Variational Iteration Method for Solving Riccati Matrix Differential Equations

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1 Indonesian Journal of Elecrical Engineering and Compuer Science Vol. 5, No. 3, March 17, pp. 673 ~ 683 DOI: /ijeecs.v5.i3.pp Variaional Ieraion Mehod for Solving Riccai Marix Differenial Equaions Khalid Hammood Al-jizani*, Noor Ainah Ahmad, Fadhel Subhi Fadhel School of Mahemaical Sciences, Universii Sains Malaysia, 118, Penang, Malaysia *Corresponding auhor, khm13_mah9@suden.usm.my Absrac Riccai marix differenial equaion has long been known o be so difficul o solve analyically and/or numerically. In his connecion, mos of he recen sudies are concerned wih he derivaion of he necessary condiions ha ensure he exisence of he soluion. herefore, in his paper, He s Variaional ieraion mehod is used o derive he general form of he ieraive approximae sequence of soluions and hen proved he convergence of he obained sequence of approximae soluions o he exac soluion. his proof is based on using he mahemaical inducion o derive a general formula for he upper bound proved o be converging o zero under cerain condiions. Keywor: Marix Riccai differenial equaion, Variaional ieraion mehod, Ieraive meho, He s ieraive mehod, Marix differenial equaions. Copyrigh 17 Insiue of Advanced Engineering and Science. All righs reserved. 1. Inroducion Mahemaical modeling of real life problems, especially in conrol problems lea o differenial equaions, inegral equaions, sysem of differenial and algebraic equaions. While he soluion of such obained models may be so difficul o be evaluaed analyically, herefore numerical and approximae meho seem o be necessary o be used o evaluae he soluion of such problems. Among he mos popular and simples accurae meho, he ieraive meho is he variaional ieraion mehod (VIM), which is proposed by Ji-Huan He in 1999 as a special approach of he homoopy analysis mehod. he VIM has been shown o solve funcional equaions efficienly. In relaed lieraure, his mehod has been referred o as a modificaion of he general Lagrange muliplier mehod [18]. Several researchers compared he VIM wih oher numerical and approximae meho, where i is shown by all ha his mehod may give more accurae resuls and faser han he oher meho. In addiion, in his mehod, he convergen o he exac soluion is considered ino a coun and proved in he research work [9]. he main feaure of he VIM is ha he soluion of he problem under consideraion wih linearizaion assumpion is used as an iniial approximae soluion o more accurae and precise approximae soluion which is obained hrough ieraive. he VIM has been used by oher researchers o solve various problems such as He in 1999 used he VIM o give approximae soluion for some well known non-linear problems, [11]. He in 7 used his mehod o solve auonomous sysems of ordinary differenial equaions, [1]. In 6 he VIM was applied for solving nonlinear inegro-differenial equaion of funcional order by Kurulay M. [13, 17]. David K. and Hadi R. G. in 1 sudied he convergence of he VIM for he soluion of he elegraph problem, [3]. Ghorbani A. and Saberi J. proved he convergence of he VIM of nonlinear oscillaors wih an illusraive example [6]. Nguyen. e al in 1 propose a new mehod o solve Riccai marix differenial equaion, which is based on he differenial Lyapunov equaion o calculae numerically he soluion of Riccai marix differenial equaion [19]. he meod of Nguyen., e al., [19] is shown o be robus and numerically efficien. While he applicaion of he proposed mehod by Nguyen. e al. had been applied [] o he singular perurbed linear quadraic opimal conrol problem, in which for his objecive, he auhor s compose he full-order opimal linear quadraic conrol problem ino an reduced order subproblems. A similar ransformaion idea can be found in he works of Glizer Received November 3, 16; Revised February 3, 17; Acceped February 1, 17

