Variation of Parameters for Causal Operator Differential Equations

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1 Applied Mahemaics, 217, 8, hp://wwwscirporg/journal/am ISSN Online: ISSN Prin: Variaion of Parameers for Causal Operaor Differenial Equaions Reza R Ahangar Mahemaics Deparmen, Texas A & M Universiy Kingsville, Kingsville, USA How o cie his paper: Ahangar, RR (217) Variaion of Parameers for Causal Operaor Differenial Equaions Applied Mahemaics, 8, hps://doiorg/14236/am Received: November 7, 217 Acceped: December 26, 217 Published: December 29, 217 Copyrigh 217 by auhor and Scienific Research Publishing Inc This work is licensed under he Creaive Commons Aribuion Inernaional License (CC BY 4) hp://creaivecommonsorg/licenses/by/4/ Open Access Absrac The operaor T from a domain D ino he space of measurable funcions is called a nonanicipaing (causal) operaor if he pas informaion is independen from he fuure oupus We will sudy he soluion x() of a nonlinear operaor differenial equaion where is changes depends on he causal operaor T, and semigroup of operaor A(), and all iniial parameers (, x ) The iniial informaion is described x = ϕ for almos all and φ( ) = φ We will sudy he nonlinear variaion of parameers (NVP) for his ype of nonanicipaing operaor differenial equaions and develop Alekseev ype of NVP Keywords Nonlinear Operaor Differenial Equaions (NODE), Variaion of Parameers, Nonanicipaing (Causal), Alekseev Theorem 1 Definiions and Example of Nonanicipaive Operaors An imporan feaure of ordinary differenial equaions is ha he fuure behavior of soluions depends only upon he presen (iniial) values of he soluion There are many physical and social phenomena which have herediary dependence Tha means he fuure sae of he sysem depends no only upon he presen sae, bu also upon pas informaion (see [1]-[6]) Twins before he ime of concepion share all of heir geneic hisory and may go o a differen pah in heir fuure life We are going o sudy he phenomenon which can be formulaed in principle ha he presen evens are independen of he fuure These kinds of evens are called nonanicipaion or causal evens Definiion 11: Coninuous Nonanicipaing Sysem A mapping of T from he space of funcions Y ino iself is said o be a DOI: 14236/am Dec 29, Applied Mahemaics

2 nonanicipaing mapping if for every fixed s in he real line R, ( Tx) = ( Ty) for all < s, whenever x = y for all < s Example 11: All of he delay operaors and inegral operaors are nonanicipaing All composiions or Caresian producs of he nonanicipaing operaors are nonanicipaing Example 12: Le I = [, a ] R be a compac subse of he real line and f a funcion from he inerval I Y ino Y The knowledge of he sae of he sysem a a given ime (, ) f( y, ) y = (11) y is sufficien o deermine is sae a any fuure ime This sysem has no afer-effec or no memory Example 13: In a dynamic sysem (,, ) y = f y T y (12) when T is a nonanicipaing operaor, o find he sae curve y( ) we need o have informaion abou he iniial funcion y = φ for < in order o deermine he sae of he soluion The following are examples of coninuous anicipaing operaors (see Naylor and Sell 1982 [7]) Definiion 12: A mapping T of Y ino iself is said o be causal if for each ineger N, whenever wo inpus x = { x n } and y = { y n } are such ha xn = yn for n N, i follows ha T x = T y, for n N n n where and = {, ( 1),, ( 1), ( 2), } T x T x T x T x T x = {, ( 1),, ( 1), ( 2), } T y T y T y T y T y In oher words, if he inpus x and y agree up o some ime N, hen he oupus T(x) and T(y) agree up o ime N In paricular, T(x) and T(y) agree up o ime N no maer wha he inpus x and y are in he fuure beyond N The evens in he pas and presen are independen from he fuure 2 Example 13: Consider Z = l ( R), and le T be a mapping of Z ino iself represened by a convoluion inegral defined of he form = ( ) d Tx h u x u u This is a nonanicipaing mapping if and only if h ( s) = for almos all s< This Volera inegral mapping shows ha ( Tx)( ) is independen of x() for > s Noice ha when a mapping is no nonanicipaing i will be an anicipaing mapping, meaning ha he pas and he presen depend on he fuure Anicipaing (anicausal) Mapping: This is a mapping ha he fuure oupu DOI: 14236/am Applied Mahemaics

