International Journal of New Technology and Research (IJNTR) ISSN: , Volume-4, Issue-2, February 2018 Pages 52-57

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1 Inernaional Journal of New Technology and Research (IJNTR) ISSN: , Volume-4, Issue-2, February 28 Pages Relaive Conrollabiliy of Fracional Inegrodifferenial Sysems in Banach Spaces wih Disribued Delays in he Conrol Dr Sir Paul Anaeodike Oraekie Absrac In his work, Fracional Inegro-differenial Sysems in Banach Spaces wih Disribued Delays in he Conrol of he form: d n x() d n = Ax d θ H, θ u θ h f, x, g, s, x s ds is presened for conrollabiliy analysis. Necessary and Sufficien Condiions for he sysem o be relaively conrollable are esablished. The Se Funcions upon which our resuls hinged were exraced. Uses were made of: Unsymmeric Fubini heorem, he Conrollabiliy Sandard and he Concep of Fracional Calculus o esablish resuls. Index Terms Relaive Conrollabiliy, Fracional Inegro-differenial Sysems,Banach Spaces,Fracional Calculus, Unsymmeric Fubini Theorem, Posiive Definie. I. INTRODUCTION According o Bonilla eal (27), fracional differenial equaions emerged as a new branch of mahemaics. Fracional differenial equaions have been used for many mahemaical models in Sciences and Engineering. The equaions are considered as an alernaive model o nonlinear differenial equaions. The heory of fracional differenial equaions has been sudied exensively by many auhors Dulbecco (996) and Lakshimilkanhan(28).While he problems of sabiliy for fracional differenial sysems are discussed in Bonne (2), Nec(27), Balachandran(29). Apar from sabiliy, anoher imporan qualiaive behavior of a dynamical sysem is conrollabiliy. Sysemaic sudy of conrollabiliy sared over years a he beginning of he sixies when he heory of conrollabiliy based on he descripion in he form of sae space for boh ime-varying and ime-invarian linear conrol sysems are carried ou. Roughly speaking, conrollabiliy generally means ha, i is possible o seer a dynamical conrol sysem from an iniial sae x(o) of he sysem o any final sae x() in some finie ime using he se of admissible conrolsoraekie(23).the concep of conrollabiliy plays a major role in boh finie and infinie dynamical sysems, hais sysems represened by ordinary differenial equaions and parial differenial equaions respecively. So i is naural o exend his concep Dr Sir Paul Anaeodike Oraekie, Deparmen of Mahemaics, Faculy of Physical Sciences Chukwuemeka Odumegwu Ojukwu Universiy, Uli-Campus. o o dynamical sysems represened by fracional differenialequaions. Many parialfracional differenial equaions and Inegro-differenial equaion can be expressed asfracional differenial equaions and Inegro-differenial equaions in Banach spaceselsayeed (966)). There exis many works on finie dimensional conrollabiliy of linear sysems Klamka (993)) and infinie dimensional sysems in absrac spaces Curain (978). The conrollabiliy problems of nonlinear sysems and Inegro-differenial sysems wih delays have been carried ou by many researchers in boh finie and infinie dimensional spaces Balachandran (989)) and Balachandran (22). Conrollabiliyfracional differenial sysems in finie dimensional space have been sudied by Chen (26) and Shamardan(2).While Balachandran (29) sudied Conrollabiliy offracionalinegro-differenial sysems in Banach spaces. In his paper, we sudy he relaive conrollabiliy of fracionalinegro-differenial sysems in Banach spaces wih disribued delays in he conrol he conrollabiliy sandard of dynamical conrol sysems and he unsymmeric Fubini heorem o esablish resuls. II. PRELIMINARIES Le n be a posiive ineger and E =, be e real line. Denoe E n = e space of real n uples called e Euclidean space wi norm denoed by.. If J =, is any inerval of E, L 2 is Lebesgue space of square inegrable funcions from J o E n wrien as L 2,, E n. Le > be posiive real number and le C,, E n be e Banac space of coninuous funcions wi norm of uniform convergence defined by φ = supφ s ; φε C,, E n. If x is a funcion from, o E n, en x is a funcion defined on e delay inerval, given as x s = x s ; s,,,. Definiion2. Balachandran(29) Te Riemann Liouville fracional inegral operaor of order β > of funcion f C n, n is defined as: I β f = ρ(β) s β f s ds Definiion 2. 2 fracional derivaive 52

