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Existence and Controllability Result for the Nonlinear First Order Fuzzy Neutral Integrodifferential Equations with Nonlocal Conditions

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In this paper, we devoted study the existence and controllability for the nonlinear fuzzy neutral integrodifferential equations with control system in E_N. Moreover we study the fuzzy solution for the ...

In this paper, we devoted study the existence and controllability for the nonlinear fuzzy neutral integrodifferential equations with control system in E_N. Moreover we study the fuzzy solution for the normal, convex, upper semicontinuous and compactly supported interval fuzzy number. The results are obtained by using the contraction principle theorem. An example to illustrate the theory.

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Existence and Controllability Result for the Nonlinear First Order Fuzzy Neutral Integrodifferential Equations with Nonlocal Conditions Presentation Transcript

  • 1. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 DOI : 10.5121/ijfls.2013.3304 39 Existence and Controllability Result for the Nonlinear First Order Fuzzy Neutral Integrodifferential Equations with Nonlocal Conditions S.Narayanamoorthy1 and M.Nagarajan2 1,2 Department of Applied Mathematics, Bharathiar University, Coimbatore, TamilNadu, India-46 1 snm_phd@yahoo.co.in 2 mnagarajanphd@gmail.com ABSTRACT In this paper, we devoted study the existence and controllability for the nonlinear fuzzy neutral integrodifferential equations with control system in E_N. Moreover we study the fuzzy solution for the normal, convex, upper semicontinuous and compactly supported interval fuzzy number. The results are obtained by using the contraction principle theorem. An example to illustrate the theory. KEYWORDS Controllability, Fixed Point Theorem, Integrodifferential Equations, Nonlocal Condition. 1. INTRODUCTION First introduced fuzzy set theory Zadeh[15] in 1965. The term "fuzzy differential equation" was coined in 1978 by Kandel and Byatt[6] . This generalization was made by Puri and Ralescu [12] and studied by Kaleva [5]. It soon appeared that the solution of fuzzy differential equation interpreted by Hukuhara derivative has a drawback: it became fuzzier as time goes. Hence, the fuzzy solution behaves quite differently from the crisp solution. To alleviate the situation, Hullermeier[4] interpreted fuzzy differential equation as a family of differential inclusions. The main short coming of using differential inclusions is that we do not have a derivative of a fuzzy number valued function. There is another approach to solve fuzzy differential equations which is known as Zadeh’s extension principle (Misukoshi, Chalco-Cano, Román-Flores, Bassanezi[9]), the basic idea of the extension principle is: consider fuzzy differential equation as a deterministic differential equation then solve the deterministic differential equation. After getting deterministic solution, the fuzzy solution can be obtained by applying extension principle to deterministic solution. But in Zadeh’s extension principle we do not have a derivative of a fuzzy number- valued function either. In [2], Bede, Rudas, and Bencsik[4], strongly generalized derivative concept was introduced. This concept allows us to solve the mentioned shortcomings and in Khastan et al. [7] authors studied higher order fuzzy differential equations with strongly generalized derivative concept. Recently, Gasilovet al. [3] proposed a new method to solve a fuzzy initial value problem for the fuzzy linear system of differential equations based on properties of linear transformations.
