1、CONTROLLABILITY OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYAbstract In this article, we give sucient conditions for controllability of some partial neutral functional dierential equations with innite delay. We suppose that the linear part is not necessarily densely dened but sat

2、ises the resolvent estimates of the Hille-Yosida theorem. The results are obtained using the integrated semigroups theory. An application is given to illustrate our abstract result.Key words Controllability; integrated semigroup; integral solution; innity delay1 IntroductionIn this article, we estab

3、lish a result about controllability to the following class of partial neutral functional dierential equations with innite delay: (1)where the state variabletakes values in a Banach spaceand the control is given in ,the Banach space of admissible control functions with U a Banach space. C is a bounde

4、d linear operator from U into E, A : D(A) E E is a linear operator on E, B is the phase space of functions mapping (, 0 into E, which will be specied later, D is a bounded linear operator from B into E dened byis a bounded linear operator from B into E and for each x : (, T E, T > 0, and t 0, T ,

5、 xt represents, as usual, the mapping from (, 0 into E dened byF is an E-valued nonlinear continuous mapping on.The problem of controllability of linear and nonlinear systems represented by ODE in nit dimensional space was extensively studied. Many authors extended the controllability concept to inn

6、ite dimensional systems in Banach space with unbounded operators. Up to now, there are a lot of works on this topic, see, for example, 4, 7, 10, 21. There are many systems that can be written as abstract neutral evolution equations with innite delay to study 23. In recent years, the theory of neutra

7、l functional dierential equations with innite delay in innite dimension was developed and it is still a eld of research (see, for instance, 2, 9, 14, 15 and the references therein). Meanwhile, the controllability problem of such systems was also discussed by many mathematicians, see, for example, 5,

8、 8. The objective of this article is to discuss the controllability for Eq. (1), where the linear part is supposed to be non-densely dened but satises the resolvent estimates of the Hille-Yosida theorem. We shall assume conditions that assure global existence and give the sucient conditions for cont

9、rollability of some partial neutral functional dierential equations with innite delay. The results are obtained using the integrated semigroups theory and Banach xed point theorem. Besides, we make use of the notion of integral solution and we do not use the analytic semigroups theory.Treating equat

10、ions with innite delay such as Eq. (1), we need to introduce the phase space B. To avoid repetitions and understand the interesting properties of the phase space, suppose that is a (semi)normed abstract linear space of functions mapping (, 0 into E, and satises the following fundamental axioms that

11、were rst introduced in 13 and widely discussed in 16.(A) There exist a positive constant H and functions K(.), M(.):,with K continuous and M locally bounded, such that, for any and ,if x : (, + a E, and is continuous on , +a, then, for every t in , +a, the following conditions hold:(i) ,(ii) ,which

12、is equivalent to or every(iii) (A) For the function in (A), t xt is a B-valued continuous function for t in , + a.(B) The space B is complete. Throughout this article, we also assume that the operator A satises the Hille-Yosida condition :(H1) There exist and ，such that and （2）Let A0 be the part of

13、operator A in dened byIt is well known that and the operator generates a strongly continuous semigroup on .Recall that 19 for all and ,one has and .We also recall that coincides on with the derivative of the locally Lipschitz integrated semigroup generated by A on E, which is, according to 3, 17, 18

14、, a family of bounded linear operators on E, that satises(i) S(0) = 0,(ii) for any y E, t S(t)y is strongly continuous with values in E,(iii) for all t, s 0, and for any > 0 there exists a constant l() > 0, such that or all t, s 0, .The C0-semigroup is exponentially bounded, that is, there exi

15、st two constants and ,such that for all t 0. Notice that the controllability of a class of non-densely dened functional dierential equations was studied in 12 in the nite delay case.2 Main Results We start with introducing the following denition.Denition 1 Let T > 0 and B. We consider the followi

16、ng denition.We say that a function x := x(., ) : (, T ) E, 0 < T +, is an integral solution of Eq. (1) if(i) x is continuous on 0, T ) ,(ii) for t 0, T ) ,(iii) for t 0, T ) ,(iv) for all t (, 0.We deduce from 1 and 22 that integral solutions of Eq. (1) are given for B, such that by the following

17、 system (3)Where.To obtain global existence and uniqueness, we supposed as in 1 that(H2).(H3)is continuous and there exists > 0, such thatfor 1, 2 B and t 0. (4)Using Theorem 7 in 1, we obtain the following result.Theorem 1Assume that (H1), (H2), and (H3) hold. Let B such that D D(A). Then, there

18、 exists a unique integral solution x(., ) of Eq. (1), dened on (,+) .Denition 2Under the above conditions, Eq. (1) is said to be controllable on theinterval J = 0, , > 0, if for every initial function B with D D(A) and for anye1 D(A), there exists a control u L2(J,U), such that the solution x(.)

