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 International Journal of Modern Nonlinear Theory and Application, 2013, 2, 153-160 http://dx.doi.org/10.4236/ijmnta.2013.23020 Published Online September 2013 (http://www.scirp.org/journal/ijmnta) Approximate Controllability of Fractional Order Retarded Semilinear Control Systems Simegne Tafesse1, Nagarajan Sukavanam2 1Department of Mathematics, Haramaya University, Dire Dawa, Ethiopia 2Department of Mathematics, Indian Institute Technology Roorkee (IITR), Roorkee, India Email: wtsimegne@gmail.com Received June 4, 2013; revised July 6, 2013; accepted July 25, 2013 Copyright © 2013 Simegne Tafesse, Nagarajan Sukavanam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper, approximate controllability of fractional order retarded semilinear systems is studied when the nonlinear term satisfies the newly formulated bounded integral contractor-type conditions. We have shown the existence and uniqueness of the mild solution for the fractional order retarded semilinear systems using an iterative procedure ap-proach. Finally, we obtain the approximate con trollab ility results of the system under simple condition. Keywords: Approximate Controllability; Fraction a l Order; Existence and Uniqueness, Retarded; Semilinear System; Integral Contractor 1. Introduction Let X and U be Hilbert spaces with the corresponding function spaces 20, :ZLX and 20, :YL U respectively. Consider the following fractional order se-milinear system   ,,0,,0CqtDxtAxtButftxtxht  (1.1) where is the fractional order qth derivative in CqtDxCaputo’s sense, 112q, A is the infinitesimal genera- tor of a Co-semigroup T(t) of bounded linear op erator on the Hilbert space X, B is a bounded linear operator from , f is a nonlinear function such that YZ:0, ,0:fCh XX , :,0txhX is de-fined as   for t,0xxt,0:Ch Xh and . The norm in X shall be denoted by  .. The corresponding linear fractional order system is given by   ,0,,0CqtDxtAxt Buttxh  (1.2) Fractional differential equations are the generalization of ordinary differential equations of arbitrary non integer orders. The fractional calculus is widely popular in the field of engineering and sciences, Shantanu [1]. Debnath [2] studied the recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelastic-ity etc. Many papers have appeared on the controllability concepts for fractional order differential systems. For instance, Wang and Zhou [3] studied complete controlla-bility of fractional evolution systems. In that paper frac-tional calculus method and fixed point theorem are used. The semigroup operator is assumed to be noncompact. Similarly controllability of fractional order impulsive neutral functional infinite delay integrodifferential sys-tems in Banach space is studied by Tai and Wang [4]. Sakthivel et al. [5] discussed the controllability of a class of control systems governed by the semilinear fractional equations in Hilbert spaces using fixed point techniques. Kumar and Sukavanam [6] studied approximate control-lability of fractional order semilinear systems with bounded delay. In that paper contraction principle and Schauder fixed point theorem are used. Zhou and Jiao [7] and El-Borai [8] studied the existence of mild solutions for fractional neutral evolution systems. The notion of integral contractor was first introduced by Altman [9] and later on it was used by many authors to study the existence and uniqueness of solution of nonlinear evolution systems. In [10] George et al. studied the existence and uniqueness of the solution and the con-trollability of the nonlinear third order dispersion equa-Copyright © 2013 SciRes. IJMNTA S. TAFESSE, N. SUKAVANAM 154 tion without delay using the bounded integr al contractor. In this paper the approximate controllability of a frac-tional order retarded semilinear system is studied. We consider the system with the nonlinear term satisfying a bounded regular integral contractor-type condition. Un-der this condition we show first the existence and uniqueness of the mild solution of the system. Then us-ing some simple condition we obtain the approximate controllability results. 2. Preliminaries and Basic Assumptions Some notions of fractional order differential equations are given as follows. Definitio n 2 .1: 1) The fractional in tegral of order  for a function f is defined as   1001d, 0, 0ttIfttsfsst provided that the right hand side is defined pointwise on [0,). Here  is the gamma function. 0tIft is called Reimann Liouvilli integr ation. 2) Riemann-Liouville derivative order of  for a func-tion :0,fR can be written as   1001d d, 0,1dtmmtmDftt sfsstmmmt  3) Let . Then the Caputo derivative of order  for a function 0,mfC:0,fR can be writte n as   101d,0,01tmCmmmtDftt sfssIfttmmm   Define the mild solution of (1.1) as [7]  1010,d2,,tqqqSttsTtsBus fsxshstqxttt ,0,10h (2.1) where  111001d, d,0qqqqqqqq qqStTtTt qTtwq    ,   111111sin,!nqnqnnqwnn  0,q q is a probability density functio defined on n0,, that is and  0, 0,q 0d1q ,  0011dd1qqqwq  (see [7]) Let M be a constant such that Tt M for all 0,t. Then the following Lemma stated as follows. Lemma 2.1: [7] Sq(t) and Tq(t) are bounded linear op-erators and  andfor all0,1qqMqSt MTttq . Definition 2.2: The system (1.1) is said to be ap-proximately controllable over a time interval [0,], if for any given 1xX and a constant  > 0, there exists a control u such that the corresponding mild solution x(t) of (1.1) satisfies 1xx. Let 0, :CC X denote the Banach space of continuous functions on 0,J with the standard norm max :0Cxxtt for x  C.By con-  sidering the nonlinear initial value problem of the form ,,0, 0xt Ftxtx  in Banach space X [9] introduced a bounded integral contractor in the following definition. Definition 2.3: [9] A function f is said to have a bounded integral contractor if :JXBLX is a bounded operator and there exists a positive number  such that for any w, y  C   00max ,,d,,tCtftwtytswsyssf twttwtyty  (2.2) Copyright © 2013 SciRes. IJMNTA S. TAFESSE, N. SUKAVANAM 155 Definition 2.4: [9] A bounded integral contractor  is said to be regular if the integral equation  0,dtztytsws yss (2.3) has a solution y in C fo r ever y w, z  C. We define a bounded integral contractor operator q for the fractional order system without delay in a similar fashion as: Definition 2.5: Suppose :qJXBLX is a bounded operator and there exists a positive number  such that for any w, y  Z we have    10,,d,,tqqqqftwtytt sTt sswsyssftwttwtytyt(2.4) Then we say that f has a bounded integral contractor with respect to the operator Tq(t). qDefinition 2.6: A bounded integral contractor q is said to be regular if the integral equation  10,tqqqztyttsTt sswsyss has a solution y in Z for every w, z  Z. Let us assume 1,,qLB XtwttJ wZ d (2.5) Let 2,:hZLhX . Now we define a new bounded integral contractor-type operator so as to make compatible with retarded system as follows. qhDefinition 2.7: Let :,qhh hJCBLCX be such