3 is independen of he pas inpu, meaning ha he mapping T : Z Z is said o be (anicausal) or anicipaing if for fixed s in I = [, a], ( Tx) = ( Ty) for > s, whenever x = y for > s Example 14: Le T be an operaor from he space of a square summable 2 2 funcion L ( R ) ino L ( R ) We can show ha he following mappings are anicausal; ( u) e y = x u du Since, for fixed real number s, he fac ha x1 x2 = for > s and means ha he fuure inpu = for > s implies y1 y2 { y : > s} will affec he pas Therefore, his is an anicausal operaor 2 Nonanicipaing Operaor Differenial Equaion Noaions Le S be he inerval of all nonposiive numbers Le I be he compac inerval [, a ], J = { R: }, and define J = S I Assume Y, Z, and U are Banach spaces Le M( IY, ) be he space of all essenially bounded Bochner measurable funcions wih respec o classical Lebesgue measure from he inerval I ino he Banach space Y Denoe by L( JY, ) he space of all Lipschizian funcions y srongly differeniable almos everywhere from J ino Y Le φ be a fixed iniial funcion from he space L( SY, ) Denoe by D( φ, Y) he subse of he lip-space L( JY, ) consising of all funcions y such ha y = φ for all in S According o hese wo definiions, D( φ, Y) L( JY, ) For any Banach space Y and Z, le Lip( I; Y, Z ) denoe he space of all funcions f( y, ) from he produc I every fixed y he funcion (, ) Y ino Z, Lipschizian in y, and for M IZ This space is called Lip-space We apply he definiion of nonanicipaive operaors in Secion 11 o he iniial domain An operaor T from he iniial domain D( φ, Y) ino M( IZ, ) will be called a nonanicipaing operaor if for every wo funcions y and z in D( φ, Y) and every poin s I, he fac ha y = z for almos all < s implies ha T( y) = T( z) for almos all < s An operaor P from a subse D of Y ino Z is said o be Lipschizian if here exiss a consan b such ha f y belongs o he space (, ) P y P y b y y (21) for every y1, y2 Y f Lip I, Y ; Z he operaor F: M I, Y M I, Z defined by F y = f, y (22) For is called he operaor induced by f and he operaor F is called Induced Operaor generaed by he funcion f Lipschizian Space (or simply he Lip-Space), denoed by Lip( K, Y ; Z ), is he DOI: 14236/am Applied Mahemaics

4 se of all funcions f : K Y Z such ha (, ) Lipschizian in y, and is measurable in Tha is, { is Lipschizian in y and f(, y) M( KZ, )} consans L will be denoed by f f y is uniformly bounded, Lip K, Y; Z = f : K Y Z f The infimum of all Lipschizian On he space M( IY, ), we shall inroduce a family of norms, called k-norm by he formula { k } y = esssup e y : I k for any fixed real number k Observe ha from his definiion follows he inequaliy y e k for almos all in I Noice ha for every k, he k norms equivalen y A Lipschizian operaor P from a subse D of M( JY, ) ino he space M( IU, ) is called an operaor of exponenial ype if for some consans b and k, P( y) P( z)) b y z k for all y and z in he domain D and all k k T y = sin, y r for a consan real Example 21: The operaor ( ) number r is an induced operaor Thus for any funcion y M( IY, ) operaor T is nonanipaing and Lipschizian k k k and are he Properies of he nonlinear operaor F in (22) induced by he funcion f have been sudied by Bogdan 1981 and 1982 In paricular, i is known ha for f Lip( I, Y ; Z ) and y M( IY, ) he funcion g: I g ()= f( y, ()) belongs o he space of measurable funcions (, ) Ahangar 1989, [1]-[6]) Z defined by M IZ (see When an operaor T is nonanicipaing, he fuure values of he inpu will have no effec on he presen sae One can prove ha he composiion and he Caresian produc of nonanicipaing and Lipschizian operaors are Nonanicipaing and Lipschizian Furhermore, he operaor F induced by he funcion f is a well defined, nonanicipaing, and Lipschizian operaor Definiion 21 (Direc Sum Operaors): Le Ti ( i= 1, 2) be operaors from he domain D M( IY, ) ino he space M ( IZ, ) Define he direc sum operaor T = T1 T2 such ha T T y = T y + T y for every y in M( JY, ) and in I Lemma 21: A direc sum operaor of wo nonanicipaing and Lipschizian operaors is nonanicipaing and Lipschizian Proof: Firs le us prove ha he direc sum operaor is a nonanicipaing operaor Assume ha wo funcions y and z are in he space of M( JY, ) and for some poin s in he inerval I we have y = z for almos all < s Since T 1 and T 2 are nonanicipaive, hen DOI: 14236/am Applied Mahemaics

5 T1( y) = T1( z) for almos all < s, T2( y) = T2( z) for almos all < s These wo equaliies will imply ha { T1( y) + T2( y) } = { T1( z) + T2( z) } for almos all < s Thus ( T1 T2)( y) = ( T1 T2)( z) for almos all s This will imply ha T( y)( ) and T( z)( ) coincide for almos all < s Now le us prove ha he operaor T = T1 T2 is Lipschizian According o he definiion < = ( 1 2) ( 1 2) (( T1)( y) ( T2)( y) ) (( T1)( z) ( T2)( z) ) (( T1)( y) ( Tz )( z) ) (( T2)( y) ( T2)( z) ) T y T z T T y T T z = + + = + Since boh operaors are Lipschizian, he righ hand side will be L y z + L y z 1 2 Noice ha represens he esssup norm in he space measurable max L = L for i = 1, 2 and ake he essenial funcions (, ) M IY If we le { } supremum norm on he lef hand side of he above relaion hen i will be i T y T z L y z (23) for all y and z in he domain D This proves ha he direc sum operaor is Lipschizian QED Example 22: Assume ha T1 ( y) = y( r) for a consan real number r and T ( y) = y( s) d s The operaor 2 T( y) = ( T ) 1 T2 y is nananicipa- ing and Lipschizian Nonanicipaing Deerminisic Dynamical Sysem: Assume ha he operaor T is nonanicipaing and Lipscizian The behavior of a dynamic sysem (,, ) y' = f y T y (24) is known as an afer effec differenial equaion wih he iniial domain D( φ, Y) Given ha f Lip( I, Y Z; Y ) here exiss a unique soluion y o he sysem (24) Equaions of his ype arise in many mahemaical modeling problems In a simples case, T as a consan delay operaor can be applied (see Hale 77 [8] and Driver 77 [9]) The following is a single species growh model wih ime delay Example 23: A single species model wih delay can be described by 1 y = ry y τ K where r is he growh rae of he species y, and K is called he environmen capaciy for y The chaoic behavior induced by ime delays was presened by Yang Kuang 1996 The global exisence of he general single species wih sage srucured model described by a sysem DOI: 14236/am Applied Mahemaics