2 674 ISSN: and Dmiriev [7, 8], in which a series of expansion mehod is proposed. Biazar J., e al., in 11 sudied he numerical soluion of cerain funcional inegral equaions by he VIM []. Geng F. used he modified VIM in 9 o solving scalar Riccai differenial equaion [4]. Lu J. solved a nonlinear sysem of second order boundary value problems using VIM in 7 [16] while Hemeda A.A. used he VIM o solve he wave parial differenial equaion [1]. he VIM has also been used o solve more advance problems movable boundaries by Ghomanjani F. and Ghaderi S. in 1 [5]. In addiion, in recen years, here is a wide occurrence of Riccai marix differenial equaion as a conrol model in which he analyical and heoreical resuls concerning his marix equaion has been esablished, bu sill is soluion seems o be difficul o be evaluaed. In he curren sudy, he VIM will be used o solve Riccai marix differenial equaion, where he convergence of he sequence of ieraed approximae soluions have been proved o be converge o he exac soluion of he problem, which is assumed o exis and be unique depending on he resuls of he oher researchers [15].. he Ricai Equaion he sudy of scalar and Riccai marix and/or differenial equaions has a grea imporance, which daes from he early days of modern mahemaical analysis, since such equaions represen one of he obained ypes of nonlinear equaions, especially in mahemaical physics. Also, wihin recen years, here is a wide occurrence of Riccai marix differenial equaions noably in variaional heory and allied areas of opimal conrol, invarian embedding and dynamic programming [1]. An aypical algebraic Riccai equaion is similar o one of he following [14, ]. he firs is a quadraic marix equaion for he unknown n n marix X of he form: A X XA XBX C (1) Where AB, and C are n n complex marices wih B and C marix ranspose. Equaion (1) is called he coninuous ime Riccai equaion. he second equaion has he funcional form for he unknown nnmarix X. 1 hermiian, and refers o he X A XA ( C B XA) ( RB XB) ( C B XA) Q () Where R and Q are mmand n n marices respecively and A, B, C are complex marices wih respec o sizes n n, n m and m n respecively. In addiion, boh equaions arise in sysems heory, differenial equaions and filer design in conrol heory [14]. More advanced and ineresing form of Riccai marix equaion is he nonlinear Riccai marix differenial equaion of he form [1]: X XA A X XBX C (3) Where AB, and C are consans n n marices such ha B B and C C. he exisence condiions of a unique soluion are imposed and saisfied on equaion (3). Soluion of equaion (3) may be achieved analyically, which is so complicaed and difficul o evaluae in some cases and herefore numerical and/or approximae meho seems o be necessary o find he soluion of equaion (3). In he nex secion, we will presen and use one of he He s ieraion meho o solve equaion (3), namely we will use he VIM. 3. he Variaional Ieraion Mehod o illusrae he basic ideas of he VIM, consider he following nonlinear equaion given in absrac form as an operaor equaion: Au( ) g( ), [a, b] (4) IJEECS Vol. 5, No. 3, March 17 :

3 IJEECS ISSN: Where A is a nonlinear operaor, g () is a given funcion, ab, and u is he unknown funcion. If i is assumed ha he operaor A may be decomposed ino wo operaors, namely; L and N ha will be rewrien equaion (4) as follows: Lu( ) Nu( ) g( ) (5) is a nonlinear operaor and g is any given funcion, which is called he nonhomogeneous erm. he analysis of he VIM is o rewrie equaion (5) as follows: Where L is a linear operaor, N Lu( ) Nu( ) g( ) (6) And if i is supposed ha u n is he h n approximae soluion of equaion (6), hen i follows ha: Lu ( ) Nu ( ) g( ) (7) n n And herefore, a correcion funcional for equaion (5) may be given by: (8) un1( ) un( ) ( s) Lun ( s) Nun ( s) g( s) Where is he general Lagrange muliplier ha can be idenified opimally via variaional heory, h he subscrip n denoes he n ieraive approximae soluion of u, and u n is considered as a resriced variaion, i.e., un, [1]. o solve equaion (8) by he VIM, he Lagrange muliplier should be deermined and evaluaed firs ha will be idenified opimally hrough inegraion by pars. hen, he successive approximaions un(), n, 1, ; of he soluion u() was obained upon applying he obained Lagrange muliplier. Now, using selecive funcion u( ) as an iniial guess for () u, he zero h approximaion u may be seleced by any funcion ha jus saisfies a leas he iniial and boundary condiions wih predeermined, hen several approximaions un(), n, 1, ; follows immediaely and consequenly he exac soluion may be arrived since: u( ) lim u ( ) (9) n n In oher wor, he correcion funcional for equaion (8) will give a sequence ieraed approximaions, and herefore he exac soluion is obained as he limi of he resuling successive approximaions [5]. 4. Approximae Soluion of Riccai Marix Differenial Equaions Using VIM I is obvious now ha he main aspecs of he VIM requires firs he deerminaive of he Lagrange muliplier (), which may be found by he mehod of inegraion by pars, i.e., for equaion (8) ha can be used: ( ) u( ) d ( ) u ( ) ( ) u ( ) d n n n Variaional Ieraion Mehod for Solving Riccai Marix Differenial (Khalid Hammood Al-jizani)