  • 2. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 40 Fuzzy optimal control for the nonlinear fuzzy differential system with nonlocal initial condition in NE was presented by Y.C.Kwun et al. [8] . Recently, the above concept has been extended to the integrodifferential equations by J.H.Park et al. [12]. We will combine these earlier work and extended the study to the following nonlinear first order fuzzy neutral integrodifferential equations ( 0)( ≡tu ) with nonlocal conditions: ])()()()[(=)))(,()(( 0 dssxstGtxtAtxthtx dt d t −+− ∫ ][0,=),())(,( bJttutxtf ∈++ (2) 0=)((0) xxgx + (3) where NEJtA →:)( is fuzzy coefficient, NE is the fuzzy set of all upper semicontinuous, convex, normal fuzzy numbers with bounded −α level intervals, NN EEJf →×: , NN EEJh →×: , NN EEg →: are all nonlinear functions, )(tG is nn× continuous matrix such that dt xtdG )( is continuous for NEx∈ and Jt ∈ with ktG ≤)( , 0>k , NEJu →: is control function. The rest of this paper is organized as follows. In section 2, some preliminaries are presented. In section 3, existence solution of fuzzy neutral integrodifferential equations. In section 4, we study on nonlocal cotroll of solutions for the neutral system. In section 5, an example. 2.PRELIMINARIES In section, we shall introduce some basic definitions, notations, lemmas and result which are used throughout this paper. A fuzzy subset of n R is defined in terms of a membership function which assigns to each point n x R∈ a grade of membership in the fuzzy set. Such a membership function is denoted by [0,1].: →n u R Throughout this paper, we assume that u maps n R onto [0,1], 0 ][u is a bounded subset of n R , u is upper semicontinuous, and u is fuzzy convex. We denote by n E the space of all fuzzy subsets u of n R which are normal, fuzzy convex, and upper semicontinuous fuzzy sets with bounded supports. In particular, 1 E denotes the space of all fuzzy subsets u of .R A fuzzy number a in real line R is a fuzzy set characterized by a membership function aχ [0,1].: →Raχ A fuzzy number a is expressed as x a a x χ ∫∈R =
  • 3. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 41 with the understanding that [0,1],)( ∈xaχ represents the grade of membership of x in a and ∫ denotes the union of . x aχ Definition 2.1 A fuzzy number R∈a is said to be convex if, for any real numbers zyx ,, in R with zyx ≤≤ , )}(),({min)( zxy aaa χχχ ≥ Definition 2.2 If the height of a fuzzy set equals one, then the fuzzy set is called normal. Thus, a fuzzy number R∈a is called normal, if the followings holds: 1.=)(max xa x χ Result 2.1 [10] Let NE be the set of all upper semicontinuous convex normal fuzzy numbers with bounded α - level intervals. This means that if NEa∈ , then −α level set 1},,0)(:{=][ ≤≤≥∈ ααα xaxa R is a closed bounded interval, which we denote by ],[=][ ααα rq aaa and there exists a R∈0t such that 1.