6 = ( ( τ )) ( ) y' f y g y has been sudied (See Kuang 1996, p173, [1]) Example 24: Le T be he operaor defined in example 22 One can verify he exisence and uniqueness of he soluion of he sysem y' = T ( y) wih he iniial daa funcion y φ r < Our goal is o invesigae = for he condiions which guaranee he soluion of he sysem (24) when here is a random perurbaion in he sysem Soluion o he Nonanicipaing Operaor differenial Equaions: The following operaor differenial equaion when G is a nonanicipaing operaor from he iniial domain D( φ, Y) o he Banach space Z is called nonanicipaing differenial equaion φ, y = G y ) > y = (25) for almos all in he inerval I We define ha a funcion y from he space M(I,Y) is a soluion o he nonanicipaing operaor differenial equaion if i is srongly differeniable and saisfies he sysem (25) (see Bogdan 1981 [11], Bogdan 1982 [12], Ahangar 1989, [1], and Ahangar 1986 [2]) We accep he following heorem wihou proof Theorem 21: Given a nonanicipaing and Lipschizian operaor G from he iniial domain D( φ, Y) ino he space of Bochner measurable funcions M( IY, ), here exiss a unique soluion y D( φ, Y) ha saisfies he nonlinear operaor sysem (25) Noe: The purpose of his paper is o develop a generalized nonlinear variaion of parameers formula, analogous o Alekseev's resul (see Alekseev 1961 [13]) The generalizaion is lised below: 1) The classical exisence and uniqueness heorem for he soluion of absrac Cauchy problems no longer holds if he underlying space is an infinie dimensional Banach space (See Lakshmikanham 1972, [14] [15] and [16]) 2) The nonlinear sysem in his paper includes all evoluionary equaions of C semigrop of operaors 3) Insead of coninuiy of he nonlinear funcions f (, y, T( y)( )), we will replace he more general form of hese funcions in Banach spaces o be Bochner measurable in and Lipschizian in y For regulaory condiions, we will assume he nonlinear operaor involved in he nonlinear sysem is nonanicipaing and lipchizian 4) The soluion funcions eiher x or y are assumed srongly differenial 3 Srong Soluion o he Perurbed Nonanicipaing Operaor Differenial Equaions Definiion 31: By Nan-Lip we mean nonanicipaing and Lipschizian operaors The operaor G in he sysem (25) is nonanicipaing and Lipschizian We DOI: 14236/am Applied Mahemaics

7 need o clarify he meaning of he soluion o he nonlinear sysem of operaor differenial Equaion (25) The imporan par is when we accep some oher principles indirecly hidden in he proof of Theorem (21) In fac we use he equivalen relaionship beween (25) and he inegral φ y = + G y s ds Noice ha his equivalen relaion requires he absolue coninuiy of funcion y and he summabiliy of he operaor G which implies he differeniabiliy of y The above nonlinear operaor sysem similarly could be presened by he following operaor differenial equaion (,, ) for almos all x' = f x T x > (31) which conain he iniial funcion φ for he pas ime inerval S { R: } = The soluion of he sysem (31) is denoed by x() which depends on he iniial ime and he iniial funcion φ and can described by x (,, φ ) which is called he srong soluion o he sysem Definiion 32: A funcion x() is said o be a srong soluion o he sysem (31) if i saisfies he following condiions: 1) x is srongly differeniable, 2) x saisfies he sysem (31) almos everywhere in he inerval I, D JY, x = φ, for almos all 3) here exiss a funcion φ such ha The following proposiion will show he exisence and uniqueness of he soluion o he perurbed operaor differenial Equaion (31) For inroducory perurbaion heory see Brauer 66 and Brauer 67 Proposiion 31: Assume ha he operaor T is Nan-Lip and funcions f and g belong o he Lip-space which is f Lip( I, Y Z; Y ) and g Lip( I, Y; Y ) 1) If g is he perurbaion o he Equaion (31) hen here is a unique srong soluion y() in he iniial domain D( φ, Y) which saisfies he perurbed sysem of differenial equaion 2) Given a soluion (,, ) (,, ) (, ) y' = f y T y + g y (32) xφ of (31) hen he soluion o he perrubed equaion will saisfy he inegral equaion ( φ ) ( ) y = x,, + g sy, s ds (33) Proof: 1) Le us assume ha he operaor P1 ( y) = f (, y, T( y) ) and P2 ( y) = g(, y ) Define he direc sum operaor G = P1 P2 By Lemma 21, he operaor G will be Nan-Lip and he differenial Equaion (32) will be in he following form y = G( y) (34) for almos all in I According o Bogdan s heorm (see Bogdan 1981 and 1982, D φ, Y o he Equaion (34) [11], [12]), here exiss a unique soluion y() in DOI: 14236/am Applied Mahemaics