4 676 ISSN: () for he ( ) u ( ) d ( ) u ( ) ( ) u ( ) ( ) u ( ) d n n n n And so on. herefore, afer having deermined (), he sequence of approximae soluions un 1, n,1,... of he soluion will be obained immediaely upon any selecive funcion u as an, iniial guess. As i is found previously in lieraures (see for example [3,4]), he general form of h n order ODE: u ( n) ( ) f (, u( ), u( ),..., u ( n1) ( )) (1) Is proved by inducion o be: n 1 ( ) ( 1) ( x) ( n 1)! n1 (11) Where i is clear ha for he firs order ODE ( ) 1, for he second order ODE ( ) 1 and so on. he generalizaion of he correcion funcional (8) wih is corresponding Lagrange muliplier for sysems of n-operaor equaions may be considered similarly by using sysem of n- correcion funcionals wih relaed n-lagrange muliplier. Hence, in connecion wih Ricaai marix differenial equaion (3), he correcion funcional may be considered for a sysem of n-equaions of he firs order ODE s as follows: Equaion (3) may be wrien in he form of equaion (1) as: X( ) F(, X( ) (1) Where, F(, X( )) AX ( ) X( ) A X( ) BX ( ) C And hence equaion (3) (or equivalenly equaion 1) may be wrien in he form of he nonlinear equaion (4), where he relaed nonlinear operaor in he lefhand side of equaion (4) is defined by: d A A A B (13) d And he nonhomogenuous funcion in he righhand side is given by: g() C (14) herefore, in connecion wih he nonlinear operaor (13) ha may be decomposed ino wo operaors, namely; a linear operaor L and a nonlinear operaor N, which are defined as: d L A A (15) d N B (16) IJEECS Vol. 5, No. 3, March 17 :

5 IJEECS ISSN: Hence, he saemen of he variaional ieraion formula (8) relaed o he Riccai marix differenial equaion: LX ( ) NX ( ) g( ) (17) Where L, N and g are defined by equaions (14), (15) and (16) respecively. he derivaion of he approximaed formula is presened and i is easily verified ha he correcion funcional akes he form as in he nex heorem. heorem (1): he correcion funcional of he Ricaai marix differenial equaion (3) has he form: Xn1( ) Xn( ) [ Xn( s) Xn( s) A A Xn( s) X( s) BX n( s) C] (18) For all n=,1, Proof: Rewrie equaion (3) in he form of equaion (1) in marix form. hen using he resuling Lagrange muliplier for equaions (1) and (11) implies ha he nxn values of he Lagrange muliplier () o be ( ) 1. In order o ensure he convergence of he sequence of approximae soluions obained by using equaion (18), which may be proved o be converge o he exac soluion Xs (). Before inroducing he proof, he nonlinear operaor M is defined by: MX XBX Which is assumed o saisfy Lipschiz condiion wih consan L, i.e., saisfy: X1BX 1 XBX L X1 X (19) Where is an appropriae marix norm. heorem (): 1 Le X1,..., Xn ( C [, b], ), n,1, ; be he exac and approximae soluions of he Riccai marix differenial equaion (1). If En( ) Xn( ) X( ) and MX XBX saisfies Lipschiz condiion wih consan L, hen he sequence of approximae soluions Xn( ), n,1, ; converges o he exac soluion X (). Proof he approximae soluion of Riccai marix equaion (1) using he VIM is given by (18) and since X is he exac soluion, hen i saisfy: X X [ X XA A X XBX C] () Now, subsrac equaion () from equaion (18) o ge: Xn1 X Xn X [ Xn XnA XnA XnBX n C X XA XA XBX C] Variaional Ieraion Mehod for Solving Riccai Marix Differenial (Khalid Hammood Al-jizani)

