 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 . 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=0d() ,()()() ,();d:=0,; |=(); =1,2,; (0)()ktt k kkxtFtxht AxtButGtxhtttJ bttxIxtkmxgxx X (1) where the linear operator ()A generates an analytic semigroup ; is a multi-valued map and 0()tTt G==() ()ttk kkxxt xt, 0()=( )limhkkxtxtkh and 0()=limhkxt()xth represent the right and left limits of xt=t at k respectively, 0t,xX)\ :(FJ XPX is a multi-valued map [ is the family of all subsets of X] and ()PX)(,gCJ2(, )uLJUUX. 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 . :()kIXDA; and . =1,2, ,km12As a model we consider the following system of heat equations; ,hh C(,JJ220(,)(,(cos, )),(cos, )(, )=(,),(cos,),(cos(,0)=(,π)=0;()()= (()),;=1, 2,,(0,)((,))=(), 0π,[0,1]kkkkkzztx Ftztxtxtxztx zutxG tztxtxxxzt ztztztI ztttkmzxgztx 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. Gg(.) = 0The existence and controllability of the following sy- stem is studied by Benchohra and Ntouyas  =d()(, )()()(,);d:= [0,];|=(()); =1,2,,;()=(), (,0).ttkttk kkxtgt xAxtButFt xttJ bttxIxtk 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  with de- ) D. N. CHALISHAJAR ET AL.1487 viating arguments in terms involving spatial partial de- rivatives. As indicated in , and reference therein, the nonlocal Cauchy problem 0(0)() =xgx x can be applied in di- fferent fields with better effect than the classical initial condition 0(0)xx. For example in , the author described the diffusion phenomenon of a small amount of gas in a transparent tube by using the formula =0()= (),piiigxcxt 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  and the references therein. In , Chalishajar discussed exact controllability of third order nonlinear integro-differential dispersion system without compactness of semigroup. ,= 0,1,,ici p, ,,.ptttb00L such that that is, there exists a con112 21212(,)( ,)AFt xAFtxtxx ,L t for any .121 20, ;,ttbxxX reover, there exists a constant ii) Mo1>0L such that the inequality 1(, )1,AFtxL x xXholds for any . lti-valued map )sa(H3) The mu,:(ccpGJ XPX tisfies the following conditions: i) for each ,tJ the function ,(,.):( )ccpGtXPX is u.s.c. and for each ,xX the function ,(.,) :()ccpGxJP Xurable. Also fo is measr each fixed y the set ,2,):()(,(())) .. GxSvJXvtGtxhtforaet 1= (LJis nonempty. positive number , there exists a poii) for each lNon lsitive function ()wl dependent such that (, )()Gtx wl supxland ()liminf= lyt l for some ()[0, ],tl b where denotes t is depen- dent on l. However one other hand we have, ()tl th01100<( )( )()(0,((0)))( ,(( )))()(,(( )d()()( )dlltlllllytTxgxFxh10,0< <( )))d( )()(()), where lttkklkl GxlttktFtxhtATtsFsxv sstsBuss h ssTt sTTttIx tvS Hence, Copyright © 2011 SciRes. AM D. N. CHALISHAJAR ET AL. 1490 011110000<<<( )()(0,((0)))(,(()))() (,(()))d( )()d( )()()d( )(())llltlttllkklkttklTtxgx AAFxhAAFtxhtATtsAFsxhs sTtsvs sTtsBus sTttIxt  02201 01111(1 )001230=1(1) (1)(1)d ()d() ()d()ttmtkklMxLlL MLlMLlCCLlswl sts tsCMM MsMdts Dividing on both sides byand taking the lower limit as we get l l101 201 11() 1bbMLLMML CL C  1.This is a contradiction with Formula (2). Hence for some positive integer Step 2: is c Indeed if) then ther()llNH H. onvex for each()Nx yyx. e exists 12,(Nx12 ,,Gxvv S such that , we have for each tJ01( )=( )()(0,((0)))iytTt xgxFxh110(, ( ()))()(,( ()))d()()d ()()()dtttiikk k000< <()(()); =1,2.ttkFtxh tATtsFsxhssTtsvs sTtsBus s Tt t Ixti Let 01. Then for each tJha we ve 11101201200<<() =((0)))(,( ()))()(,( ()))d() ()(1)()d() ()(1)()d()(())tttkk kttkt hFtxh tATtsFsxh ssTtsv svssTtsBu sussTtt Ixt   12 0(1)()()(0,yyTtxgxFx Sinc convex becausehas convex v( )y xe isalues, ,GxS 12) G (1 yNNx . Step 3: is closed for each Let x such that in () yNx. yy0()nn n. Then y 1g the fact that Gt valuesato a sury to ge and hence ytJ and there exists , ,nGvSx such that for ever011000< <()=()()(0,((0)))(,(()))()( ,(( )))d()(())ntttttkytTtxgxFxhFtxh tATts Fsx hssTTt txt  0()()d ()()()dntsvssTtsBu ssI kk kUsin has compac, we my pass bsequence if necessat that vn conver-ges to 1(,J)vL X,GxvS. Then for each ,tJ 0111000()()=()()(0,((0)))(,( ()))()(,(()))d()()d()()()d()(());.ntttkkkytytTtxgxFxh0< ,(),nnnAnD AnNwhere 2()=(); =1,2,πnxsinnxn is the orthogonal set of eigenvector of A. The eigenvalues are b’) The operator 2,.nn N 12A 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 pro12=A=0<,>nnnn  on the space 2=1()=( <,>nnnDA 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 ifunction u(.) is given inWe assume that the fo 2([0, π], )LU. llowing conditions holrol d: unction b is measurable and i) The fππ20100sup(,,)dd<.tbtyxyx  ii) The function 2(,bteasurable, (, ,0)= (, ,π)=0,btybt y and 2, )yxx is m1222ππ (,, )=supdd<.bt yxNyx101200tx 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12,(.,,zXhzz) is measurable. 