8 Proof of 2) The equivalen inegral equaion of (34) will be φ y = + G y s ds (35) Applying he direc sum operaors P 1 and P 2 we ge he conclusion which is (33) QED Subsiue for unperurbed soluion x (,, φ) = φ + f( sy, ( s), T( y)( s) ) ds in (33) as a soluion of (31) we will ge he following = φ + (,, ) d + (, ) d (36) y' f sys T y s s gsys s This complees he proof of par (ii)qed 4 Generalized Operaor Differenial Equaions Inroducion o he mild (Weak) soluions: For he definiion of srong soluion in he previous secion, i was assumed equivalen relaions beween he differenial and inegral forms This assumpion required he differeniabiliy of he soluion This condiion may no be rue in a large class parial differenial equaions We are going o review he difficulies of applying he conceps of srong soluion o he operaor differenial equaions The following are some examples The collecion of soluions of he problem of free oscillaions of an infinie sring expressible in he form 2 2 u 2 u = c 2 2 x u, x = φ x + c + ψ x c, where φ and ψ are wice akes he form differeniable funcions Noice ha a he verices of hese soluions, u( x, ) will no be differeniable Noice also he Lipschizian condiion for he nonlinear operaor G which is required for he unique soluion o he sysem (25) may no hold for unbounded operaors in evoluionary equaions Thus, we need o have a new concep which includes he nondiffereniable soluions for unbodied operaors We are going o demonsrae his sudy by a linear sysem of absrac Cauchy problem, f L 2 = + (41) u α 1 u f u 1 1 for u H Ω, where α may be an unbounded operaor in he 1 space X Assume ha he domain of his operaor is denoed by D( α ) X We are looking for a soluion space Y X One way o o ge he soluion space Y is o work from A and show ha i generaes a C -semigroup When he operaor is PDE, i may be unbounded, hus he soluion in (41) may no be well defined 1 We use a es funcion φ H such ha 1, =, +, (42) u φ α u φ f u φ We define a weak soluion mild soluion u such ha boh relaions (42) and he following are equivalen DOI: 14236/am Applied Mahemaics

9 A ( s ) ( ) A u = e u + e f u s ds (43) Mos of he physical models can be described by a PDE sysem wih evoluion equaions One can inerpre he soluion as an ODE soluion in an appropriae infinie dimensional space Nonlinear Operaor Differenial Equaions(NODE): Suppose X is a Banach space, A: D( A) X X is he generaor of a C -semigroup on X, U R X is open and f : U X be a coninuous funcion such ha x f( x, ) is differeniable and ( x, ) Df x ( x, ) is a coninuous in U For (, φ ) U, we denoe by x (,, φ ) he mild soluion o he Cauchy problem ( ) φ x = Ax + f, x, T x, for > x =, for (44) which has no been defined ye We can define i by employing a similar argumen and using he inegral form of he sysem (44) A φ ( ) A ( s) = e + e,, d (45) x f sxs T x s s Definiion 41: We define he funcion x() o be a mild soluion o he sysem (44) on I = [, a] if i saisfies (45) and x D( A) for all in I Lemma 41: Every semigroup of operaors generaed by he operaor A is a nonanicipaing and Lipschizian operaor Proof: Assume ha he semigroup T generaed by A is given Thus by he definiion of semigroup, for every y in D(A) ( ξ) = = ( + ξ), for, ξ T y TT y y ξ =, for every ξ Thus =, for all ξ This proves ha he semigroup operaor T is nonanicipaing Remarks: 1) The converse is no rue There may be a nonanicipaing operaor which may no be a semigroup 2) I would be ineresing o find ou wha condiions we may impose on he nonanicipaing operaors o generae a semigroup? Theorem 42: (Exisence and Uniqueness of he Soluion) Le he operaor A be a semigroup operaor and T nonanicipaing and Lipschizian Assume ha f Lip( I, Y, Z ) Then he sysem (44) has a unique soluion in he space of iniial domain D( φ, Y) Proof: The homogeneous soluion is guaraneed by he semigroup of operaors and i will be equal o e A φ The unique soluion of he enire sysem (44) will be obained by he nonanicipaing and Lipschizian properies of T and he Theorem 21 These ypes of problems arise in a variey of physical models like hea conducion, populaion dynamics, and chemical reacions Suppose ha for y and y in D(A) hen y( ξ) y( ξ) he equaliy y( + ξ) = y( + ξ) implies ha T ( y)( ξ) T ( y)( ξ) DOI: 14236/am Applied Mahemaics