=)( 0ta Result 2.2 [10] Two fuzzy numbers a and b are called equal ba = , if ),(=)( xx ba χχ for all R∈x . It follows that (0,1].,][=][= ∈⇔ ααα allforbaba Result 2.3 [10] A fuzzy number a may be decomposed into its level sets through the resolution identity ,][= 1 0 α α aa ∫ where α α ][a is the product of a scalar α with the set α ][a and ∫ is the union of α ][a with α ranging from 0 to 1. . Definition 2.3 A fuzzy number R∈a is said to be positive if 21 <<0 aa holds for the support ],[= 21 aaaΓ of a, that is, aΓ is in the positive real line. Similarly, a is called negative if 0<21 aa ≤ and zero if 21 0 aa ≤≤ .
  • 4. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 42 Lemma: 2.1 [14] If NEba ∈, , then for (0,1]∈α , ],,[=][ ααααα rrqq bababa +++ ),,=,(}],{max},{min[=][ rqjibabaab iiii ααααα ].,[=][ ααααα rrqq bababa −−− Lemma: 2.2 [14] Let ],[ αα rq aa , 1<0 ≤α , be a given family of nonempty intervals. If ,<0],[],[ βαααββ ≤⊂ foraaaa rqrq ],,[=]lim,lim[ αααα rq k r k k q k aaaa ∞→∞→ whenever )( kα is nondecreasing sequence converting to (0,1]∈α , then the family ],[ αα rq aa , 1<0 ≤α , are the α -level sets of a fuzzy number NEa∈ . Let x be a point in n R and A be a nonempty subsets of n R . We define the Hausdroff separation of B from A by }.:{inf=),( AaaxAxd ∈− Now let A and B be nonempty subsets of n R . We define the Hausdroff separation of B from A by }.:),({sup=),(* BbAbdABdH ∈ In general, ).,(),( ** ABdBAd HH ≠ We define the Hausdroff distance between nonempty subsets of A and B of n R by )}.,(),,({max=),( ** ABdBAdBAd HHH This is now symmetric in A and B . Consequently, 1. 0=),(0),( BAdwithBAd HH ≥ if and only if ;= BA 2. );,(=),( ABdBAd HH
  • 5. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 43 3. );,(),(),( BCdCAdBAd HHH +≤ for any nonempty subsets of A , B and C of n R . The Hausdroff distance is a metric, the Hausdroff metric. The supremum metric ∞d on n E is defined by ,,(0,1]},:)][,]([{sup=),( n H Evuallforvudvud ∈∈∞ ααα and is obviously metric on n E . The supremum metric 1H on ),( n EJC is defined by )}.:(,),:)(),(({sup=),(1 n EJyxallforJttytxdyxH C∈∈∞ We assume the following conditions to prove the existence of solution of the equation (1.2). (H1). The nonlinear function NN EEJg →×: is a continuous function and satisfies the inequality )(.)][,(.)]([)(.))]([,(.))](([ αααα δ yxdygxgd HgH ≤ (H2). The inhomomogeneous term NN EEJf →×: is continuous function and satisfies a global Libschitz )(.)][,(.)]([)))](,([,))](,(([ αααα δ yxdsysfsxsfd HfH ≤ (H3.) The nonlinear function NN EEJh →×: is continuous function and satisfies the global lipschitz condition )(.)][,(.)]([)(.))],([,(.))],(([ αααα δ yxdyshxshd HhH ≤ (H4). S(t) is the fuzzy number satisfies for NEy∈ , ),(),( NN EJCEJCyS ∩∈′ the equation ))()()()((=)( 0 ydssSstGytStAytS dt d t −+ ∫ JtdssGsAstSytStA t ∈−+ ∫ ,)()()()()(= 0 such that ],[=)]([ ααα rq SStS and )(tSi α , rqi ,= are continuous. That is, a postive constant sδ such that si tS δα <)( (H5) )( 21 b∆+∆ <1, where hghhgs δδδδδδ +++∆ )(= 11 and )(=2 hAfs M δδδ +∆
