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Applied Mathematics, 2011, 2, 1486-1496 doi:10.4236/am.2011.212211 Published Online December 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Controllability of Neutral Impulsive Differential Inclusions with Non-Local Conditions Dimplekumar N. Chalishajar1, Falguni S. Acharya2* 1Department of Mathematics and Computer Science, Virginia Military Institute (VMI), Lexington, USA 2Department of Applied Sciences Humanities, Institute of Technology and Management Universe, Vadodara, India E-mail: *falguni 69@yahoo.co.in, dipu17370@yahoo.com, chalishajardn@vmi.edu Received July 14, 2011; revised September 27, 2011; accepted October 7, 2011 Abstract In this short article, we have studied the controllability result for neutral impulsive differential inclusions with nonlocal conditions by using the fixed point theorem for condensing multi-valued map due to Martelli [1]. The system considered here follows the P.D.E involving spatial partial derivatives with α-norms. Keywords: Controllability, Neutral Impulsive Differential Inclusions, Spatial Partial Derivative, Martelli Fixed Point Theorem 1. Introduction In this paper we have discussed the controllability of nonlocal Cauchy problem for neutral impulsive differen- tial inclusions of the form 12 =0 d() ,()()() ,(); d :=0,; |=(); =1,2,; (0)() k tt k k k x tFtxht AxtButGtxht t tJ btt x Ixtkmxgxx X (1) where the linear operator () A generates an analytic semigroup ; is a multi-valued map and 0 ()t Tt G ==() () ttk k k x xt xt , 0 ()=( ) limh kk x txt k h and 0 ()= limh k x t () xth represent the right and left limits of x t=t at k respectively, 0 t, x X )\ :(FJ XPX is a multi-valued map [ is the family of all subsets of X] and ()PX )(, g CJ 2(, )uLJU U X. Also the control function , a Banach space of admissible control functions with U as a Banach space. B is a bounded linear operator from to X and X is a separable Banach space with norm . :() k I XDA; and . =1,2, ,km12 As a model we consider the following system of heat equations; ,hh C(,JJ 2 2 0 (,)(,(cos, )),(cos, ) (, ) =(,),(cos,),(cos (,0)=(,π)=0; ()()= (()), ;=1, 2,, (0,)((,))=(), 0π,[0,1] kkkk k z ztx Ftztxtx tx ztx z utxG tztxtx x x zt zt ztztI zt ttkm zxgztx zxxt ,), (2) Since F and involve spatial partial derivative, the results obtained by other authors cannot be applied to our system even if . This is the main motivation of this paper. G g(.) = 0 The existence and controllability of the following sy- stem is studied by Benchohra and Ntouyas [2] = d()(, )()()(,); d := [0,]; |=(()); =1,2,,;()=(), (,0). tt k ttk k k xtgt xAxtButFt x t tJ btt xIxtk mxttt (3) Here authors have proved exact controllability by using fixed point theorem for condensing multi-valued maps due to Martelli. In this paper, we have discussed controllability results with α-norms as in [3] with de- ) D. N. CHALISHAJAR ET AL.1487 viating arguments in terms involving spatial partial de- rivatives. As indicated in [4], and reference therein, the nonlocal Cauchy problem 0 (0)() = x gx x can be applied in di- fferent fields with better effect than the classical initial condition 0 (0) x x. For example in [5], the author described the diffusion phenomenon of a small amount of gas in a transparent tube by using the formula =0 ()= (), p ii i g xcxt where are given constants and 01 In this case the above equation allows the additional measurement at i, . In the past several years theorems about controllability of differential, integro-differential, fractional differential systems and inclusions with nonlocal conditions have been studied by Chalishajar and Acharya [6-9], Ben- chohra and Ntouyas [10,11], and Hernandez, Rabello and Henriquez [12] and the references therein. In [13], Chalishajar