10 5 Variaion of Parameers for Perurbed Operaor Differenial Equaions Suppose X is a Banach space, A: D( A) X X is a generaor of a C -semigroup on X, U R X Y is open and f : U Y be a coninuous funcion such ha x f( xz,, ) is differeniable and (, x, φ) Dx f (, φ) is coninuous in U where z = T( x) and z = T( x) = φ For (, x, z ) U, we denoe by (,, ) following Cauchy problem x φ he mild soluion o he x xs (,,, φ ) = A x + f ( sx,, (,, φ )), for almos all > x = φ, for almos all Assume also ha y() is a soluion o he following perurbed sysem ( ) ( ) (,, φ ) (,, φ), y = A x+ f, y, T y + g, y, > y = x (51) (52) These soluions in he sysem (51) are hen relaed by he evoluionary propery ( ;, φ) = xsxs ( ;, ( ;, φ) ) x for all s The iniial funcion φ depends on,, and x I is denoed φ,, x The soluion o he sysem says ha he fuure is deermined by compleely by he presen, wih he pas being involved only in ha i deermines he presen This is a deerminisic version of he Markov propery We make use of he following heorem in developing he variaion formula for nonlinear differenial equaions The Alekseev s formula for C -semigroups was generalized by Hale 1992 [17] In addiion, F Bruaer 1966 [18] and 1967 [19] sudied he perurbaion of Nonlinear Sysems of Differenial Equaions [1], [11] We will use he same approach o develop he Nonlinear Variaion of Parameer (NVP) for operaor differenial equaions Le X be a Banach space, operaor A: D( A) X Y is generaor of a C -semigroup on X, f Lip( I, Y Z; Y ) is coninuously differeniable wih respec o x Le us summarize our condiions o presen he following hypohesis; (H1) The operaor A in (51) and (52) is a Semigroup (H2) Assume ha funcions f and g belong o he following Lip spaces Tha is hey are Bochner measurable on he firs variable and Lipschizian on he oher variables (H3) Assume ha (,, ) (,, ), ( ;, ) f Lip J Y Z Y g Lip J Y Y x φ is a mild soluion o he following unperurbed operaor differenial Equaion (51) (H4) also le (,, ) y φ be a soluion o he following perurbed nonlinear operaor differenial Equaion (52) DOI: 14236/am Applied Mahemaics

11 Lemma (51): Assume ha all condiions for he exisence of he soluion o he nonlinear operaor sysem of he unpeurbed equaion hold Then 1) The derivaive x (,, x, φ) U(,, x, φ) exiss and i is denoed by x (,, x, φ ) 2x as parial derivaive on variaion wih respec o he second parameer x I saisfies he following nonlinear operaor equaion du U = = fx x, (,, x, ) U for d φ > (54) U = I for all The relaion (54) shows how fas he unperurbed soluion x() changes wih respec o is iniial posiion x, and is iniial funcion φ This is a parial derivaive wih respec he variable x(s) for new iniial value x( s = ) 2) Also assume ha he funcion x() is Freche differeniable wih respec he firs parameer variable x (,, x, φ ) V(,, x, φ ) exiss and i is denoed by x (,, x, φ ) 1 I saisfies he second kind of operaor differenial equaion x ( φ ) ( φ ) V = f x,,, x, V for > V = f, x, for all (56) Furhermore (,,, φ ) (,,, φ ) (,, φ ) V x = U x f x (57) Proof: Par 1): We are assuming ha he ransformaion T will be applied on he soluion funcion x() and will produce a funcion a (, ) x which will be he iniial funcion φ Though he unperurbed soluion can be described by x = x,, x, φ Le us ake he derivaive of boh sides of (53) wr variable : d d d U x x x x x d d x x d (,,, φ ) = (,,, φ ) = (,,, φ ) x = fx (,, T( x) = f( x,, T( x) ) x x x Subsiue is equivalen from (53) hen we can conclude: (,,, φ ), (,,, φ ) U x = fx x x U for > where he second par of he relaion (54) can be inerpreed as an ideniy marix: d U U(,, x, φ) = I U = (58) d 2) Noice ha, a he saring poin = we can re-evluae he rae of change of he soluion wih respec o he iniial momen DOI: 14236/am Applied Mahemaics

12 (,,, φ ) (,,, φ ) x x V x A few noes are imporan: For a vecor soluion x x = x = x x = I (,,, φ ) φ(, ) φ (,,, φ ) x (59) Similar o (58) d V (,, x, φ) = I V (,, x, φ) = d Noice ha d d d V x x x x x d d d (,,, φ ) = (,,, φ ) = (,,, φ ) f dx = f(,, x, φ ) = x d x (,,, φ ) (,,, φ ) = f x V x for all d V,, x, φ = fx,, x, φ V,, x, φ d This complees he proof of he firs par of (b) To prove he second par of (b), we can assume ha [ ] s, Le us ake he derivaive of boh sides of (59) wih respec : x x( x φ ) = x + x = x,,, x x x x + = + = = (,, ) x I V V x u d s has a variaion on d x(,, x, φ ) V ( ) f (, x, φ ) = = (51) This complees he second par of he resul in (2) Proof of he las par of (2): using he definiion of operaors U and V: V x = x x (,,, φ ) (,,, φ ) Subsiing (51) yields dx = x (,, x, φ ) x = U(,, x, φ ) x (,,, φ ) = (,,, φ ) (,, φ ) V x U x f x d Theorem (51): Alekseev Type Variaion of Parameers Theorem for NODE Sysems: DOI: 14236/am Applied Mahemaics