  • 6. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 44 3.EXISTENCE AND UNIQUENESS OF FUZZY SOLUTION The solution of the equations (2)-(3) ( 0≡u ) is of the form dssxshstSsAtxthxgxhxgxtStx t ))(,()()())(,())]((0,)()[(=)( 0 00 −++−−− ∫γ dssxsfstS t ))(,()( 0 −+ ∫ (4) where γ is a continuous function from );( NEJC to itself. Theorem: 3.1 Suppose that hypotheses (H1)-(H5) are satisfied. Then the equation (4) has unique fixed point in ),( NEJC . Proof. For );(, NEJCyx ∈ , ))]([,)](([ αα γγ tytxd H ))(,())]((0,)()[(([= 00 txthxgxhxgxtSdH +−−− α ]))(,()())(,()()( 00 dssxsfstSdssxshstSsA tt −+−+ ∫∫ dssyshstSsAtythygxhygxtS t ))(,()()())(,())]((0,)()[([ 0 00 −++−−− ∫ )]))(,()( 0 α dssysfstS t −+ ∫ )))]]((0,)()([,))]((0,)()[(([ 0000 αα ygxhygxtSxgxhxgxtSdH −−−−−−≤ )))](,([,))](,(([ αα tythtxthdH+ )]))(,()()([,]))(,()()(([ 00 αα dssyshstSsAdssxshstSsAd tt H −−+ ∫∫ )]))(,()([,]))(,()(([ 00 αα dssysfstSdssxsfstSd tt H −−+ ∫∫ )()][,()]([)))]((0,)([,))]((0,)(([ 00 αααα δδ yxdygxhygxgxhxgd HhHs +−+−+≤ dtsysfsxsfddtsyshsxshdM H t sH t As )))](,([,))](,(([)))](,([,))](,(([ 00 αααα δδ ∫∫ ++ ))]([,)](([))( 1 αα δδδδδδ tytxdHhghhgs +++≤ dtyxdM H t hAfs )()][,()]([)( 0 αα δδδ ∫++ dttytxdtytxd H t H ))]([,)](([))]([,)](([= 0 21 αααα ∫∆+∆ where, hghhgs δδδδδδ +++∆ )(= 11 and )(=2 hAfs M δδδ +∆ . Therefore, ))]([,)](([sup=))(),(( [0,1] αα α γγγγ tytxdtytxd H ∈ ∞
  • 7. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 45 ))]([,)](([sup [0,1] 1 αα α tytxdH ∈ ∆≤ dttytxdH t ))]([,)](([sup 0[0,1] 2 αα α ∫∈ ∆+ dttytxdtytxd t ))(),(())(),(( 0 21 ∞∞ ∫∆+∆≤ Hence, ))(),((sup=))(),(( ][0,= 1 tytxdtytxH bJt γγγγ ∞ ∈ dtyxdtytxd bt t Jt ())(),(sup))(),((sup ][0,0 21 ∞ ∈ ∞ ∈ ∫∆+∆≤ ))(),(())(),(( 1211 tytxbHtytxH ∆+∆≤ ))(),(()(= 121 tytxHb∆+∆ Then by hypotheses, γ is a contraction mapping. By using Banach fixed point theorem, equations (2)-(3) have a unique fixed point, ),( NEJCx∈ 4. CONTROLLABILITY OF FUZZY SOLUTION In this secetion, we show for the controller term in equation(2)-(3), and the solution of the form dssxshsAstStxthxgxhxgxtStx t ))(,()()())(,())]((0,)()[(=)( 0 00 −++−−− ∫ dssustSdssxsfstS tt )()())(,()( 00 −+−+ ∫∫ (5) Definition 4.1 [13] The equation (4.1) is controllable if there exists )(tu such that the fuzzy solution )(tx of (5) satisfies )(=)( 1 xgxbx − , that is αα )]([=)]([ 1 xgxbx − , where 1 x is a target set. The linear controll system is nonlocal controolable. That is, dssusbSxgxbSbx b )()()]()[(=)( 0 0 −+− ∫ )(= 1 xgx − αα ])()()]()[([=)]([ 0 0 dssusbSxgxbSbx b −+− ∫ ])()()(,)()()]()[([= 0 0 0 0 dssusbSxbSdssusbSxgxbS rr b rrqq b qqq ααααααααα −+−+− ∫∫ )]()(),()[(= 0 11 xgxxgx rqq αααα −− α )]([= 1 xgx − . We Define new fuzzy mapping NERP →)(:ζ by