discussed exact controllability of third order nonlinear integro-differential dispersion system without compactness of semigroup. ,= 0,1,, i ci p , ,,. p tttb0< t=0,1, ,ip Xianlong Fu and Yueju Cao [14], has discussed the existence of mild solution for neutral partial differential inclusions involving spatial partial derivative with - norms in Banach space. However in their work authors impose some severe assumptions on the operator family generator by () A , i.e. ():() A DA XX ()Tt is an infinitesimal generator of a compact analytic semigroup of a uniformly bounded linear operator 0t, which imply that underlying space X has finite dimension and so the example considered in [14], and subsequently in Section 4 is ordinary differential equation but not partial differential equation which shows lack of exi- stence (exact controllability) in abstract (control) system (refer [15]). This fact and several other applications of neutral equation (inclusions) are the main motivation of this paper. In Section 3 (followed by Preliminaries) of present paper we discuss the controllability of neutral impulsive differential inclusion with nonlocal condition with devi- ating arguments with α-norm, which is the genera- lization of [14], in a finite dimensional space. The ex- ample is given in Section 4 to support the theory. In Section 5 we study exact controllability of same system in infinite dimension space by dropping the compactness assumption of semigroup 0 ()t Tt e generalized the result proved in Section 3. . Here w 2. Preliminaries In this section, we shall introduce some basic definitions, notations and lemmas which are used throughout this paper. Let ,X be a Banach space. is the Banach space of continuous functions from (, )CJX J into X with the norm defined by := sup():. J x xtt J Let be the Banach space of bounded linear operators from ()BX X into X with standard norm () :=sup():= 1. BX NNxx A measurable function : x JX is Bochner inte- grable if and only if x is Lebesgue integrable. (For properties of the Bochner integral see [16]). Let denotes the Banach space of Bochner inte- grable functions 1(, )LJX : x JX with norm 10 =()d b L x xt t for all 1(,) x LJX X . We use the notations ()=2:PX YY , closed()=(): cl PXYPXY, ounded()=(): b PXYPXYb, onvex()=(): c PXYPXYc, and Ycompact :2 X GX ()=(): cp PXY PX. A multi-valued map is convex (respec- tively closed) valued if is convex (respectively closed) for all ()Gx x X . The map is bounded on bounded sets if G ()= () xB GB Gx is bounded in X for any bounded set of B X . ..sup:( )< supxB iexxGx . G is called upper semi-continuous (u.s.c.) on X if for each 0 x X , the set 0 is a nonempty closed subset of ()Gx X and if for each open set of B X containing 0, there exists an open neighborhood ()Gx A of 0 x such that ()GA B. The map is said to be completely continuous if is relatively compact for every bounded subset G ()GB BX. If the multi-valued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph, That is, if 00 , G G G , nn x xyy where nn then 00 (yGx)()yGx . has a fixed point if there is G x X such that () x Gx . A multi-valued map is said to be measurable, if for each :BC()GJ CX x X, the distance function defined by :YJ R ()=,() =inf:()Yt dxGtxzzGt is measurable. An upper semi-continuous map is said to be condensing, if for any bounded subset , with :2 X GX BX () 0B , we have , where < ()B ()GB denotes the Kuratowski measure of non-compactness. Copyright © 2011 SciRes. AM D. N. CHALISHAJAR ET AL. 1488 We remark that a completely continuous multi-valued map is the easiest example of a condensing map. For more details on multivalued maps see the books of Deimling [17]. Throughout this paper, :() A DA XX will be the infinitesimal generator of a compact