13 Le x (,, x, φ ) and (,,, ) and (52) hrough he iniial condiions (,, ) y x φ be soluions of he NODE sysems: (51) x φ respecively Then for y,, x = x,, x, φ + U, s, y s,, x g s, y s,, x ds y(,, x ) = x(,, x ) + x, s, y( s,, x, φ ) g s, y( s,, x, φ ) ds (511) x Noice: As we see in Equaions (51) and (52), he perurbaion causes he changes on he iniial condiions a = and x = x and on he iniial funcion φ Up o he iniial condiion boh funcions x( ) and y( ) have he pas hisory and hey will be idenical a = Proof: Variaions of unperurbed soluion x() and perurbed soluion y() when he iniial condiions of he moving objec change wih respec o he independen variable s [, ] can be demonsraed by he following chain rule formula d x x dy xsys,, ds = + s y ds = xsys,, + xsys,, y ( s) s v = xsys,, + xsvs,, y ( s) s y Subsiue (53) and he perurbed soluion y'(s) from (52) = V sys,, + xsys,, f( sys, ) + gsys (, ) v Subsiue for V(s) by (57) = U sxs,, f( sxs, ) + U sys,, f( sys, ) + gsys (, ) As a resul of hese subsiuions we can inegrae he following relaion on s [, ] : d xsys,, = U sys,, gsys (, ) ds Now inegrae = ( ) xsys,, d s U sys,, gsys, ds x,, y x[,, x] = x, s, y( s,, u) g( s, y( s) ) ds x x,, y = x[,, x, φ] + x, s, y( s,, x, φ) g( s, y( s) ) ds x Now he quesion is his: wha is x,, y? The unperurbed soluion x() wih he perurbed soluion as he iniial condiions = and x = u = y Thus x y,, = y (,, u), and he above relaion will be concluded as follows: y[,, x, φ] = x[,, x, φ] + x, s, y( s,, x, φ) g( s, y( s) ) ds x This is a conclusion of he Alekseev ype Theorem for Nonlinear Operaor DOI: 14236/am Applied Mahemaics

14 Differenial Equaions 6 Generalized Alekseev s VOP of NODE wih Iniial Funcions When he operaor A is unbounded, one canno expec o derive he same resul for any x X since x (,, x, φ ) in general is no differeniable wih respec o We also need he differeniabiliy of he soluion x (,, x, φ ) wih respec o he parameers (, x, φ ) The variaion of parameers was invesigaed wih respec o he parameers (, x ) in he previous secion and i will be invesigaed in his secion wih respec o φ The relaion (52) has been generalized in Hale 1992 for infinie dimensional 1 variaional operaor when f C 1x(,, x, ) = 3x(,, x, ) Ax + f (, x, T( x) ) Assume ha y (,, x, φ) W(,, x, φ) exiss, hen φ dw d y dy f y,, φ φ ( ) W = = = = Ax + f y T y = = f y W d d φ φ d φ y φ DOI: 14236/am Applied Mahemaics (61) This argumen can lead o he fac ha if he operaor f Lip( I, Y Z; Y ) hen he soluion o he sysem ( φ ) W = fy y,,, W, W = I, for y, (62) has a unique soluion The sysem (62) is called he variaional equaion Noice ha for all, y (,, φ) = φ hen U,, φ = φ y,, φ = φ φ = I Using he chain rule for absrac funcions, we ge d y,, φ + s( ψ φ) = U,, φ + s( ψ φ) ψ φ ds Thus by inegraing he sysem, 1 ( ψ) ( φ) = ( φ + ( ψ φ) )( ψ φ) y,, y,, U,, s ds (63) Proposiion 61 (Alekseev's Theorem for Operaor Differenial Equaions): Suppose f : U R X X and g: U R X X are of class C 1 If x (,, φ ) is he soluion of Equaion (51) hrough he iniial sae, (, x, φ ) and y (,, x, ψ ) is he perurbed soluion of y = Ay+ f (, y, Ty ) + g(, y ), > (64) y = φ, for hrough (, φ ), hen, for any φ D( A) D( φ Y), we have y(,, x, φ) = x(,, x, φ) + x( sy,, ( s,, φ) ) g( sy, ( s,, φ) ) ds (65) φ Proof: For φ D( A) D( φ Y) assume ys xs,, φ ( s), = Differeniaing

15 wih respec o he firs parameer ( s, φ ( s) ), xs (,, φ( s) ) xs (,, φ( s) ) φ ( s) y' ( s) = + s v s = 1xs (,, φ( s) ) +2xs (,, φ( s) ) x ( s) Using he relaion (53) (,, ) = 1 (,, ) +2 (,, ) = 2x(, s, y( s) ) Ay( s) + f ( s, y( s), T( y)( s) ) + 2xsys (,, ) Ay+ f( sys,, T( y)( s) ) + gsys (, ) =2xsys (,, ) gsys (, ) x'sys xsys xsys y s Inegraing from o, we will conclude ha Therefore, ( ) ( ) = ( ) ( ) 2 xy,, x,, y xsys,, gsys, ds ( φ) = ( φ) + ( ) ( ) 2 y,, x,, xsy,, s g sy, s ds This proves he heorem for φ D( A) D( φ, Y) Assume ha for he iniial funcion φ Y he maximal inerval is [, a ) for he soluion y (,, φ ) For τ le us define ξ B = { y Y : y φ δ δτ ( τ) } and operaors F1( : ) B C( [, τ], Y), F2( : ) B C( [, τ], Y) δτ by he following relaions F( φ ) = x (,, φ ) + xsys (,, ) gsys (, ) ds 1 2 A ( ) A ( s) ( φ ) = φ + ( ) + ( ) F2 e e f sys,, T y s gsys, ds Since boh operaors are well defined and coninuous on B δτ and coincide on D( A) B δτ, hey mus coincide on B δτ This proves he heorem The nex heorem will provide he variaion of parameers formula for operaor differenial equaions Theorem 61 (Variaion of Parameers for NODE): The soluion of he sysems (51) and (52) saisfy he following y (,, x, φ ) = x (,, x, φ ) + W sy,, ( s,, φ ) g( s, y( s,, φ )) ds x (,, φ ) where W s,, y( s,, φ ) = (,, ) 1 W v exiss φ δτ (66) and assume he inverse marix Proof: In a variaion of parameers, we will deermine a funcion v o saisfy he differenial equaion for perurbed soluion y such ha DOI: 14236/am Applied Mahemaics