2 Ife funcion f C m and m is posiive ineger, en we can define e fracional derivaive of f in e Capuo sense as: d n f() d n = ρ mn s mn f m s ds ; m < n m. If m =, en m < n m becomes < n. Ten d n f d n = ρ n = = s n f s ds ρ n ρ n = ρ n s n f s ds s n f s ds f s s n ds, were f s = df(s) and f is an absrac funcion wi values in X. ds 2.. VARIATION OF CONSTANT FORMULA Consider e following sysem represened by e fracional Inegro differenial equaions in Banac spaces wi disribued delays in e conrol of e form: d n f d n = Ax d θ H, θ u θ f, x, g, s, x s ds. x = x ; J =,. were e sae x. akes values in e Banac space X, < n <, e conrol funcion u L 2,, U, a Banac space of admissible conrol funcions wi U as a Banac space. H, θ is an nxm marix funcion coninuous a and of bounded variaion in θ on,, h > for eac, ; >. The inegral is in he Lebesgue Sieljes sense and is denoed by he symbol d θ. And he nonlinear operaors f: JxXxX X, g: xx X are coninuous; =, s : s. If, Gx = g, s, x s ds, hen he equaion. becomes equivalen o he following nonliear inegral equaion x = x s n Ax s ds θ ds s n d θ H, θ s n f, x, Gx s ds And he mild soluion of he sysem (.) is given by x = T x s s n T d θ H, θ s n T u θ ds u s f, x, Gx s ds (.2) which is similar o he concep defined in he book of Pazy(983). For e limiing case, n, e above sysem.2 reprsenaion becomes x = T x T s d θ H, θ u θ ds T s f, x, Gx s ds (.3). Wic is e mild soluion of dx d = Ax d θ H, θ u θ f, x, Gx s Wi iniial condiion x = x X. Analogus o e convenional conrollabiliy concep. Acareful observaion of e soluion of e sysem. given as sysem.2 sows a e values of e conrol funcion u for, ener e definiion of complee sae ereby creaing e need for an explici variaion of consan formula. Te conrol in e 2nd erm of e formula.2, erefore, as o be separaed in e inervals, and,. To acieve is a 2nd erm of sysem.2 as o be ransformed by applying e meod of Klamka as conained in Chukwu 992. Finally, we inercange e order of inegraion using e Unsymmeric Fubuni eorem o ave 53

3 Inernaional Journal of New Technology and Research (IJNTR) ISSN: , Volume-4, Issue-2, February 28 Pages x = T x d Hθ s n T s n T s d θ H s s H s, θ u s θ ds s n T s f, x, Gx s ds (2.) x = T x d Hθ θ θ s n T s H s θ, θ u s θ θ ds s n T s f, x, Gx s ds 2.. Simplifying sysem 2., we ave x = T x d Hθ d Hθ s n T s f, x, Gx s ds θ θ s n T s H s θ, θ u s ds s n T s H s θ, θ u s ds (2.2) Using again e Unsymmeric Fubuni Teorm on e cange of e order of inegraion and incorporaing H as defined below: H H s θ, θ, for s s θ, θ = (2.3), fors Sysem (.2.2) becomes x = T x d Hθ s n T s f, x, Gx s ds θ θ, θ u s ds s n T s H s θ, θ u s ds (2.4) Inegraion is sill in e Lebesgue Sieljes sense in e variable θ in H. For breviy, le α, s = T x β, s = d Hθ μ, s = s n T s f, x, Gx s ds (2.5) θ s n T s H s θ, θ u s ds (2.6) s n T s d θ H s θ, θ u s ds (2.7) Subsiuing equaions 2.5, 2.6 and 2.7 in equaion 2.4, we ave a precise variaion of consan formula for e sysem (.) as: x, x, u = α, s β, s μ, s ds BASIC SET FUNCTIONS AND PROPERTIES Definiion 2.2. (Reachable se) Te reacable se of e sysem. denoed by R(, ) is given as R(, ) = s n T s d θ H s θ, θ u s ds u U; u j ; j =,2,, m Were U = u L 2,, E m Definiion 2.2.2(Aainable se) Te aainable se of e sysem. denoed by A, is given as A, = x, x, u : u U; u j ; j =,2,, m, were U = u L 2,, E m Definiion (Targe se) Te Targe se for e sysem(.) denoed by G, is given by G, = x, x, u τ >, for some fixed τ and u U Definiion (Conrollabiliy grammian or Map) Te conrollabiliy grammian orconrollabiliy map of e sysem. denoed by W, 54