6 678 ISSN: Since E n X n X, hen: En1( ) En ( ) [ En ( s) En( s) A A En( s) ( Xn( s) BX n( s) X( s) BX ( s))] n n n n n n E ( ) E ( s) E ( s) A A E ( s) ( X ( s) BX ( s) X ( s) BX ( s)) (1) And upon using he mehod of inegraion by pars for he firs inegral of equaion (1), o ge: En1( ) En ( ) En ( ) En() En( s) A A En( s) ( Xn( s) BX n( s) X( s) BX ( s))] () Hence aking he suprumum norm o he boh sides of equaion () yiel o: En1( ) En ( s) A A En ( s) Xn( s) BX n( s) X( s) BX ( s) n n n E ( s) A A E ( s) L E ( s) ( A L) En( s) herefore: En1( ) ( A L) En( s) Now, if n=, hen: E1( ) ( A L) E( s) ( ) sup ( ) s[, b] A L E s ( A L) sup E ( s) s[, b] If n=1, hen: E( ) ( A L) E1( s) IJEECS Vol. 5, No. 3, March 17 :

7 IJEECS ISSN: ( A L) [ A L) s sup E ( s) ] ( ) sup ( ) s[, b] s[, b] A L E s s s[, b] ( A L) sup E ( s ) If n=, hen: E3( ) ( A L) E( s) ( A L) ( ) sup ( ) A L s E s 33 ( A L) 3! s[, b] sup E ( s) s[, b] And so on in general for any naural number n. n n ( A L) En ( s) sup E ( s) n! n s[, b] n ( A L) b sup En ( s) n! s[, b] 1 Since ( A L) b 1 is a consan and as n hen which implies n! En( ) as n, i.e., Xn( ) X( s) as n, which means ha he sequence of approximae soluions using he variaional ieraion formula equaion (18) converge o he exac soluion of Riccai marix differenial equaion. 5. Illusraive Examples In his secion, wo illusraive examples will be considered, where i is noable ha he marix C may be considered as a marix funcion of which has no effec on he derivaion of he Lagrange mulipliers () (see heorem (1)). he firs example is for he one-dimensional Riccai differenial equaion while he second example is for marix Riccai differenial equaion. In each example a comparison have been made wih he exac soluion. Example (1): Consider he firs order Riccai differenial equaion: y( x) 1 x y, x [,1] Wih iniial condiion y() 1, where comparison wih equaion (3), give: A, B 1 and C 1 x Variaional Ieraion Mehod for Solving Riccai Marix Differenial (Khalid Hammood Al-jizani)

8 68 ISSN: he exac soluion for comparison purpose is given by: y() x x e 1 e x d Hence, saring wih he iniial soluion y ( x) 1, and by applying he variaional ieraion formula (18), we ge he firs hree approximae soluions: y1( x) y( x) [ y ( x) 1 s y ( s)] 3 x y ( x) y ( x) [ y ( x) 1 s y ( s)] x x (x 1) y ( x) y ( x) [ y ( x) 1 s y ( s)] x x (88x 31 x 54 x 1617 x x 3497) x (x 1) Also, numerical resuls and is comparison wih he exac soluion are given in able 1. able 1. Absolue error beween he exac and approximae soluions of example (1) x y 1(x) y(x) y (x) y(x) y 3(x) y(x).1 1.6E E-7..47E-4.E E E E-3 6.1E E E E E Example (): Consider he Riccai marix differenial equaion wih: 1 A 1, 1 B 1, 4 1 C () 3 3 Which has he exac soluion: X () 1 IJEECS Vol. 5, No. 3, March 17 :

9 IJEECS ISSN: hen using he variaional ieraion formula o find X1, X,..., X 5 wih iniial soluion 1 X. he resuls of he absolue error for each componens wih he exac soluion had given in able -5. X, X,..., X able. Absolue error of he firs componens soluions 1 5 X 1 X X 3 X 4 X E E-4 1.9E-5.89E E-3 5.8E-4.66E E-3.6E E X, X,..., X able 3. Absolue error of he second componens soluions 1 5 X 1 X X 3 X 4 X E E E E E-3 9.8E-5.47E E E-3 1.6E E E E E-3.67E E X, X,..., X able 4. Absolue error of he hird componens soluions 1 5 X 1 X X 3 X 4 X E-4 1.3E-3.4E-4 1.5E E E E-3 3.4E E-4.13E E-3 3.7E E X, X,..., X able 5. Absolue error of he fourh componens soluions 1 5 X 1 X X 3 X 4 X E-4 6.6E E-5 4.9E-6 4.8E-7..67E E E E-5.55E E E-3 3.9E-4.14E E-4 7.6E E E E Variaional Ieraion Mehod for Solving Riccai Marix Differenial (Khalid Hammood Al-jizani)