16 ( φ ) ( ) φ y,, = x,, v, for> v =, for (67) is a soluion process for he sysem (57) From he sysem (57) and differenia- ion of (59) we will ge Since (,, ) (,, ) (,, ) x v x v v y = + v = Ay+ f, y, T y + g, y, > ( ) ( ) x v is a soluion of (55), hen (, ) g y (,, v ) x = v φ 1 I can be observed ha he inverse marix W (,, v ) 1 v W (,, v ) g x, (,, v ) By inegraing we will obain exiss, hen (68) (69) = (61) 1 φ ( ) ( ) v = + W s,, v s g sx, s,, v s ds (611) Differeniaion wih respec o he second independen variable s when s implies ha ( ) ( ) v( s) ( ) d x,, v s x,, v s x,, v = = v ds v s φ (,, ) = W v s v Subsiuing (61) for v' ( ) we ge he following for he righ hand side 1 = W(,, v( s) ) W ( s,, v( s) ) g s, x( s,, v( s) ) which implies 1 (,, ) = (,, φ ) + (,, ) (,, ) g( sxs, (,, v( s) ) ds x v s x W v s W s v s Using variaion definiion (67) in he above relaion, we will now ge he variaion of parameers for nonlinear operaor differenial Equaion (65) 1 (,, ) = (,, φ ) + (,, ) (,, ) g s, y( s,, φ ) ds y x x W v s W s v s (612) The operaor T in he differenial Equaion (51) and (57) could be any delay, inegral, composiion, or Caresian produc of nonanicipaing and Lipschizian operaors which will affec he nonperurbed soluion (,,, φ ) x x hrough hese changes The variaion formula (514) will be effeced by he operaor T Assuming ha he variaion of parameers is given, we will invesigae some of he properies of his formula hrough he following conclusions for paricular DOI: 14236/am Applied Mahemaics

17 cases Corollary 61: Suppose ha he condiions of Theorem 61 saisfy and guaranee he exisence and uniqueness of he soluion of he sysem (52) Assume also φ( ) = φ is he iniial sae of he sysem x = Ax Then he relaion (57) will be (,, φ) = (,, φ) + (,, ) d + ( ( φ) ) ( ( φ) ) y x f sx s T x s s Proof: Assuming ha (,, ) x sy,, s,, x, g sy, s,, x, ds x (613) x φ is a soluion o he homogeneous equaion x Ax =, hen by he direc inegraion of he sysem (51) and applying he variaion of parameers formula (53) o he nonlinear sysem (52), we will ge he formula (613) Corollary 62: Suppose ha he condiions of H1 hrough H4 guaranee he exisence and uniqueness of he soluion of (51) and (56) Assume also a paricular case when f and gx (, ) g, hen he Alekseev s formula (57) deduces he variaion of parameers formula =Φ + Φ x, x s, g s ds (616) for linear differenial equaion: x' = A x + g Proof: Assuming ha (,, ) x A x x x is a soluion o he homogeneous =, hen he fundamenal marix of he homogeneous sysem will be By considering he following (,, ) = Φ (, ) x x x (,, ) =Φ (, ) = and x x x x (,,,, ) = Φ (, ) (,, ) x sy s x s y s x we conclude ha xsys,, (,, x) =Φ s, ys,, x =Φ s, x =Φ s, x x x {I can be verified ha (,, ) (617) y s x = x} Noice ha he deerminisic funcion f is idenically equal o zero f This concludes he variaion of consans for linear sysem (617) Corollary 63: Suppose ha in he differenial Equaion (51) A =, hen he general soluion of (616) abou he equilibrium soluion y will be (,, ) = +, d (, (,, )) d + (618) A =, hen he soluion x (,, x) = x is a y x x f s x s s g s y s x s Proof: Since he operaor consan funcion Therefore x (,, x) = I To find he perurbed soluion x of he sysem (56), we use he conclusion of he Proposiion 51 for unperurbed soluion of he sysem (51) o obain he relaion (618) DOI: 14236/am Applied Mahemaics