analytic semi- group of uniformly bounded linear operator Let ().Tt 0() A , then it is possible to define the fractional power , as a closed linear operator on its domain , 01Afor ()DA . Furthermore, the subspace ()DA is dense in X and the expression =; () x AxxDA defines a norm on ()DA . Hereafter we denote by , X the Banach space ()DA normed with x . Then for each 0< 1 , X is a Banach space, and XXf0<or< 1 and the imbedding is compact whenever the resolvent operator of A is compact. Semigroup satisfies the following properties: ()t Tt 10 a) there is a M such that () forall 0Tt Mt ; b) for any 0<1, there exists a positive constant C such that (); 0< C ATt t t For more details about the above preliminaries, we re- fer to ([18,19]). In order to define the solution of the system (1) we shall consider the space 0 =:[0,];(,);=0,1,, andthereexist ()and (); =0,1,, with ()=( ),(0)()=, kk kk kk x bXxCJXk m xtxt km xtxtxg xx which is a Banach space with the norm =;=0,1, kJk , x max xkm where k x is the restriction of x to 1,=0,1,, k =, kk J tt km and =()upxs. ksJk Jk k xs For the system (1) we assume that the following hypo- theses are satisfied for some (0,1) : (H1) Let 2 :(,)WLJU X be the linear operator defined by 0 =( )()d b WuT bs Bu ss The 2 :(,)WLJU kerWX induces a bounded invertible operator and there exists positive con- 1 W 1 2 WM . (H2) i) there exists a constant (0,1) such that :[0, ] F bX X is a continution, and ous func :[0, ] A FbX X satisfies the Lipschitz condition, stant >0L such that that is, there exists a con 112 21212 (,)( ,)AFt xAFtxtxx ,L t for any . 121 2 0, ;,ttbxxX reover, there exists a constant ii) Mo1>0L such that the inequality 1 (, )1,AFtxL x x X holds for any . lti-valued map ) sa (H3) The mu, :( ccp GJ XPX tisfies the following conditions: i) for each ,tJ the function , (,.):( ) ccp GtXPX is u.s.c. and for each , x X the function , (.,) :() ccp GxJP Xurable. Also fo is measr each fixed y the set ,2 ,):()(,(())) .. Gx SvJXvtGtxhtforaet 1 = (LJ is nonempty. positive number , there exists a po ii) for each lN on lsitive function ()wl dependent such that (, )()Gtx wl sup xl and () liminf= < l wl l where (,) =sup:(,)Gtxv vGtx, 0 =( sup s) x xs . (H4) (,),=1,2. i hCJJig X: and satisfies that is continuous positive constants and i) there exists 2 L2 L such that 22 () forall .gyL yLy ii) A g is completely continuous map. (H5) (,),=1,2,,, stant 1 M and 2 M such that and 1 BM and k I CX Xkm are all bounded, that is, there exist constants ,=1,2 k dk , ,,m such that () , kk I xd f x X . or each Now we define the mild solusystem (1). - lo tion for the DEFINITION 2.1 The system (1) is said to be non cally controllable on the interval J if for every (0)( )() x gx DA and 01 , x zX, there exists a con- 0, :=,uL bLLJU 0, 2π such that the 22 2 trol of (1) satcorresponding solution () x isfies i) 1 () ()= x bgx z w0 (0)( );ith x gx x 1, 2,,; ii) )); =ttk k k |( ( x Ixt k m there exists a function 1 vL iii) J(, )X such that Copyright © 2011 SciRes. AM D. N. CHALISHAJAR ET AL.1489 1 2 ( )(,(( )))tGt xht a.e. on J and ( )(0,((0))) v 0 0< < ( )=( ) ( () t tt k 01 1 00 , ( ,(( ))) )( ,(( )))d()( )d ()()d ()(()); ,. tt kk kGx x tTtxgFtxhtxFxh A Tt Tt s Tt The following lem ma [2 sFsxhssTtsvss Bu ss t I xttJvS (2.1) mas are crucial in the proof o 0] Let f our in theorem. LEMMA 2.2 X be a Banach space. Let ,, :() bclc J XPX satisfies that G i) For each x X wi t a , ,,tx Gt is measurable x each tJ, th respect to for (, )(,)tx Gtx is u.s.c. with respect to x . ii) For each fixed nd , x CJX , the set 1 ,2 =, (()),fo xvLJX ht:,r.. vtGtx aet G S is nonempty. a 1) L :,OS C ():=xOSx (, )(, )CJXCJX LEMMA 2.3 [1 set J r continuous mapping from d graph operator in . et be a bounded and convex Let be a line (,JX to (, )CJX then the operator ,, c J XCJX, Gcp ,GGx S is a clo P se 7] L F in Banach space X. :2\ be an upper semi-continuous and condeed map. If for every x, () nsing multi-valu F x is closed and convex set in , then F has a fixed point in . 