  • 8. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 46     Γ⊂−∫ otherwise vdssvstS v u t 0, ,,)()( =)( 0 α ζ Then there exists ),=( rqii α ζ such that ],[,)()(=)( 1 0 uuvdssvstSv qqqq t qq ααα ζ ∈−∫ ],[,)()(=)( 1 0 ααα ζ rrrr t rr uuvdssvstSv ∈−∫ We assume that iζ 's are bijective mappings. Hence the α -set of )(su are )](),([=)]([ sususu rq ααα ))]((0,))[(()()(()[(= 00 11 xgxhgxbSxgx qqqqqqq −−−−−− αααααα α ζ dssxthsAstStxth qqq t q ))(,()()())(,( 0 αααα −−− ∫ ,))(,()( 0 dssxtfstS qq t αα −− ∫ ))]((0,)())[(()()(()( 00 11 xgxhxgxbSxgx rrrrrrr −−−−−− αααααα α ζ dssxthsAstStxth rrr t r ))(,()()())(,( 0 αααα −−− ∫ dssxtfstS rr t ))(,()( 0 αα −− ∫ Then substituting this expression into equation (5) yields α - level set of )(bx α )]([ bx ))(,())]((0,)())[(([= 00 sxshxgxhxgxbS qqqqq ααααα +−−− dssxsfsbSdssxthsAsbS qq b qqq b ))(,()())(,()()( 00 ααααα −+−+ ∫∫ αααα α α ζ qqqqqq b xbSxgxsbS ))[(()()(())(( 0 11 0 −−−+ − ∫ ))(,()](0,)( 0 sxshxhxg qqq ααα −−− ,))(,()())(,()()( 00 dssxsfsbSdssxshsAsbS qq b qqq b ααααα −−−− ∫∫ ))]((0,)())[(( 00 xgxhxgxbS rrrr −−− αααα dssxsfsbSdssxshsAsbStxth rr b rrr b r ))(,()())(,()()())(,( 00 αααααα −+−++ ∫∫ ))]((0,)())[(()()(())(( 00 11 0 xgxhxgxbSxgxsbS rrrrrrrr b −−−−−−+ − ∫ αααααα α α ζ
  • 9. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 47 dssxshsAsbSsxsh rqr b r ))(,()()())(,( 0 αααα −−− ∫ ]))(,()( 0 dssxsfsbS rr b αα −− ∫ dssxshsAsbSsxshxgxhxgxbS qqq b qqqq ))(,()()())(,())]((0,)())[(([= 0 00 ααααααα −++−−− ∫ αααα α ααα ζζ qqqqqqqq b xbSxgxdssxsfsbS ))[(()()(()())(,()( 0 11 0 −−+−+ − ∫ dssxshsAsbSsushxgxhxg qqq b qqq ))(,()()())(,())]((0,)( 0 0 αααααα −−−−−− ∫ ))]((0,)())[((,))(,()( 00 0 xgxhxgxbSdssxsfsbS rqrrqq b −−−−− ∫ αααααα dssxsfsbSdssxshsAsbSsush rr b rrr b r ))(,()())(,()()())(,( 00 αααααα −+−++ ∫∫ dssxshsAsbSxhxgxbSxgx rrr b rrrrrrrr ))(,()()()](0,)())[(()()(()( 0 00 11 ααααααααα α α ζζ −−−−−−+ ∫ − dssxsfsbS rr b ))(,()( 0 αα −− ∫ )]()(),()[(= 11 xgxxgx rrqq αααα −− .)]([= 1 α xgx − We now set dssxthsAstStuthxgxhxgxtStx t ))(,()()())(,())]((0,)()[(=)( 0 00 −++−−−Ω ∫ dssxsfstS t ))(,()( 0 −+ ∫ ))]((0,)()[()(()( 00 1 1 0 xgxhxgxbSxgxstS t −−−−−−+ − ∫ ζ dssxsfsbSdssxshsAsbStxth bb ))(,()())(,()()())(,( 00 −−−−− ∫∫ where the fuzzy mapping 1− ζ satisfied above statement. Now notice that ),(=)( 1 xgxTx −Ω which means that the control )(tu steers the equation(5) from the origin to 1 x in the time b provided we can obtain a fixed point of the nonlinear operator Ω . Assume that the hypotheses (H6)The system (4.1) is linear 0≡f is nonlocal controllable. (H7) 1.))((( ≤+++ hkfsh b δδδδδ
  • 10. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 48 Theorem: 4.1 Suppose that the hypotheses (H1)-(H7) are satisfied. Then the equation (5) is a nonlocal controllable. Proof. We can easily check that Ω is continuous from ):]([0, NEbC to itself. For ),:]([0,, NEbCyx ∈ ))]([,)](([ αα tytxdH ΩΩ ))(,())]((0,)()[(([= 00 txthxgxhxgxtSdH +−−− dssxsfstSdssxshsAstS tt ))(,()())(,()()( 00 −+−+ ∫∫ )()[()(()( 0 1 1 0 xgxbSxgxsbS b −−−−+ − ∫ ζ dssxshsAsbStxthxgxh b ))(,()()())(,()]((0, 0 0 −−−−− ∫ ,))(,()( 0 dssxsfsbS b −− ∫ dssyshsAstStythygxhygxtS t ))(,()()())(,())]((0,)()[([ 0 00 −++−−− ∫ dssysfstS t ))(,()( 0 −+ ∫ ))(,())]((0,)()[(()( 00 1 1 0 tythygxhygxsbSxsbS b −−−−−−−+ − ∫ ζ )])))(,()())(,()()( 00 α dsdssysfsbSdssyshsAsbS bb −−−− ∫∫ dssxshsAstStxthxgxhxgtSd t H ))(,()()())(,())]((0,)()[(([ 0 0 −++−+≤ ∫ ,))](()())(,()( <<0 0 α− −+−+ ∑∫ kkk ktt t txIttSdssxsfstS dssyshsAstStythygxhygtS t ))(,()()())(,())((0,)()[([ 0 0 −++−+ ∫ dssysfstS t ))(,()( 0 −+ ∫ ))(,())((0,)()[()(()(([ 0 1 1 0 sxshxgxhxgbSxgxstSd t H −−+−−−+ − ∫ ζ ,))(,()())(,()()( 00 dssxsfsbSdssxshsAstS bb −−−− ∫∫ ))(,())]((0,)()[()(()([ 0 1 1 0 syshygxhygbSygxstS t −−+−−− − ∫ ζ dssysfsbSdssyshsAsbS bb ))(,()())(,()()( 00 −−−− ∫∫ )][,)](([))(( 1 αα δδδδδδ ysxdHhghhgs +++≤
  • 11. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 49 ))]([,)](([)(( 0 αα δδδ sysxdM H t fhAs ∫++ ))]([,)](([)(()][,)](([ 0 αααα δδδδ sysxdysxd H b fhsHh ∫+++ Let hghhgs δδδδδδκ +++ )(= 11 , and )(=2 fhAs M δδδκ + then we have )))]([,)](([))]([,)](([()][,)](([2 00 21 αααααα κκ sysxdsysxdysxd H b H t H ∫∫ ++≤ Therefore ))]([,)](([sup=))(),(( [0,1] αα α tytxdtytxd H ΩΩΩΩ ∈ ∞ )][,)](([2 1 αα κ ysxd∞≤ )))]([,)](([))]([,)](([( 00 2 αααα κ sysxdsysxd bt ∞∞ ∫∫ ++ Hence ))]([,)](([sup=))(),(( ][0, 1 αα tytxdtytxH H bt ΩΩΩΩ ∈ ),()2(2 121 yxHbκκ +≤ ),())(2(= 121 yxHbκκ + By hypotheses )( 6H , we take sufficiently small b , Ω is a contraction mapping. By Banach fixed point theorem, equation (5) has a unique fixed point ).:]([0, NEbCx∈ 5.EXAMPLE Consider the fuzzy solution of the nonlinear fuzzy neutral integrodifferential equation of the form: ,),()(3])()(2[=)))(2)(( 2)( 0 2 Jttuttxdssxetxttxtx dt d st t ∈++−− −− ∫ (6) ,0=)((0) 1= Nkk n k Etxcx ∈+ ∑ (7) where 1 x is target set, and the α - level set of fuzzy number 0,2 and 3 are [0,1]],1,1[=[0] ∈−− αααα for [0,1]],1,3[=[2] ∈−+ αααα for [0,1].],2,4[=[3] ∈−+ αααα for Let .)(2=))(,(,)(3=))(,(,=)( 22)( ttututhttututfestG st−− − Then α - level set of )(=)( 1= kk n k txcxg ∑ is
  • 12. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 50 αα )]([=)]([ 1= kk n k txcxg ∑ )](),([= 1=1= krk n k kqk n k txctxc αα ∑∑ ))]([,)](([=))]([,)](([ 1=1= αααα kk n k kk n k HH tyctxcdygxgd ∑∑ )])(),([)],(),(([= 1=1=1=1= krk n k kqk n k krk n k kqk n k H tyctyctxctxcd αααα ∑∑∑∑ })()(,)()({max 1= |||||||| krkrkqkq k k n k tytxtytxc αααα −−≤ ∑ )()][,()]([ αα δ yxdHg≤ where, |||| k n kg c∑ 1= =δ ,where )](),([=)]([ txtxtx rq ααα and [0,1]],2,4[=[3] ∈−+ αααα for and the α - level set of ))(,( txtf is αα ])([3=))](,([ 2 ttxtxtf αα ])([[3]= 2 txt ],))()((4),)(2)([(= 22 txtxt rq αα αα −+ )))](,([,))](,(([ αα tytftxtfdH ])))()((4),)(2)([(],))()((4),)(2)([((= 2222 tytyttxtxtd rqrqH αααα αααα −+−+ }|))(())(()(4,|))(())((2){(max= 2222 tytxtytxt rrqq αααα αα −−−+ || }|))(())((||))(())((|,))(())((||))(())(({|max)(4 tytxtytxtytxtytxb rrrrqqqq αααααααα α +−+−−≤ | }))(())((|,))(())(({|max|))(())((|4 || tytxtytxtytxb rrqqrr αααααα −−+≤ ))]([,)](([= αα δ tytxdHf where, |))(())((|4= tytxb rrf αα δ + )](),([=)]([ txtxtx rq ααα and [0,1]],2,4[=[3] ∈−+ αααα for and the α - level set of ))(,( txth is αα ])([2=))](,([ 2 ttxtxth αα ])([[2]= 2 txt ]))((,))(][(1,3[= 22 txtxt rq αα αα −+ ].))()((3,))(1)([(= 22 txtxt rq αα αα −+ we introduced the −α set of Equation (5.2)(5.1) − Thus, )))](,([,))](,(([ αα tythtxthdH