10 68 ISSN: Conclusion In he curren sudy, we have applied He s VIM o solve Riccai marix differenial equaion, which is a powerful ool ha has he abiliy o solve differen ypes of linear and nonlinear operaor equaions. he general form of he approximae soluion has been derived depending on he general crieria of he VIM for solving n-h order ODE s. Also, he convergence of he obained approximae soluion have also been proved by proving he convergence of he error erm beween he approximae and exac soluions o be end o zero as he sequence of ieraions increased. he obained resuls of he illusraive examples seem o be reliable in which he absolue error beween he approximae soluion and he exac soluion has been used also o check he accuracy of he obained resuls. In addiion, differen approach based on he heory of marix algebra may be used o derive he approximae soluion however wih differen rae of convergence o he exac soluion and wih more resricions on he marices ABand, C of he Riccai marix differenial equaion, where he approximae soluion in his case has he form as follows: ( AA ) ( AA ) s n n n X 1( ) X ( ) e e [ X ( s) X( s) A A X( s) X( s) BX ( s) C] n=,1. Acknowledgemens his work is suppored by he Fundamenal Research Gran Scheme (FRGS) under he MInisry of Higher Educaion, Malaysia. References [1] Baiha B, Noorani MS, Hashim I. Numerical Soluions of he Nonlinear Inegro-Differenial Equaions. Journal of Open Problems Comp. Mah. 8; 1(1): [] Biazar J, Porshokouhi MG, Ghanbari B. Numerical soluion of funcional inegral equaions by he variaional ieraion mehod. Journal of compuaional and applied mahemaics. 11; 35: [3] Davod KS, Hadi RG. Convergence of he variaional ieraion mehod for he elegraph equaion wih inegral condiions. Wiley Iner Science. 1; 7: , [4] Geng F, Lin Y, Cui M. A piecewise variaional ieraion mehod for Riccai differenial equaions. Compuers and Mahemaics wih Applicaions. 9; 58: [5] Ghomanjani F, Ghaderi S. Variaional Ieraive Mehod Applied o Variaional Problems wih Moving Boundaries. Applied Mahemaics. 1; 3: [6] Ghorbani A, Saberi-Nadjafi J. Convergence of He s variaional ieraion mehod for nonlinear oscillaors. Nonlinear Sci. Le. A. 1; 1(4): [7] Glizer VY, Dmiriev MG. Solving perurbaions in a linear opimal conrol problem wih quadraic funcional. Sovie Mahemaics Doklady. 1975; 16: [8] Glizer VY, Dmiriev MG. A sympoic properies of soluion of singularly perurbed Cauchy problem encounered in opimal conrol heory. Differ. Equaion. 1978; 14(4): [9] He JH. he Variaional Ieraion Mehod: Reliable, Efficien, and Promising. An Inernaional Journal, Compuers and Mahemaics wih Applicaions. 7; 54: [1] He JH, Variaional Ieraion Mehod; Some Recen Resuls and New Inerpreaions. Journal of Compuaional and Applied Mahemaics. 7; 7: [11] He JH. Variaional Ieraion Mehod; A Kind of Non-Linear Analyical echnique; Some Examples. Inernaional Journal of Non-Linear Mechanics. 1999; 34: [1] Hemeda AA. Variaional ieraion mehod for solving wave equaion. Compuers and Mahemaics wih Applicaions. 8; 56: [13] Kurulay M, Secer A. Variaional Ieraion Mehod for Solving Nonlinear Fracional Inegro-Differenial Equaions. Inernaional Journal of Compuer Science and Emerging echnologies. 11; : 18-. [14] Lancaser P, Rodman L. Soluions of he Coninuous and Discree ime Algebraic Riccai Equaions: A Review. In: Sergio B, Alan L, lan CW. Ediors. he Riccai Equaion. Springer-Verlag; [15] Lin MN. Exisence Condiion on Soluions o he Algebraic Riccai Equaion. Aca Auomaica SINICA. 8; 34(1). IJEECS Vol. 5, No. 3, March 17 :

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