3. Controlla e are now ab .1 bility Result le to state and prove our main contro- Let W llability result. THEOREM 30. x X If the hypotheses (H1)- (H5) are satisfied, then the system (1) is controllable provided 001 1 :1 1M MC (3.1) LL 01201 1 11 11 1 1 MLL MML CL C (3.2) and 31 1022 1 01 1 1 =1 =(1)( (1) (1)(1) () , 1 m k k Mz MxMLlL C 0=MA . ) M ML lL lb CwlbM d (3.3) where, :=( ,())CCJDA Proof. Let denote the Banac of continuous functions from h space to ()DA J normed by =(): C x supx ttJ Using (.) x hypothesis (H5) for an arbitrary function and 1(zDAtrol 1 () =()()((0,((0)))utWzgx Tbgxh ) define the con 110 11 0 0 ) (,( ()))()(,( ()))d x b kk k xxF 0< < () ()d ()(()) b tt k F bxh bATbsFsxh ss Tb svs sTtt Ixt Using the above control, define a multi-valued map :2N by 01 11 0 , 0<< ()=:()=( )()(0,((0))) (,(()))()(, (()))d ()(());, . t kk kGx tt k NxyytTtxgxFxh 00 () ()d ()()()d tt F txh tATtsFsxhss tt IxttJvS Tt svs sTtsBuss T ,, k F gI and the fact that 0 x X , By assumption on it is obvious that ()Xyt . ints of Clearly the fiN are mild solu We shall show that N satisfies the hypotheses of Lemma 2.3. The proof will be given in several steps. Step 1: There exists a positive number such that xed potions to (1). lN (), ll H H where N ),0.lt=:( l Hx xt For each positive number l, l H is clearly a bounded closed convex set in . We claim that there exists a positive integer l such that () , l NH H where ()=(). l NHNx If it is ue, then fo er l, the l tr the functi lx positive integ Hnot r each eons re xist(.) ll x H and () ll yNx, but (.) , ll yH that is ()> l yt l for some ()[0, ],tl b where denotes t is depen- dent on l. However one other hand we have, ()tl th 01 1 00 <( )( )()(0,((0))) ( ,(( )))()(,( ( )d()()( )d ll t ll ll lytTxgxFxh 1 0 , 0< < ( )))d ( ) ()(()), where l tt kklkl Gx l tt k t F txhtATtsFsx v sstsBuss h ss Tt sT TttIx tvS Hence, Copyright © 2011 SciRes. AM D. N. CHALISHAJAR ET AL. 1490 01 1 1 1 0 00 0<< <( )()(0,((0))) (,(())) () (,(()))d ( )()d( )()()d ( )(()) ll l t l tt ll kklk tt k lTtxgx AAFxh AAFtxht ATtsAFsxhs s Ttsvs sTtsBus s TttIxt 02201 01 1 11 (1 ) 00 123 0=1 (1) (1) (1)d ()d () () d () tt m t k k lMxLlL MLlMLl CC Llswl s ts ts CMM MsMd ts Dividing on both sides byand taking the lower limit as we get l l 1 01 201 11 () 1 bb MLLMML CL C 1. This is a contradiction with Formula (2). Hence for some positive integer Step 2: is c Indeed if) then ther () ll NH H. onvex for each ()Nx yy x. e exists 12 ,(Nx12 , ,Gx vv S such that , we have for each tJ 01 ( )=( )()(0,((0))) i ytTt xgxFxh 11 0 (, ( ()))()(,( ()))d ()()d ()()()d t tt ii kk k 00 0< < () (()); =1,2. tt k F txh tATtsFsxhss Ttsvs sTtsBus s Tt t Ixti Let 01. Then for each tJha we ve 1 11 0 12 0 12 0 0<< () =((0))) (,( ()))()(,( ()))d () ()(1)()d () ()(1)()d ()(()) t t t kk k tt k t h Ftxh tATtsFsxh ss Ttsv svss TtsBu suss Ttt Ixt 12 0 (1)()()(0,yyTtxgxFx Sinc convex becausehas convex v ( )y x e isalues, ,Gx S 12 ) G (1 yN Nx . Step 3: is closed for each Let x such that in () yN x. yy 0() nn n. Then y 1 g the fact that Gt valuesa to a sury to ge and hence y tJ and there exists , ,nG vSx such that for ever 01 1 0 0 0< < ()=()()(0,((0)))(,(())) ()( ,(( )))d ()(()) n t tt tt k ytTtxgxFxhFtxh t ATts Fsx hss T Tt txt 0 () ()d ()()()d n tsvssTtsBu ss I kk k Usin has compac, we my pass bsequence if necessat that vn conver-ges to 1(,J)vL X,Gx vS . Then for each ,tJ 01 11 0 00 ()()=()()(0,((0))) (,( ()))()(,(()))d ()()d()()()d ()(());. n t tt kkk ytytTtxgxFxh 0< <tt k F txh tATtsFsxhss