可控的无穷时滞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:(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可控的无穷时滞中立型泛函微分方程摘要在这篇文章中，我们给一些偏中性无限时滞泛函微分方程的可控性的充分条件。我们假设线性部分不一定密集定义，但满足的Hille- Yosida定理解估计。使用积分半群

29、理论得到的结果。为了说明我们给出了一下抽象结论。关键词：可控性;积分半群;解决方法 无穷极限一， 引言 在这篇文章中，我们建立一个关于可控的结果偏中性与无限时滞泛函微分方程的下面的类： (1)状态变量在空间值和控制用受理控制范围的Banach空间，Banach空间。 C是一个有界的线性算子从U到E，A：A : D(A) E E上的线性算子，B是函数的映射相空间（ - ，0在E，将在后面D是有界的线性算子从B到E为是从B到E的线性算子有界，每个x : (, T E, T > 0,，和t0，T，xt表示为像往常一样，从（映射 - ，0到由E定义为F是一个E值非线性连续映射在。ODE的代表在三

30、维空间中的线性和非线性系统的可控性问题进行了广泛的研究。许多作者延长无限维系统的可控性概念，在Banach空间无限算子。到现在，也有很多关于这一主题的作品，看到的，例如，4，7，10，21。有许多方程可以无限延迟的研究23为抽象的中性演化方程的书面。近年来，中立与无限时滞泛函微分方程理论在无限维度仍然是一个研究领域（见，例如，2，9，14，15和其中的参考文献）。同时，这种系统的可控性问题也受到许多数学家讨论可以看到的，例如，5,8。本文的目的是讨论方程的可控性。 （1），其中线性部分是应该被非密集的定义，但满足的Hille- Yosida定理解估计。我们应当保证全局存在的条件，并给一些偏中性

31、无限时滞泛函微分方程的可控性的充分条件。结果获得的积分半群理论和Banach不动点定理。此外，我们使用的整体解决方案的概念和我们不使用半群的理论分析。方程式，如无限时滞方程。 （1），我们需要引入相空间B.为了避免重复和了解的相空间的有趣的性质，假设是（半）赋范抽象线性空间函数的映射（ - ，0到E满足首次在13介绍了以下的基本公理和广泛16进行了讨论。（一） 存在一个正的常数H和功能K，M：连续与K和M，局部有界，例如，对于任何，如果x : (, + a E,，和是在 ，+ A 连续的，那么，每一个在T，+ A，下列条件成立：(i) ,(ii) ,等同与或者对伊(iii) （a）对于函数在A

32、中，t xt是B值连续函数在, + a.（b）空间B是封闭的整篇文章中，我们还假定算子A满足的Hille- Yosida条件：(1) 在和，和 （2）设A0是算子的部分一个由定义为这是众所周知的，和算子对于具有连续半群。回想一下，19所有和。.我们还知道在，这是一个关于电子所产生的局部Lipschitz积分半群的衍生，按3，17，18，一个有界线性算子的E系列，满足(iv) S(0) = 0,(v) for any y E, t S(t)y判断为E,(vi) for all t, s 0, 对于 > 0这里存在一个常数l() > 0, s所以或者 t, s 0, .C0 -半群指数有界，即