  • 13. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 51 ])))()((3,))(1)([(],))()((3,))(1)([((= 2222 tytyttxtxtd rqrqH αααα αααα −+−+ }))(())(()(3,))(())((1)({max= 2222 |||| tytxttytxt rrqq αααα αα −−−+ }))(())((|))(())((|,))(())((|))(())(({|max)(3 |||| tytxtytxtytxtytxb rrrrqqqq αααααααα α +−+−−≤ }))(())((|,))(())(({|max))(())((|3 ||| tytxtytxtytxb rrqqrr αααααα −−+≤ ))]([,)](([= αα δ tytxdHg where, |))(())((|3= tytxb rrg αα δ + Next we prove the nonlocal controllability parts, let us take target set, 2=1 x α )]([ su ],[= αα rq uu 2 1= 1 )1)(()()((1[= αα ααζ qkqk n k q xttxc +−−+ ∑ − dssxtsbSdssxtsbS qq b qq b )()2)(()()()1)(()( 2 0 2 0 αααα αα +−−+−− ∫∫ )()((3),)()1)(()( 1= 1 2 0 krk n k rqq b txcdssxtsbS ααα αζα ∑∫ −−+−− − dssxtsbSdssxtsbSxt rr b rr b r )()2)(()()()1)(()()1)(( 2 0 2 0 2 ααααα ααα +−−+−−+− ∫∫ )])()1)(()( 2 0 dssxtsbS rr b αα α +−− ∫ Then substituting this expression into the integral system with respect to (6)-(7) yields −α level set of )(bx . α )]([ bx )()1)(()](1))[(([= 2 1= txttxcbS qkk n k q αα αα ++−− ∑ dstxttstS qq t q )()(1)()( 2αα α +−+ ∫ ))()1)(()())2)(()( 2 0 2 0 dstxtstSdsxtstS qq t qq t αααα αα +−++−+ ∫∫ dstxtsbStxtsbS qq b qqq b )()1)(()()()1)((1)(())(( 2 0 21 0 ααα α α αααζ +−−+−+−+ ∫∫ − ))))(1)(()()()2)(()( 2 0 2 0 dsdssxtstSdssxtstS qq t qq t αααα αα +−−+−− ∫∫ dstxttsbStxttbS rr b rr )()()(3)()()(1)()])(1([ 2 0 2 αααα ααα −−++− ∫
  • 14. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 52 ))())((3)()))((4)( 2 0 2 0 dstxtstSdsxtstS rq t rr t αααα αα −−+−−+ ∫∫ dstxtsbStxtsbS rr b rrr b )())((3)()()1)(()((3))(( 2 0 21 0 ααα α α αααζ −−−+−−−+ ∫∫ − )])))()((3)()())((4)( 2 0 2 0 dsdssxtsbSdssxtsbS rr b rr b αααα αα −−−−−− ∫∫ )]()(3),(1)[(= 1=1= krk n k kqk n k txctxc αα αα ∑∑ −−−+ α )]([2= 1= kk n k txc∑− )]([= 1 xgx − Then all condition stated in theorem4.1 are satisfied, so the system (6)-(7) is nonlocal controllable on ].[0,b 6. CONCLUSION In this paper, by using the concept of fuzzy number in NE , we study the existence and controllability for the nonlinear fuzzy neutral integrodifferential with nonlocal controll system in NE and find the sufficient conditions of controllability for the controll system (2)-(3). ACKNOWLEDGEMENTS The author’s are thankful to the anonymous referees for the improvement of this paper. REFERENCES [1] K.Balachandran and E.R.Anandhi,(2003),”Neutral functional integrodifferential control systems in Banach Spaces”, Kybernetika,Vol.39, pp.359-367. [2] B.Bede, I.J.Rudas, A.L.Bencsik,(2007),”First order linear differential equations under generalized Differentiability”, Information Sciences, Vol.177, pp. 1648-1662. [3] N.Gasilov, S.E.Amrahov and