Ttsvs sTtsBus s Ttt IxttJ Hence ()yNx . ext we showe operator N c Step 4: that this u.s. and condensing. For this purpose, we decompose N as N = N1 + N2, where the operators N1, N2 are defined on Hl respectively by 1 N 11 1 0 ()( )=( ,()( ))()(0,((0)) ()( ,(( )))d t NxtFt xhtTt Fxh ATtsFsx hss 00 () ( )d()()()d ()(()); . tt 20 , 0< < =:()=()()(0,((0) ( kk k Gx tt k NxyytTt xgxFxh 1 1 )) 1 0 (,(( )))()(,()))d t F txh tATtsFsxhss Tt svs sTtsBuss Ttt IxtvS We will verify that N1 is a contraction while 2 is a completely continuous operator. To prove that N1 is a contraction, we take N 12 ,l x xH (H2), arbitrarily. Then for each and by condition we have that tJ 1112 11 21 11 21 21 11 ()()( )() (,(()))(,( ())) ()[(0,( (0)))(0,((0)))] (0,( (0)) Nx tNxt Ftxh tFtxh t TtF xhF xh Fx hAF 11 0()(,(( )))(,(( ))) d tAT tsFs xhsFs xhss 11 2 1 =( ,(()))(,(())) ( ) AAFtxh tAFtxht TtA A 21 1 0 11 21 1 012 1 00 01 12 0 012 0 (0,((0))) () ( ,(( )))( ,(()))d (1)d. ()() sup () 1 (1)()() sup =()() sup t t s s s xh ATts AFsxhsAF sxhss C M MLLsxsxs ts LMMCxsxs Lxsxs Copyright © 2011 SciRes. AM D. N. CHALISHAJAR ET AL.1491 Thus, 112 2012 NxN xLxx tion 0 0< <1L, 1 N is a contraction. Next we show that . Therefore by assump s.c. and condensing. i) ii) is equi-continuous. Let 2 is clearly bounded. N is u. 2() l NH ()NH 2l 121 2 ,,<J . Let l x H and 2()yNx . Then thsucr eac ere exists , vSGx h that foh tJ , we have ) Then, 00 00<< ()=()()()( )d ()()()d( )(() t t kk k tt k ytTtxgxTtsvs s Tts Bu ssTttIxt 10 1 ()(0,((0)))zgx xFxh 21 210 12 21 2 01 11 21 0 0 () () [()()](( )) [() ()]()d()()d ()() )()( ( b b yy TTxgx TsTsvss Tsvss TsTsBW Tbxg Tb 1 0A1 (,(()))()(, (()))dFbxh bTbFxh ) (v 01 2 1 () tk Ts BW 1 2 10 1 11 0 12 )( ()) d ( )()( )(0,((0))) (, (()))()(, (())) ()()d ()(())d kk k b b kk k tk t Ixts zgxTbxgxFxh FbxhbATbFxhd vTtt Ixts 0Tb )d (Tt The right hand side tends to zero as 0 21 () , compactness in the uniform ope- uous on . l since is strongly continuous and the of implies the continuity rator . Thus is equi-contin ()Tt 0 ()t Tt topol ogy 2(.)N H act for each ,tJ iii) 2 ()( lt )is relatively compNH where 22 ()=():() ll yt yNH. condition (4)(),( ()NH t Obviously, by 2)() l H iiN Ht is rela- tively compact in X for =0t. Let 0<tb be fixed and 0< <t . For l x H and 2()yN ch that x, there exists a ) Define, ) Since is compact, the set function ,Gx vS su 00 0 0<< ()=()[()]()( )d () ()d()()()d ()()()d( )(() t tt t t kk k ttt k ytTtxgxTtsvss Ttsvs sTtsBus s Tts Bu ssTttIxt 00 00< < ()=()()() ( )d ()()()d( )(() t t kkk tt k ytTtxgxTt svss Tts Bu ssTttIxt 00 0 0< < ()=()()()() ( )d ()()( )()d ()(()) t t kk k tt k ytTtxgxTTtsvss TTtsBuss Ttt Ixt ()Tt 2 ()=( ) l YtyNH():y t is relatively compact in X for,0 << tevery . Moreorevery yN 2( ), l H over,f 12 1 01 ()( )(0,((0))) ( ,(( ))) t Tb xgxF xh Fbxhb 1 1 0 0< < () () =()()d ()()()d ()d () )d ( tt tt t t t tt k yty t TtsvssTts Buss Mwls MM Mzgx T 0() ( )(() b kk k Tb v tt Ixt ()( ,(()))d bATbFx h 12122 022 01 1 01 1 (1 ) 0 0=1 123 123 )d () () ()(1) (1) (1)d () ()d d () () () t t b m b k k s Mw l MMMzLlL MxLlLMLl C MLlLl b CwlM ds b Mw lMMMM Mwl MMM Therefore, letting 0 act sets arb , we see that there re relatively compitrarily close to the set a 2 ():( ) l ytyN H is relatively comp . Hence the set 2 ():( ) l ytyN H act in X . As a consequence of (i),(ii), (iii) and together with the can conclude that : Arzela-Ascoli theorem we 22 H l l NH map and, th is a comply continuous multi-valued erefore, a condensing mu valued map. lete lti- Copyright © 2011 SciRes. AM D. N. CHALISHAJAR ET AL. Copyright © 2011 SciRes. AM 1492 iv) has a closed graph. Froe above steps we can see, for every 2 N m th We must prove that there exists such that ) where ,Gx vS , l x H which can pact set. 2()Nx be