 3 5 5 5 3 4 5 x (m) Fgure 7: Curve of traectory. (m s ).5.5 3 4 5 6 5 Fgure : Curve of veocty. 훼 ( ) 5 q (Pa) 5 3 4 5 6 5 3 4 5 6 Fgure : Curve of dynamc pressure. Fgure 8: Curve of ange of attac. 4 4 3 n (g) 휃 ( ) 4 6 3 4 5 6 Fgure 9: Curve of fght path ange. but. In auson to the traectory optmzaton probems wth parametrc dscontnuty and poor fttng effect of state andcontroatthenonsmoothpontsmut-ntervamesh refnement method was ntroduced n ths paper. Compared wth h method ths method can effectvey decrease the dmensonaty of NLP probems and mprove the convergence rate wth the same precson. Its effectveness was verfed by smuaton. 3 4 5 6 Confcts of Interest Fgure : Curve of overoad. The authors decare no potenta confcts of nterest wth respect to the research authorshp and/or pubcaton of ths artce. Acnowedgments Ths study was cosupported by the Natona Natura Scence Foundaton of Chna (Grant no. 6573374 and no. 65348)
8 Mathematca Probems n Engneerng and Aeronautca Scence Foundaton of Chna (Grant no. 4964 and no. 5966). References [] R. Narayana K. Sudesh G. Gra et a. Stuaton and threat assessment n BVR combat n Proceedngs of the AIAA GudanceNavgatonandControConferencePortandOreUSA August. [] K. Sarar P. Kar A. Muheree et a. Range extenson of an arto-ar engagement by offne traectory optmzaton n ProceedngsoftheAIAAAtmosphercFghtMechancsConference PortandOreUSAAugust. [3]A.D.SandersC.G.JenstaR.H.Arrowoodeta. Mutdscpnary desgn of a supersonc ong-range ar-superorty msse through parametrc desgn space exporaton and physcs-based modeng n Proceedngs of the 5th AIAA/ ASME/SAE/ASEE Jont Propuson Conference and exhbt Ceveand Oho USA Juy 4. [4]J.T.Betts Surveyofnumercamethodsfortraectoryoptmzaton Journa of Gudance Contro and Dynamcs vo. no. pp. 93 7 998. [5] G.HuangY.LuandY.Nan Asurveyofnumercaagorthms for traectory optmzaton of fght vehces Scence Chna Technoogca Scencesvo.55no.9pp.538 56. [6] P. Han J. Shan and X. Meng Re-entry traectory optmzaton usng an hp-adaptve Radau pseudospectra method Proceedngs of the Insttuton of Mechanca Engneers Part G: Journa of Aerospace Engneerngvo.7no.pp.63 636. [7]P.E.GE.WongW.MurrayandM.A.SaundersUser s Gude for SNOPT Verson 7.4: Software for Large-Scae Nonnear Programmng Unversty of Caforna San Dego Caf USA 5. [8] I. M. Ross and M. Karpeno A revew of pseudospectra optma contro: from theory to fght Annua Revews n Controvo.36no.pp.8 97. [9] X.-J. Tang J.-L. We and K. Chen A Chebyshev-Gauss pseudospectra method for sovng optma contro probems Acta Automatca Sncavo.4no.pp.778 7875. [] G.T.HuntngtonD.BensonandA.V.Rao Acomparsonof accuracy and computatona effcency of three pseudsopectra methods n Proceedngs of the AIAA Gudance Navgaton and Contro Conference and Exhbt pp. 5AIAAPressSouth Carona SC USA 7. [] X. Jang and S. L Mars atmospherc entry traectory optmzaton va partce swarm optmzaton and Gauss pseudo-spectra method Proceedngs of the Insttuton of Mechanca Engneers Part G: Journa of Aerospace Engneerng vo.3no.pp. 3 39 5. [] H. Zhou T. Rahman D. Wang and W. Chen Onboard pseudospectra gudance for re-entry vehce Proceedngs of the Insttuton of Mechanca Engneers Part G: Journa of Aerospace Engneerngvo.8no.pp.95 9363. [3] S. Zhang J. Yu and H. Sun UAV path pannng va Legendre pseudospectra method mproved by dfferenta fatness n Proceedngs of the 7th Chnese Contro and Decson Conference (CCDC 5) pp. 58 584 Qngdao Chna May 5. [4] Y. Zhao and P. Tsotras Densty functons for mesh refnement n numerca optma contro Journa of Gudance Contro and Dynamcsvo.34no.pp.7 77. [5] Q. Gong F. Fahroo and I. M. Ross Spectra agorthm for pseudospectra methods n optma contro Journa of Gudance Contro and Dynamcs vo.3no.3pp.46 47 8. [6]F.LuW.W.HagerandA.V.Rao Anhpmeshrefnement method for optma contro usng dscontnuty detecton and mesh sze reducton n Proceedngs of the 53rd IEEE Annua Conference on Decson and Contro (CDC 4) pp.5868 5873 Los Angees Caf USA December 4. [7] R. PachónR.B.PatteandL.N.Trefethen Pecewse-smooth chebfuns IMA Journa of Numerca Anayssvo.3no.4pp. 898 96. [8] H. Hou Convergence Anayss of Orthogona Coocaton Methods for Unconstraned Optma Contro Unversty of Forda Forda Fa USA 3.
Advances n Operatons Research Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Advances n Decson Scences Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Journa of Apped Mathematcs Agebra Hndaw Pubshng Corporaton http://www.hndaw.com Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Journa of Probabty and Statstcs Voume 4 The Scentfc Word Journa Hndaw Pubshng Corporaton http://www.hndaw.com Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Internatona Journa of Dfferenta Equatons Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Voume 4 Submt your manuscrpts at https://www.hndaw.com Internatona Journa of Advances n Combnatorcs Hndaw Pubshng Corporaton http://www.hndaw.com Mathematca Physcs Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Journa of Compex Anayss Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Internatona Journa of Mathematcs and Mathematca Scences Mathematca Probems n Engneerng Journa of Mathematcs Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Voume 4 Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 #HRBQDSDĮ@SGDL@SHBR Journa of Voume Hndaw Pubshng Corporaton http://www.hndaw.com Dscrete Dynamcs n Nature and Socety Journa of Functon Spaces Hndaw Pubshng Corporaton http://www.hndaw.com Abstract and Apped Anayss Voume 4 Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Internatona Journa of Journa of Stochastc Anayss Optmzaton Hndaw Pubshng Corporaton http://www.hndaw.com Hndaw Pubshng Corporaton http://www.hndaw.com Voume 4 Voume 4