3) There is a positive number 1 suchthatc 1(, ,),gtzzc z for all 12( ,)[0,1].tz X iv) 1122,,=1,kICXX k ,=1,,,kdk m such that ,and there exist cons- tants 1122() , .kkIzdzX Here we choose 1==.2 According to paper , we know that, if 12,zX then z is absolutely ous , In view of this result, for continu ,zX and (0) =(π)=0.zz12( ,)tz [0,1] X,  ( is defined as in Section 3), we can define respectively that π0(, )()(,,)( )( )d.=Ftzxbtyxzyz yy(,)()=(, (()),Gtzxhtzxz x and 1(())= (),piigtKs ),=0,i where 1122:iKXX is completely continuous  such that π0()()=(, )()diiKzx kyxzyy and 12:[0,1]GXX It is easy to see that 1211 1122 22:[0,1], :[0,1]FXXAFXX for each [0,1],t we have In fact, π0(, ),12 (,,)=()()d,cos()πnFtzbt yxzyz yynxnx also, ,2π12 (,,)=(bt yx220(, ),)()d,sin().πnFtzzyzyynxnx This shows that F and 12AF both take values in 12X e the in terms of properties (a’) and (b’), and thereforfunction g. Since, for any 12 12,xxX, 2222212121=, ,nn=0 =02nnxxxxznxxz1212.xxThis inequality alongwith condition (ii) says that (H2) is satisfied. Also G satisfies and g satisfies (H4). (3)HBy (i), (, )Ftz is a bounaded liner operator on X. (),( 3),(1),(24),(5)HHH 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 dimensionapplications 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 Inficannot be recIf the semigroction 2 is valid only inal space so the ordinary differential control system but not to partial differential equations (refer ). We have tried toction 3 for the inclusion (1). Here we present anay of exact controllability result of t ([0, ]),)X be a space formed by normalized piecewise continuous function([0, ],).X Let  h tha  suc=:kkVV; where tively compact if ane space Assume that the function F and G ve mptions 1(); ,()(); =kkkkkVttttVtVtt t The set  is relad only if each set k is relatively compact in th ).C 1([ ,];kktt XTHEOREMrify the assu 5.2 (1)H and (2)H respectively an ndiare fulfilled: d suppose that the following coa1) For every >0r and alltions >0 there are com- pact sets su,1, 2,irUX(, )Fs, =i1Uch that ,r()TA and 2,() rTG U(, )s for every ():(0, ).rFJ b1) Conditions (3)H and (4)H e are satisfied. Then there exists a mof thProof 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 isCopyright © 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 TtxgxFxhtxh t 1F (2) ()=(,(( )))dtNtATsxhss (3)0( )=(,()()dtNtz TsxhsBuss (4)0() =ttkNt()iN10()tsF20()( )))d(ttsGs s Tt()(()),foreach .kk kTttIxt 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 ()iNx((0,))r=(0rr 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 rVNb] and continuo(.)Tr the 1,)) ;krVT AU  it follows that kV 1[,kkttt()r is relatively comp1( )(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(())rNt is relati.tJtb=0t it is tri0<2 < and let 1,rU pothesis (1abe the com the hy. Sincepact set introduced in) (.)AT is strongly cs on [,]bontinuou, it fo11llows that  is relatively com- pact in ,=():[,],rUATsxsbxU.X usinNow g mean value theorem for Bochner inteal, we can write gr2(2)N110112()=()()(,(())d() (,(())d(2 )()()(0,)tttrxtATtsTAFsxhs sATtsAFsxhs stcoUX  for each where (),rx()co U denotes convex hull of U and *11=21 .rCLxThus 2()()rNt is relavepact in . tily comX tinuous.Next we show that (2) i-c()rN. Then is equon Let 000.t Thus (2) ()rN is equi-continuous and hence (2) ()rN is relatively compact in . Step 3: By using same argument as in Step 2 we can prove that the set (3)().rN is relatively compact in . Step 4: The relatively compactness of (4)().rN is consequence of assumption (H4) and Lemnce thcompactnesption o.2 (a1) and growth condition (H2) (5). If the maps F and ma 5.1 Hee proof. Remark Throughout Section 5 we have used s assumf Theorem 5ii) and (H,=1, 2,,kIkm (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.  C. C. Travis anebb, “Existence, Stab α-Norm for Partial s,” Tran of Aearem,” 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  M. Mi, “A Rothe’s Type Theorem for NonedAcyclic-Valu Map,” Bollettino dell’Unione Mathe-matica Italiana, Vol. 2, 1975, pp. 70-76.  M. Benchohra and S. K. Ntouyas, “Existence Results for Nondensely Defined Impulsive Semilinear FunctionDifferential Incn with Infielay,” Journal of Fixed pplications, Vd G. F. 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