testified as i Let 00 00< < ()=()()() ()d ()()()d( )(() t t kk k tt k yxTtxgxTtsvss TtsBussTttIx t is relatively compact and closed set, n Step 3. Hence is a com 2()Nx , n x x ) 2nln n ,( x HyNx and .yy n We m that there ust show that exists 22 (N for each ();yNxy ,nGx n vS such that , ) nn x means .tJ 00 00< < ()=()()()( )d ()()()d( )(() nn n t nkk tt k ytTt xgxTtsvss Tts BussTttIxt t ) nk where 1 nk 1 10 11 0 00<< ()=()()()(0,((0))) (, (()))()(, (()))d ()()d ()(()) xnn n b b nkk tt k ut Wzgx TbxgxF xh F bxh bATbsFsxhss Tbsv ssTttIx t 1 10 1 11 0 00<< () =( )()()(0,((0))) (,(()))()( ,(( )))d ()()d ()(()) x b b kk k tt k ut WzgxTbxgxFxh F bxh bATbsFsxh ss TbsvssTttIx t ,=1, 2,, k I km and g Clearly, since are continu- ous we have that 000<< 000<< ()()()()()()d()(()) ()()()()()()d()(())0,as . t nnn kknk tt k t kk k tt k ytTt xgxTtsBussTttIxt yxTtxgxTtsBussTttIx tn Consider the linear continuous operator . From Lemma (H3) it follows that is a closed graph operator. Moreover, we obtain that Gx Since G OS 1 0 :, (,),=()d t LJXCJXvvtTt svss 00 (( )()()( nn n ytxgxTt sBu , 0< < )())d()(()) (). t kknk n tt k TtssTttIxtS , n x x it follows from (H3) that has a i u-val p value, c. On the other handis a con- traction. nce is u.s.c. and ng By La fixed point 00 ()()( )()()y tTtxgxTtsBu that is, there must exist a , () Gx vt S such that , 0< < ()d()(()) () t kk kGx tt k ssTttIxtS 0 ()( )yTtxgx 00<< () ()()d( )(()) t kk k tt k t Tt s BussTttIx t 0 = Therefore, 2 Nclosed graph. Since s a completely continuous mltiued map with cact () (())= ()()d. t vtTtsvs s is u.s. 2.3, 2 N om nsi ) 2 N He emma 1 N conde (. 12 =NNN there exists . x for N on . l H Therefore, the nonlocal Cauchym ffect (1) is controllable on proble with impulsive e. J Particularly, if is a single-valued map, then the system (1) will become 12 = 0 d(),()=()(),() ; d := 0,; |=(()); =1,2,,; ; (0)() = ttk kk k x tFtxht AxtButGtxht t tJ b xIxtk mtt xgxx (4) by using Sadovskii’s fixed-point theorem for condensing map, we can analogously study the controllability of the system (4). (H3)’ The function satisfies the fo- llowing conditions: i) for each :GJ XX 2 (, (()))Gtxh t ,tJ the function is (,.):Gt XX D. N. CHALISHAJAR ET AL.1493 continuous; and for each , x X the ion funct is stroble. positive num there exists a such that (.,) :GxJ X ii) for each positive function ngly measura ber dependent ,lN on l()wl () (,)() and li l l m= < wl supG txwl where 0 =( sup s xx).s THEOREM 3.2 Let 0 x X . If the hypotheses (1),(2),(3),(4)and( 5) H HHH the system (4) is controllabl H e on are satisfied, then J provided (1), (2) and (1) hold. Proof The mild solution of the system (4) is given by 1 01 2 00 0< < ()=()( )(0,((0))) )))d ( )(,(())d()()()d ()(()); . tt kk k tt k xtT txg xFxh 1 0 ( ,(()))( )(,(( t F txh tATtsFsx h ss TtsGsxh ssTts Bu ss Ttt IxttJ We define the operator :N by Then we can decompose as where 1 an ve epoorem can be applied to the operato ws th (4) is controllable on the interval 01 11 0 20 0<< ( () (,( ()))()(,( ()))d ()(,(()))d ()()()d ()(()) t tt kkk tt k NT tx FtxhtATtsFsxhss TtsGsxh ssTtsBu ss Ttt Ixt 0 )()=() (0,((0)))xtgxFx h N12 =,NNN 11 1 0 ()( )=(,()( ))()(0,((0)) ()(,(()))d t NxtFt xhtTtFxh ATtsFsx hss d 20 2 0 ()( )=( )()()(,(()))d )d t t Nx tTtxgxTtsGsxhss Tt s 0 and rify that 1 N is a contraction while 2 N is a compact operator, thus Sadovskii’s fixd- () ()(s Bu s int the at syste r N and hence N has atleast a fixed point on , which sho . m J T π 0 2 2 ( ,)( ,,)(sin ,)(sin ,)d (, ) =(,),(sin,),(sin, 01, 0π, , =1,2,,; ), = 0; ()( =1,2, , (0, ) k kk z ztxbtyx ztytyy ty ztx z utyhtztxtx x x txttkm ztztz km zx (,0)=(, π)) =(()), kk zt ztI t π 0 0 =0 (,)(,)d=(), 0π p ii i kyxzsyyzxx (4.1) where p is a positive integer, and is defi 01 0<< <<<1, p sss 2 0 ;( )=([0,π]).zxXL A 12 0< <<<<1 m tt t ned by =A with the domain 2 () =([0,]) =(.):, ous, ()=0 DA HX 0 are absolutelycontinu ,(0)=X Then A generates a strongly continuous semigroup which is compact, analytic and self-adjoint. a’) Also A has a discrete spectrum reresentation ; (.)T p 2 =1 =()<, >,(), nn n A nD AnN where 2 ()=(); =1,2, π nxsinnxn is the orthogonal set of eigenvector of A. The eigenvalues are b’) The operator 2,.nn N 1 2 A is given by he desireof is similar to Step 4 of Theorem 3.1. 