4 is given as W, = μ, s μ, s T, were T denoes marix ranspose Definiion 2, 2.5 (Posiive Definie) Te conrollabiliy grammian or map W is said o be posiive definie if W varnises only a e origin and W x > for all x, x D, were D = x E n x r ; r > E n 2.3. RELATIONSHIP BETWEEN THE SET FUNCTIONS We sall firs esablis e relaionsip beween e aainable se and e reacable se, o enable us see a once a propery as been proved for one se funcion, en i is applicable o e oer. From equaion (2.4), A, = η R(, ), for u U;,, were, η = α, s β, s. This means ha he aainable se is he ranslaion of he reachable se hrough he origin η E n. Using e aainable se, erefore, i is easy o sow a e se funcions possess e properies of convexiy, closedness, and compacness. No alone, e se funcions are con enuous on, o e meric space of compac subses of E n CHUKWU 988 andgyori 982 gave impeus for adapaions of e proofs of ese properies for sysem(.). Definiion 2.3. (Relaive conrollabiliy) Te sysem. is relaively conrollable on e inerval [, ] if A, G, φ, > [, ] Definiion 2.3. (Properness) Te sysem. is proper in E n on e inerval, if spanr, = E n i. e. if, s n T s d θ H s θ, θ = a. e, C = ; C E n. Definiion 2.3. (Complee sae) We denoe e complee sae of sysem. a ime by z = x, u. Ten, e iniial complee sae of sysm. a ime is given by z = x, u III. MAIN RESULTS Te issue of relaive conrollabiliy of Neural Volerra Inegro differenial Equaions ave been seled in Balachandran 992, Balachandran 989, Balachandran 997. From e resuls of ese sudies e following equivalen saemens emerge. Theorem 3..(Necessary condiions) Consider he sysem d n x() d n = Ax d θ H, θ u θ f, x, g, s, x s ds (3.) x = x ; J =,. wi e same condiions on e sysem sparameers as in e sysem., en e following saemens are equivalen. Sysem(3.)is relaively conrollable on e inerval J =,. 2. Te conrollabiliy grammian W, of sysem 3. is non singular. 3. Sysem(3.) is proper on e inerval J =,. PROOF: = 2. Recall: Te conrollabiliy grammian W, of e sysem 3. is non singular, is equivalen o saying a W, is posiive definie, wic in urn is equivalen o saying a e conrollabiliy index of e sysem 3. is equal o zero almos everywere on e inerval,, implying a C =. i. e. s n T s d θ H s θ, θ = a. e, C = ; C E n, wic is properness of e sysem 3. since e inegral is non negaive. Tis, erefore, sowsed a is equivalen o 2, or = 2. To show ha 2 and 3 are equivalen. By e definiion of properness of e sysem 3., we ave 2 given as s n T s d θ H s θ, θ = a. e, C = ; C E n, for eac s,, en = s n T s d θ H s θ, θ s n T s d θ H s u s ds θ, θ u s ds =, for u L I follows from his las equaion (3.2) ha C is orhogonal 55

5 Inernaional Journal of New Technology and Research (IJNTR) ISSN: , Volume-4, Issue-2, February 28 Pages o he reachable se R(, ) = s n T s d θ H s θ, θ u s ds u U; u j ; j =,2,, m If we assume e relaive conrollabiliy of e sysem 3. now, R, = E n, soa C =, sowing a 3 implies 2. Or is equivalen o 2 and 2 is equivalen o 3 and vis a vis 3 o 2 o. Conversely, assume a sysem 3. is no conrollable, soa e reacable R, E n for >. Ten, ere exiss C, C E n, suc a R, =. Ifollows a for all admissible conrols u L 2 a Hence, = = s n T s d θ H s θ, θ u s ds, for u L 2 θ, θ u s ds s n T s d θ H s s n T s d θ H s θ, θ u s ds =, a. e ; s,, C. By definiion of properness, i implies a e sysem 3. is no proper, since c. Hence e sysem 3. is relaively conrollable. Theorem 3.2. (Sufficiencondiions) Consider he sysem d n x() d n = Ax d θ H, θ u θ f, x, g, s, x s ds (3.2) x = x ; J =,. wi e same condiions on e sysems` parameers as in e sysem., en e sysem 3.2 is relaively conrollable on e inerval J =, if and only if zero is in e inerior of e reacable se. PROOF Te reacable se R, is closed and convex subse of E n. Terefore, a poin y E n on e boundary implies a ere is a suppor plane π of R, roug y. i. e. y y, for eac y R,, were C is an ouward normal o e e suppor plane π. If u is e corresponding conrol o y, we ave s n T s d θ H s θ, θ s d θ H s s n T θ, θ u s ds (3.2) u s ds Foreach u U and since U is a uni sphere, he inequaliy 3. 2 becomes = sgn s n T s d θ H s θ, θ u s ds s n T s d θ H s θ, θ. ds s n T s d θ H s θ, θ s n T s d θ H s θ, θ Comparing 3.2 wi 3.3, we ave u = sgn θ, θ 3.4. s n T s d θ H s 3.3. More so, as y is on e boundary since we always ave R,. If zero were no in e inerior of e reacable se R,, en i is on e boundary. Hence, from e preceding argumen, i implies a = Soa, s n T s d θ H s θ, θ 56