that for ,0h  10,d,00, ,0tqqqhssqttttsTtsswystwyth  (2.6) If for any w, y Zh  ,,,qqthftwt hythwytftwt htwt h yt hyt  , (2.7) then f is said to have an integral contractor-type operator qh. It can be seen easily if , then 0qh 0qh and f is Lipschitz continuous. Let us assume that 2,,,,hqtwht hLB CXtJ wZ and 12max ,. Similar to Equation (2.5) con- sider the integral equation of the form   10,d, 0,,, ,0tqqtqhsstytsTtsswystzwztt   tthCh (2.8) Definition 2.8: If is bounded and the integral Equation (2.8) has a solution yt in Ch for every zt, wt  Ch, then is called regular on Ch. qhqh Now we assume the following conditions: 1) The semigroup T(t), t  0, is compact and T(t) = 0 for t  [–h,0) 2) f has a bounded regular integral contractor-type qh on Ch i.e. ,,,qqtttttthtt tftwyw yftwtw yy 3) f is uniformly bounded, i.e. there exists M1 > 0 such that 1,tftx M 4) The fractional order linear system corresponding to (1.1) is approximate controllable 5) For all y  Y there exists a constant k>0 such that Byky Lemma 2.2: [7] If the assumption (1) is satisfied, then qSt and qTt are also compact operators for every t>0. 3. Main Result Define the solution mapping W: Y  Z by Wu = x where x(t) is the unique mild solution of (1.1) corresponding to Copyright © 2013 SciRes. IJMNTA S. TAFESSE, N. SUKAVANAM 156 the control u(t). Lemma 3.1: The solution mapping W is compact. The procedure of the proof is quite similar to Lemma 1 in [11]. Theorem 3.1: Under the assumptions (1) and (2) the abstract fractional order semilinear system (1.1) has a unique mild solution if 11qMq . Proof: First we show the existence of the mild solution. Consider the following iteration procedure to produce sequences {xn(t)} and {yn(t)} in X. For –h  t      1001000d,0,,,d,00, ,0tqqqtqnqsnnStts TtsBusstxtttxttsTtsfsx shsxttytth  0h (3.1)      1101100011000,d,d ,d,d, dtqqnnqhsn snnttqqqnqhsnsnnqsnttqqqqsn qhsnsnxtxttsTts sxysytxttsTt ssxysxttsTtsfsxsxtt sTt sfsxst sTt ssxys xt    (3.2)     111 0101001010,d,d ,,dtqnnq sntqqqsntsn sntqqsnytxttsTtsfsxsxttsTtsfsxssxy txtts Ttsfsxsxt     (3.3) For every t  [0,] and x  C we can define xt  Ch such that , 0txxth. Hence, we consider the following formul ation for the sequences {xtn} and {ytn}.  1000d,,tqqqtStts TtsBusstxtt   ,00h   1001,d()tqtntnqsn tqtntnttn tntnyx tsTtsfsxsxxxy xy    (3.4) Then from (3.3) we get    111001010,d, d,d,,,,ttqqqnqsnqhsnsntqqqsnsntsn sntqqqqsnsntsn snsnhsn sny tts TtsfsxstsTtssxystsTtsfsxyxystsTtsfsxysx yfsxsx ys    d Applying the definition (2.7) with yt: –ysn and wt: xsn we obtain the following inequality. Copyright © 2013 SciRes. IJMNTA S. TAFESSE, N. SUKAVANAM 157      1101011000,,,,,dmax11htqqqnqsnsntsn snsnhsn sntqqqqtttttthttttqqsn nChyttsTtsfsxyxyfsxsxystsTtsf swywyfswswysMq Mqts ystsyssqq   ,ddd      11021100maxmax d1max max11max1tqnnht htqqnnht htnqhMqytyt tssqMMytytqqMytq     (3.5) Since 11qMq , as n   w e ha ve 110lim01nqnnMythq,  Hence the for all t  [0,]. lim 0nny Now we show the convergence of the sequence xn(t) to the mild solution of the system (1.1). From (3.2) we have 11qtntnttntntnqnn ttntnnxxyxyxtxt ytxy    Note that if 0t then 10nnxt xt  and 0nyt tThus for 0     110100,d,max dhtqqnn nqhsnsnCtqqnqhsnnhxtxt yttsTtssxyytts Ttssxyss    s Define :tt then we get        1100100100,max dmax d11max1(1)tqqnn nqhsnnttqnntnqqtxtxtytts TtssxytsMqytyt tssqMM ytqq     Consider the sequence of {xn(t)} in X. For a positive integers m and n, assume m < n. Then from the above proce-dures we have      