4. Example As an application of Theorem 3.2, we study the following impulsive partial function differential system with nonlocal condition d pro 1 2=A =0 <,> nn n n on the space 2 =1 ()=( <,> nn n DA n perator 2 :(,)BLJXX 1 .) :.X X The control o is defined by ()()()=(,); (0,π)Butyu tyy which satisfies condition (H5). Here B is dentity operator and the contan i function u(.) is given in We assume that the fo 2([0, π], )LU. llowing conditions hol rol d: unction b is measurable and i) The f ππ 2 01 00 sup(,,)dd<. tbtyxyx ii) The function 2(,bteasurable, (, ,0)= (, ,π)=0,btybt y and 2 , )yx x is m 1 22 2 ππ (,, ) =supdd<. bt yx Nyx 10 12 00 tx Copyright © 2011 SciRes. AM D. N. CHALISHAJAR ET AL. 1494 iii) For the function R the fo- llowing three conditions are satisfied: 1) For each is continuous. 2) For each :[0,1]hRR [0,1],( ,.,.)tht 1 2 ,(.,,zXhzz) is measurable. 3) There is a positive number 1 suchthatc 1 (, ,), g tzzc z for all 1 2 ( ,)[0,1].tz X iv) 11 22 ,,=1, k ICXX k ,=1,,, k dk m such that ,and there exist cons- tants 11 22 () , . kk I zdzX Here we choose 1 ==. 2 According to paper [21], we know that, if 1 2 ,zX then z is absolutely ous , In view of this result, for continu ,zX and (0) =(π)=0.zz 1 2 ( ,)tz [0,1] X, ( is defined as in Section 3), we can define respectively that π 0 (, )()(,,)( )( )d.= F tzxbtyxzyz yy (,)()=(, (()),Gtzxhtzxz x and 1 (())= (), p ii gtKs ), =0 , i where 11 22 : i K XX is completely continuous [16] such that π 0 ()()=(, )()d ii K zx kyxzyy and 1 2 :[0,1]GXX It is easy to see that 1 2 11 11 22 22 :[0,1], :[0,1]FXXAFXX for each [0,1],t we have In fact, π 0 (, ), 12 (,,) =()()d,cos() π n Ftz bt yxzyz yynx nx also, , 2 π 12 (,,) =( bt yx 22 0 (, ), )()d,sin(). π n Ftz zyzyynx nx This shows that F and 1 2 A F both take values in 1 2 X e the in terms of properties (a’) and (b’), and therefor function g. Since, for any 12 1 2 , x xX, 22 22 212121 =, , nn =0 =0 2 nn x xxxznxxz 1 21 2 .xx This inequality alongwith condition (ii) says that (H2) is satisfied. Also G satisfies and g satisfies (H4). (3)H By (i), (, ) F tz is a bounaded liner operator on X . (),( 3),(1),(24),(5) H HH HH Thus are satisfied and 5.lla nite Dimensional Space It has been observed that the example in ([2-11,22]) overed as special case of the abstract result. up is compact then the assumption (H1) in Se finite dimension applications are restricted to overcome to this problem in Seother whe system (4) in infinite dimension space. LEMMA 5.1 Let the system (1) is controllable on [0,1]. Exact Controbility in Infi cannot be rec If the semigro ction 2 is valid only inal space so the ordinary differential control system but not to partial differential equations (refer [15]). We have tried to ction 3 for the inclusion (1). Here we present an ay of exact controllability result of t ([0, ]),) X be a space formed by normalized piecewise continuous function ([0, ],).X Let h tha suc =: kk VV ; where tively compact if an e space Assume that the function F and G ve mptions 1 (); , () (); = kk k kk Vtttt Vt Vtt t The set is relad only if each set k is relatively compact in th ).C 1 ([ ,]; kk tt X THEOREM rify the assu 5.2 (1) H and (2)H respectively an ndiare fulfilled: d suppose that the following co a1) For