A.G.Fatullayev,(2011) ”Ageometric approach to solve fuzzy linear systems of differential equations”, Applied Mathematics and Information Sciences, Vol.5(3), pp.484- 499. [4] E.Hullermeier,(1997) “An approach to modeling and simulation of uncertain dynamical systems” International Journal of Uncertainty, Fuzziness and Knoeledge-Based Systems, Vol.5(2), pp.117-137. [5] O.Kaleva, (1987),”Fuzzy differential equations” Fuzzy Sets and Systems,Vol.24, pp.301-317. [6] A.Kandel and W.J.Byatt,(1978),”Fuzzy differential equations” In: Proceedings of the International Conferences On Cybernetics and Society, Tokyo, pp.1213-1216. [7] A.Khastan, F.Bahrami and K.Ivaz,(2009) ”New results on multiple solutions forth-order Fuzzy differential equations under generalized differentiability ” Boundary Value Problems, doi:10.1155/2009/395714. [8] Y.C.Kwun and D.G..Park,(1998),”Optimal control problems for fuzzy differential equations ”Proceeddings of the Korea-Vietnam Joint Seminar, pp.103-114.
  • 15. International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013 53 [9] M.Misukoshi, Y.Chalco-Cano, H.Roman-Flores and R.C.Bassanezi,(2007) ”Fuzzy Differential Equations and extension principle”, information Sciences,Vol.177,pp.3627-3635. [10] M.Mizumpto and K.Tanaka,(1979), Some Properties of Fuzzy Numbers, North Holland. [11] J.H.Park, J.S.Park and Y.C.Kwun (2006) “Controllability for the Semilinear Fuzzy Integro differential Equations with Nonlocal Conditions ”FSKD-Springer-Verlag LNAI 4223, pp.221-230. [12] M.L.Puri and D.A.Raleesu,(1983),”Differentials for fuzzy functions ”, Journal of mathematical Analysis Applications, Vol.91. [13] B.Radhakrishnan and K.Balachandran,(2012),”Controllability results for semi linear Impulsive integrodifferential systems with nonlocal conditions”, Journal of Control Theory and Applications, Vol.10, pp.28-34. [14] S.Seikkala,(1987),”On The Fuzzy Initial Value Problem”, Fuzzy Sets and Systems, Vol.24,pp.319- 330. [15] L.A.Zadeh,(1965), ”Fuzzy Sets”, Information and Control,Vol.8(3),pp.338-353. AUTHORS Dr.S.Narayanamoorthy received his doctoral degree from the Department of Mathematics, Loyola College (Autonomous), Chennai-34, Affiliated to the University of Madras and he obtained his graduate degree, post graduate degree and degree of Master of Philosophy from the same institution. Currently he is working as Assistant Professor of Applied Mathematics at Bharathiar University. His current research interests are in the fields of Applications of Fuzzy Mathematics, Fuzzy Optimization, Fuzzy Differential Equations. Email: snm_phd@yahoo.co.in M. Nagarajan received the M.Sc. and M.Phil . degrees in Mathematics at Bharathiar University, Coimbatore, India in 2007 and 2009, respectively, and is currently pursuing the Ph.D, degree in Applied Mathematics at Bharathiar University, Coimbatore, India. His current research interests are in the fields of Fuzzy Differential equations. Email: mnagarajanphd@gmail.com