可控的無窮時滯中立型泛函微分方程

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:1 / 10(i) ,(ii) ,

12、which 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 pa

13、rt of 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,

14、17, 18, 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, the

15、re exist 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 f

16、ollowing 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 fol

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

18、Then, there exists a unique integral solution x(., ) of Eq. (1), dened on (,+) .Denition 2 Under the above conditions, Eq. (1) is said to be controllable on the interval J = 0, , > 0, if for every initial function B with D D(A) and for any e1 D(A), there exists a control u L2(J,U), such that the

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

20、nd satisfying and ，then, Eq. (1) is controllable on J provided that ， （6）Where .Proof Following 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 and only if

21、This implies that, by use of (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 argume

22、nts as in 1, we can see hat, 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 integral olution x(., ) on (, that veries x() = e1.Remark 1 Suppose that all linear operators W from U in

23、to D(A) dened by 0 a < b T, T > 0, induce invertible operators on ,such that there exist positive constants N1 and N2 satisfying and ,taking ,N large enough and following 1. A similar argument as the above proof can be used inductively in ,to see that Eq. (1) is controllable on 0, T for all T

24、> 0.Acknowledgements The authors would like to thank Prof. Khalil Ezzinbi and Prof. Pierre Magal for the fruitful discussions.References 1 Adimy M, Bouzahir H, Ezzinbi K. Existence and stability for some partial neutral functional dierential equations with innite delay. J Math Anal Appl, 2004, 29

25、4: 4384612 Adimy M, Ezzinbi K. A class of linear partial neutral functional dierential equations with nondense domain. 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 sem

26、ilinear integrodierential equations with nonlocal 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

27、 neutral stochastic functional dierential inclusionswith 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

28、Balasubramaniam P, Loganathan C. Controllability 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。

29、我們假設(shè)線性部分不一定密集定義，但滿足的Hille- Yosida定理解估計。使用積分半群理論得到的結(jié)果。為了說明我們給出了一下抽象結(jié)論。關(guān)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(guān)于可控的結(jié)果偏中性與無限時滯泛函微分方程的下面的類： (1)狀態(tài)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(shù)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

30、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(xù)映射在。ODE的代表在三維空間中的線性和非線性系統(tǒng)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(tǒng)的可控性概念，在Banach空間無限算子。到現(xiàn)在，也有很多關(guān)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(lǐng)域（見，例如，2，9，14，15和其中的參考文獻）。同時，這種系統(tǒng)的可控性問題也受到許多數(shù)學(xué)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(yīng)該被非密集的定

31、義，但滿足的Hille- Yosida定理解估計。我們應(yīng)當保證全局存在的條件，并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。結(jié)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(fù)和了解的相空間的有趣的性質(zhì)，假設(shè)是（半）賦范抽象線性空間函數(shù)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(shù)H和功能K，M：連續(xù)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(xù)的，那么，每一個在

32、T，+ A，下列條件成立： (i) ,(ii) ,等同與 或者對伊(iii) （a）對于函數(shù)在A中，t xt是B值連續(xù)函數(shù)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(shè)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(xù)半群。回想一下，19所有和。.我們還知道在，這是一個關(guān)于電子所產(chǎn)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(shù)l() > 0, s所以 或者 t, s 0, .C0 -半群指數(shù)有界，即存在兩個常數(shù)和 ，例如對所有的t0。一類非密集定義泛函微分方程的可控性12研究在有限的延誤。2 Mai