every >0r and all tions >0 there are com- pact sets su ,1, 2, i r UX (, )Fs , =i 1 U ch that ,r ()TA and 2 , () r TG U(, )s for every ():(0, ). r FJ b1) Conditions (3) H and (4) H e are satisfied. Then there exists a mof th Proof Consider the system (3.4). As a main portion of eorem, we prov completely continuous operator. ild solution system (3.4). the the that N is Copyright © 2011 SciRes. AM D. N. CHALISHAJAR ET AL.1495 The mild solution given in Theorem (3.2) can be splitted up into following four parts: (1) 1 () =()(0)()(0,((0))) (, (())) Nt TtxgxFxh txh t 1 F (2) ()=(,(( )))d t NtATsxhss (3) 0 ( )=(,()()d t Ntz TsxhsBuss (4) 0 () = tt k Nt ()i N 1 0()tsF 20 ()( )))d( t tsGs s Tt ()(()),foreach . kk k TttIxt tJ Obviously each is continuous. To prove that is compact operatwe will show separately that is relatively compact in for every . is relatively com or () N ()i N x ((0,)) r =(0 rr 1: Let pact in ,) (1) NStep r. oundLet is uniformly bed on [0,us fonorm of the operator in (0,b], we can observe that the sets is relatively compact for every act in (1) =(). Let r VN b] and continuo (.)T r the 1 , )) ; kr VT AU it follows that k V 1 [, kk ttt () r is relatively comp 1 ( )(0,((0)tFxh ],=0,1,2,,.k m Step 2: Let (2) N. ompWe first shvely cact for each For vial. Assume that ow that 2(()) r Nt is relati .tJ tb =0t it is tri 0<2 < and let 1 ,r U pothesis (1a be the com the hy. Since pact set introduced in) (.) A T is strongly cs on [,]b ontinuou , it fo 11 llows that is relatively com- pact in , =():[,], r UATsxsbxU . X usinNow g mean value theorem for Bochner inteal, we can write gr 2 (2) N1 1 0 1 1 2 ()=()()(,(())d () (,(())d (2 )()()(0,) t t t r xtATtsTAFsxhs s ATtsAFsxhs s tcoUX for each where (), r x()co U denotes convex hull of U and * 11 =21 .rCLx Thus 2()() r Nt is relavepact in . tily com X tinuous.Next we show that (2) i-c() r N . Then is equon Let 0 0<ttb (2) (1) 1 0 0 0 () () =()( ,(( )))d ()( ,(()))d ( t t Nxt Nxt 01 0 (2) 00 1 =( ( ))) ()(,(()))d t t A TtsFsxh ss A TtsFsxh ss xtITtt N A TtsFsxh ss Sinc for ; r x (2) 0 ();Nxt e the elements are included in a compact set, it follows that th on ght hand side convergey to zer e first term o as he fun ris uniforml0.tt ctionSimilarly it follows from (1)b that t 1 ()( ,(( ))),r tsFsx hsxAT are equi-int rm on ri egrable, which nd sideimply that the second teght ha also converges uniformly to zero as 0.tt This show that (2)() r N is equi-continuous from the right at 0.t Similarly it can be prove that 2() r N is equi-continuous from the left at 0>0.t Thus (2) () r N is equi-continuous and hence (2) () r N is relatively compact in . Step 3: By using same argument as in Step 2 we can prove that the set (3)(). r N is relatively compact in . Step 4: The relatively compactness of (4)(). r N is consequence of assumption (H4) and Lemnce th compactnesption o.2 (a1) and growth condition (H2) (5). If the maps F and ma 5.1 He e proof. Remark Throughout Section 5 we have used s assumf Theorem 5 ii) and (H ,=1, 2,, k I km (i) and (H5) instead satisfy some Lipschitz conditions (H2) of compactness in (a1) then also we can prove controllability result. 6. References artellcompact al lusionite D Point Theory and Aol. 2, No. 1, 2007, pp. 11-51. [3] C. C. Travis anebb, “Existence, Stab α-Norm for Partial s,” Tran of A [4] ear em,” Journal of Mathematical Analysis and Appli- utions of Semilinear Paprabolic Equations with Nonlocal Initial Conditions,” Journal Mathematical Analysis and Application, Vol. 179, 1993, pp. 630-637. doi:10.1006/jmaa.1993.1373 [1] M. Mi, “A Rothe’s Type Theorem for Non edAcyclic-Valu Map,” Bollettino dell’Unione Mathe- matica Italiana, Vol. 2, 1975, pp. 70-76. [2] M. Benchohra and S. K. 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