6 s n T s d θ H s θ, θ = a. e., since e inegral is no zero. Tis, by e definiion of properness implies a e sysem 3.2 is no proper since C. However, if IneriorR, for > ; >, s n T s d θ H s θ, θ = a. e C = Wic is e properness of e sysem and by e equivalence in eorem 3., e relaive conrollabiliy of e sysem 3. on e inerval J =, is esablised. IV. CONCLUSION The explici variaion of consan formula for he sysem (.) visa-à-via sysem (3.) was esablished using he Unsymmeric Fubini heorem. The se funcions upon which our sudies hinged were exraced from he Mild Soluion. We esablished he necessary condiion for he sysem (.) o be relaively conrollable. This is saed and proved in heorem (3.). While he sufficien condiion for he sysem (.) o be relaively conrollable is saed and proved in heorem (3.2).Tha is, we esablished ha- a Fracional Inegrodifferenial Sysems in a Banach Space wih Disribued Delays in he Conrol, is relaively conrollable on he inerval J = [, ] if and only if zero is in he inerior of he reachable se of he sysem (.). [3]. Y.Q.Chea, H.S.Ahu, D.Xue, Robus (26), Conrollabiliy of Inegrodifferenial Sysems, Signal Processing, vol.86, pp [4]. A. Abdel-Chaan, M.K.A.Moubarak, A.B. Shamardan (2), Inegrodifferenial Nonlinear Conrol Sysems, Journal of Fracional Calculus, vol. 7.Pp [5]. A, Pazy (983), Semi-groups of Linear Operaors and Applicaions o Parial Differenial Equaions, Springer-Verlag, New York. [6].Chukwu (992), Conrol of Global Economic Growh wih Cenre Hold, Ordinary and Delay Differenial Equaion (ed.j. Weiner and J. K. Hale) London Scienific and Technical pp9-25. [7]. K. Balachandran (992), Conrollabiliy of Neural Volerra Inegrodifferenial Sysems, Journal of Ausralian Mahemaical Sociey, vol34, pp8-25. [8]. P. Balasubramaniam (997), Asympoic Neural Volerra Inegrodifferenial Sysems,Journal of Mahemaical Sysems Esimaion and Conrol, vol.7 pp-4 [9]. V. A. Iheagwam and J. U. Onwuau(25), Relaive Conrollabiliy and Null Conrollabiliy of Generalized Sysems wih Disribued Delays in he Sae and Conrol, Journal of Nigerian Associaion of Mahemaical Physics, vol.9,pp [2]. O. Hajek (994), Dualiy for Differenial Games and Opimal Conrol, Journal of Mahemaical Sysems Theory, vol.8. [2]. M. Heyman and J. R.iov (994), On Linear Pursui Games wih an Unknown Trap, Journal of Opimizaion Theory and Applicaions, vol.42, REFERENCES []. B.Bonilla, M.Rodiguez-germa, J.J.Trujillo (27), Fracional Inegrodifferenial Equaions asalernaive Models o Nonlinear Differenial Equaions, Applied Mahemaics and Compuaion vol.87 page79-88 [2]. D.Delbosco, L.Rodino(996), Exisence and Uniqueness for a FracionalDifferenial Equaions, Journal Mahemaical Analysis and Applicaions.vol.24 pp [3]. V.Lakshmilkanham (28), Theory of FracionalFuncional Differenial Equaions, Journal of Nonlinear Analysis.vol.69 pp [4]. C.Bonne, J.R.Paringon(2), Oprime Facorizaions and Sabiliy of Fracional Differenial sysems, Sysems and Conrol Leers vol.42, pp67-74 [5]. Y.Nec, A.A.Npomnyashcha(27), Linear Sabiliy of Fracional Inegrodifferenial Sysems Mahemaical Modelling of Naural Phenomena,vol.2 pp77-5 [6]. K. Balachandran, J.Y.Park(29), Conrollabiliy offracional Inegrodifferenial Sysems inbanach Spaces, Journal of Nonlinear Analysis vol.3 pp [7]. P.A.Oraekie (23), Relaive Conrollabiliy of Neural Volerra Inegrodifferenial Sysems wih Zero in he Inerior of Reachable Se, vol.4 No. pp [8]. M. A.A.El-sayeed (966), Fracional Order Diffusion Wave Equaion Inegrodifferenial, Journal of Theoreical Physics, vol.35.pp [9]. J. Klamka (993), Conrollabiliy of Dynamical Sysems, DowdiesWaver Academic Publisher []. R.F Curain, A.J. Prichard (978), Infinie Dimensional Linear Sysems Theory, Springer- Verlag, New York. []. K.Balachandran and J.P. Dauer(989), Relaive Conrollabiliy of Perurbaions of Nonlinear Sysems, Journal of Opimizaion Theoryand Applicaions vol.6, pp [2]. K.Balachandran and J.P. Dauer (22), Conrollabiliy of Nonlinear Sysems, Journal of Opimizaion Theory and Applicaions, vol.5, pp