112 111001max11nmnnn nm mkqqntkmxt xtxtxtxtx txtxtMMytqq          Copyright © 2013 SciRes. IJMNTA S. TAFESSE, N. SUKAVANAM 158 Clearly the right hand side is the tail of a convergent series for sufficiently large m and n since 11qMq . Thus the sequence xn is a Cauchy sequence in C hence the sequence converges to say x in X. Therefore from (3.1) we have      100100100limlimlim, d,0,0,d,dtqnnq snnnntqqstqqsytxtts Ttsfsxsxttxtt sTt sfsxsxtxtxtt sTt sfsxs    Hence x(t) is a mild solution of th e system (1.1). Now let us show the uniqueness of a mild solution. Let x1(t) and x2(t) be the two mild solutions of (1.1) with control u. By the regularity of the integral co ntractor type qh with 2tt tzxx1 there exists yt in Ch such that     110121 1011,d,dtqqttq hsstqqtttqhsqtthttzy tsTtssxyssxxy tsTtssxyxy xy     s But   1212 10111 110110,,d,,,dtqqsstqqqqsstssshstqqqhssxtxttsTts fsxfsxstsTtsfsxyx yfsxsxystsTtssx ys ,ds     1211 10111011000,,,d ,ddd11max1hhtqqqsstsstqqqhssq hssttqqssCCqnh1sxtxtt sTt sfsxyxyfsxsxystsTtssxysMq Mqtsys tsysqqMysq        Note that if t +   [–h,0], then 1xtt xt2   0ty and we put . Moreover, and . 0yt  112 20, 0ttxt xxt xyt Hence  121 10,dtqqqhssxtxtyttsTts sxy ss Then we have   12110,dtqqqhsytxtxttsTt ssxys Copyright © 2013 SciRes. IJMNTA S. TAFESSE, N. SUKAVANAM Copyright © 2013 SciRes. IJMNTA 159This implies that   121 1010100,, ,1d1max d1htqqsshstqsCtqnhMqyttsf sxfsxsxysqMq ts y sqMq tsys sq  ds Since the integrand in the right hand side is positive then the integral is an increasing function of t. Hence the abstract retarded semilinear control system (1.1) is approximate controllable. Proof: Let w(t) be the solution of the linear control system (1.2) corresponding to the control v and consider the following system.   10maxmax d1tqnhrt hstMqyrtsy ssq   By Gronwall’s inequality y(t) becomes zero for all t  0. Since  ,,,0,,0CqtttDxtAxtftxBvt ftwtxh  (3.6)  0max maxhtChhtyyt yt  This implies that yt is zero. Therefore, x2 = x1 whic h means that the mild solution of (1.1) is unique. This completes the proof of the theorem. Note that the above system is the same as the system (1.1) in which Bu is replaced by ,tBvtftw. We define the mild solution of the linear system (1.2) as Theorem 3.2: Consider the assumptions (1)-(5), then   100d,,tqqqStts TtsBvsstwttt,0,0h  (3.7) And the mild solution of the system (3.6) is   100,,,,tqqq ssSttsTtsBvsfswfsxs txtttd,0,,0h  (3.8) But by the regularity condition with there exists yt such that ttzwxt 1110,dqtttttssqttttsstqqttqh sszwxy xyywx xywxtsTtssxys  Taking the norms we get 10d1hhtqttt sCCMqhCywxts y sq   By Gronwall’s inequality exp 1hhqtttCCMywxq  (3.9) Subtracting (3.8) from (3.7) and taking norm on both sides we get S. TAFESSE, N. SUKAVANAM 160   1010101010,, ,,d,, ,1d1d1hhhtqqttqssh ssCtqqqhsstqqsshsstqsCtqsCwxtsTtsfswfsxsxy sts TtssxysMq tsf swfsxsxysqMq ts ysqMq ts y sq   dd (3.10) From (3.9) and (3.10) it follows that    10100exp d11exp maxd11h hhtqqtt ssC CtqqChMq Mwxts wxsqqMqM tsws xssqq     Since the integrand in the right hand side is positive then the integral is an increasing function of t. He nce    10 00supexpsup d11tqqttrt stMq Mwxts wsxssqq    Again Gronwall’s inequality implies wt xt for all t [–h,]. Under condition (5) the equation has a solution u(t). Therefore, the fra- ctional order retarded semilinear control system (1.1) is approximately controllable with control u. ,tBuBvft xREFERENCES [1] S. 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