18 We will sudy he variaion of parameers for operaor differenial equaions disurbed force operaor funcions These nonlinear operaors can involve he following ypes: delay, inegrals, composiion, or caresian producs of all nonanicipaing and Lipschizian operaors 7 Conclusions Assume ha A is a marix funcion on I Y ino he space M ( IZ, ) Suppose ha Φ represens he fundamenal marix soluion process of a differenial equaion Then, x = A x x = x (71) A Φ = Φ, Φ = uni marix (72) exp deφ = ra( s) d s, I (73) A mehod of variaion of parameers for he sysems (71) - (73) is presened by G S Ladde and V Lakshmikanham, 198 Suppose A is Lebesgue summable from I ino M ( IZ, ) and le f Lip( I, Y; Z ) be a perurbaion in he sysem (61) hen he soluion process y = y (,, y) of he following nonlinear sysem y = A y + f, y, y = y (74) ( ) will saisfy he following inegral equaion 1 = ( ) + Φ Φ ( ) y x,, x s f sy, s ds (75) for all Furher sudy of his general form of he variaion of parameers for nonlinear operaor differenial equaions should be very ineresing These nonlinear operaors can involve varieies of many ypes of operaors like: delay, inegrals, composiion, or Caresian producs of all nonanicipaing and Lipschizian operaors A classical nonlinear sysem ype y' = f (, y ) for > and y = y in (11) is well known and exensively sudied The variaion of parameers discovered by Alekseeve is a grea ools o sudy his kind of nonlinear sysem and use his conclusion for sabiliy and asympoic behavior of a nonlinear sysem The soluions o a nonlinear operaor differenial equaions of ype (12) which include all operaors T saisfying nonanicipaing and lipschizian condiions also reviewed here, have a huge range of applicaion For operaor in his paper we proved and demonsraed a general form of Alekseeve Theorem when a non linear sysem (51) includes a C -semigroup of opearor A All imporan condiions in (H1) hrough (H4) are connecing he nonanicipaing propery of T, semigroup propery of A, and Lipschizian propery of f The variaion of parameers helped us o find he soluion o he pururbed sysem This perurbed soluion for nonanicipaing dynamic sysems DOI: 14236/am Applied Mahemaics

19 will help us in he fuure o sudy he sabiliy and asympoic behavior of he sysem Two major issues relaed o he Variaiion of Parameers can be developed for Nonlinear Operaor Differenial Equaions Firs, is he numerical algorihm and compuaional program o produce he soluion o such a general form of nonlinear variaional of parameers mehod Second, generalize he sabiliy applicaion o nonlinear sysem o operaor differenial equaions References [1] Ahangar, RR (1989) Nonanicipaing Dynamical Model and Opimal Conrol Applied Mahemaics Leer, 2, hps://doiorg/1116/ (89)916-7 [2] Ahangar, RR (1986) Exisence of Opimal Conrols for Generalized Dynamical Sysems Saisfying Nonanicipaing-Operaor Differenial Equaions Disseraion, Deparmen of Mahemaics, The Caholic Universiy of America, Washingon DC [3] Ahangar, RR and Salehi, E (22) Auomaic Conrols for Nonlinear Dynamical Sysems wih Lipschizian Trajecories Journal of Mahemaical Analysis and Applicaion, 268, 4-45 hps://doiorg/116/jmaa276 [4] Ahangar, RR (25) Opimal Conrol Soluion o Operaor Differenial Equaions using Dynamic Programming Proceedings of he 25 Inernaional Conference on Scienific Compuing, Las Vegas, 2-23 June 25, [5] Ahangar, RR and Salehi, E (26) Opimal Auomaic Conrols Soluion o Nonlinear Operaor Dynamical Sysems Proceeding of he Inernaional Conference on Scienific Compuing, Las Vegas, June 26 [6] Ahangar, RR (28) Opimal Conrol Soluion o Nonlinear Causal Operaor Sysems wih Targe Sae FCS (Foundaions of Compuer Science), WORLD COMP, [7] Naylor Arch, W and Sell, GR (1982) Linear Operaor Theory in Engineering and Science Applied Mahemaical Sciences, Vol 4, Springer-Verlag, Berlin [8] Hale Jack, J (1977) Theory of Funcional Differenial Equaions Spring Verlag, Berlin [9] Driver, RD (1977) Ordinary and Delay Differenial Equaions Applied Mahemaical Sciences, Vol 2, Springer-Verlag, Berlin hps://doiorg/117/ [1] Yang, K (1996) Delay Differenial Equaions wih Applicaions in Populaion Dynamics Mahemaics in Science and Engineering, Vol 191, Academic Press, Inc, Cambridge [11] Bogdan, VM (1981) Exisence and Uniqueness of Soluion for a Class of Nonlinear Operaor Differenial Equaions Arising in Auomaic Spaceship Navigaion NASA Technical Adminisraion, Springfield [12] Bogdan, VM (1982) Exisence and Uniqueness of Soluion o Nonlinear Operaor Differenial Equaions Generalizing Dynamical Sysems of Auomaic Spaceship Navigaion Nonlinear Phenomena in Mahemaical Sciences, Academic Press, Cambridge, [13] Alekseev, VM (1961) An Esimae for he Perurbaions of he Soluion of Ordinary Differenial Equaions Vesn Mosk Univ Ser 1, Mah Mek, No 2, [14] Lakshmikanham, V and Leela, S (1981) Nonlinear Differenial Equaions in Absrac Spaces Pergamon Press, Oxford DOI: 14236/am Applied Mahemaics

20 [15] Lakshmikanham, V and Ladas, GE (1972) Differenial Equaions in Absrac Space Academic Press, Cambridge [16] Lashmikanham, V and Ladde, GS (198) Random Differenial Equaions Academic Press, Cambridge [17] Hale, J, Arriea, J and Carvalho, AN (1992) A Damped Hyperbolic Equaion wih Criical Exponen Communicaions in Parial Differenial Equaions, 17, [18] Brauer, F (1966) Perurbaions of Nonlinear Sysems of Differenial Equaions Journal of Mahemaical Analysis and Applicaions, 14, hps://doiorg/1116/22-247x(66)921-7 [19] Brauer, F (1967) Perurbaion of Nonlinear Sysems of Differenial Equaions II Journal of Mahemaical Analysis and Applicaions, 17, hps://doiorg/1116/22-247x(67) DOI: 14236/am Applied Mahemaics