19、of Eq. (1) satises.Theorem 2Suppose that(H1), (H2), and (H3) hold. Let x(.) be the integral solution ofEq. (1) on (, ) , > 0, and assume that (see 20) the linear operator W from U into D(A)dened by， （5）nduces an invertible operatoron ，such that there exist positive constantsand satisfyingand ，the

20、n, Eq. (1) is controllable on J providedthat， （6）Where.ProofFollowing 1, when the integral solution x(.) of Eq. (1) exists on (, ) , > 0, it is given for all t 0, by Or Then, an arbitrary integral solution x(.) of Eq. (1) on (, ) , > 0, satises x() = e1 if andonly ifThis implies that, by use o

21、f (5), it suces to take, for all t J,in order to have x() = e1. Hence, we must take the control as above, and consequently, the proof is reduced to the existence of the integral solution given for all t 0, byWithout loss of generality, suppose that 0. Using similar arguments as in 1, we can see hat,

22、 for every,and t 0, ,As K is continuous and,we can choose > 0 small enough, such that.Then, P is a strict contraction in,and the xed point of P gives the unique integralolution x(., ) on (, that veries x() = e1.Remark 1Suppose that all linear operators W from U into D(A) dened by0 a < b T, T &

23、gt; 0, induce invertible operators on,such that thereexist positive constants N1 and N2 satisfying and ,taking,N large enough and following 1. A similar argument as the above proof can be used inductivelyin,to see that Eq. (1) is controllable on 0, T for all T > 0.AcknowledgementsThe authors woul

24、d like to thank Prof. Khalil Ezzinbi and Prof.Pierre Magal for the fruitful discussions.References1 Adimy M, Bouzahir H, Ezzinbi K. Existence and stability for some partial neutral functional dierentialequations with innite delay. J Math Anal Appl, 2004, 294: 4384612 Adimy M, Ezzinbi K. A class of l

25、inear partial neutral functional dierential equations with nondensedomain. J Dif Eq, 1998, 147: 2853323 Arendt W. Resolvent positive operators and integrated semigroups. Proc London Math Soc, 1987, 54(3):3213494 Atmania R, Mazouzi S. Controllability of semilinear integrodierential equations with non

26、local conditions.Electronic J of Di Eq, 2005, 2005: 195 Balachandran K, Anandhi E R. Controllability of neutral integrodierential innite delay systems in Banach spaces. Taiwanese J Math, 2004, 8: 6897026 Balasubramaniam P, Ntouyas S K. Controllability for neutral stochastic functional dierential inc

27、lusionswith innite delay in abstract space. J Math Anal Appl, 2006, 324(1): 1611767 Balachandran K, Balasubramaniam P, Dauer J P. Local null controllability of nonlinear functional dier-ential systems in Banach space. J Optim Theory Appl, 1996, 88: 61758 Balasubramaniam P, Loganathan C. Controllabil

28、ity of functional dierential equations with unboundeddelay in Banach space. J Indian Math Soc, 2001, 68: 1912039 Bouzahir H. On neutral functional dierential equations. Fixed Point Theory, 2005, 5: 1121可控的無窮時滯中立型泛函微分方程摘要在這篇文章中，我們給一些偏中性無限時滯泛函微分方程的可控性的充分條件。我們假設線性部分不一定密集定義，但滿足的Hille- Yosida定理解估計。使用積分半群

29、理論得到的結果。為了說明我們給出了一下抽象結論。關鍵詞：可控性;積分半群;解決方法 無窮極限一， 引言 在這篇文章中，我們建立一個關于可控的結果偏中性與無限時滯泛函微分方程的下面的類： (1)狀態變量在空間值和控制用受理控制范圍的Banach空間，Banach空間。 C是一個有界的線性算子從U到E，A：A : D(A) E E上的線性算子，B是函數的映射相空間（ - ，0在E，將在后面D是有界的線性算子從B到E為是從B到E的線性算子有界，每個x : (, T E, T > 0,，和t0，T，xt表示為像往常一樣，從（映射 - ，0到由E定義為F是一個E值非線性連續映射在。ODE的代表在三

30、維空間中的線性和非線性系統的可控性問題進行了廣泛的研究。許多作者延長無限維系統的可控性概念，在Banach空間無限算子。到現在，也有很多關于這一主題的作品，看到的，例如，4，7，10，21。有許多方程可以無限延遲的研究23為抽象的中性演化方程的書面。近年來，中立與無限時滯泛函微分方程理論在無限維度仍然是一個研究領域（見，例如，2，9，14，15和其中的參考文獻）。同時，這種系統的可控性問題也受到許多數學家討論可以看到的，例如，5,8。本文的目的是討論方程的可控性。 （1），其中線性部分是應該被非密集的定義，但滿足的Hille- Yosida定理解估計。我們應當保證全局存在的條件，并給一些偏中性

31、無限時滯泛函微分方程的可控性的充分條件。結果獲得的積分半群理論和Banach不動點定理。此外，我們使用的整體解決方案的概念和我們不使用半群的理論分析。方程式，如無限時滯方程。 （1），我們需要引入相空間B.為了避免重復和了解的相空間的有趣的性質，假設是（半）賦范抽象線性空間函數的映射（ - ，0到E滿足首次在13介紹了以下的基本公理和廣泛16進行了討論。（一） 存在一個正的常數H和功能K，M：連續與K和M，局部有界，例如，對于任何，如果x : (, + a E,，和是在 ，+ A 連續的，那么，每一個在T，+ A，下列條件成立：(i) ,(ii) ,等同與或者對伊(iii) （a）對于函數在A

32、中，t xt是B值連續函數在, + a.（b）空間B是封閉的整篇文章中，我們還假定算子A滿足的Hille- Yosida條件：(1) 在和，和 （2）設A0是算子的部分一個由定義為這是眾所周知的，和算子對于具有連續半群。回想一下，19所有和。.我們還知道在，這是一個關于電子所產生的局部Lipschitz積分半群的衍生，按3，17，18，一個有界線性算子的E系列，滿足(iv) S(0) = 0,(v) for any y E, t S(t)y判斷為E,(vi) for all t, s 0, 對于 > 0這里存在一個常數l() > 0, s所以或者 t, s 0, .C0 -半群指數有界，即