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Differential Calculus/ D/ x* p2 R6 r$ $Newton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insurmountable problems could be solved by more or less routine methods.The successful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch of calculus,differential calculus.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 satises 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 establish 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 bounded 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 , 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 innite 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 neutral 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. 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 controllability 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 equations 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 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 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 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, 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 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 following denition.We say that a function x := x(., ) : (, T ) E, 0 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). 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 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 and 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 ifThis 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 arguments 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 into D(A) dened by 0 a 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 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, 294: 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 semilinear 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 neutral stochastic functional dierential inclusionswith innite delay in abstract space. J Math Anal Appl, 2006, 324(1): 161176、7 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. 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: 1121The study of differential equations is one part of mathematics that, perhaps more than any other, has been directly inspired by mechanics, astronomy, and mathematical physics. Its history began in the 17th century when Newton, Leibniz, and the Bernoullis solved some simple differential equation arising from problems in geometry and mechanics. There early discoveries, beginning about 1690, gradually led to the development of a lot of “special tricks” for solving certain special kinds of differential equations. Although these special tricks are applicable in mechanics and geometry, so their study is of practical importance.微分方程牛顿和莱布尼茨，完全相互独立，主要负责开发积分学思想的地步，迄今无法解决的问题可以解决更多或更少的常规方法。这些成功的人主要是由于他们能够将积分学和微分融合在一起的事实。中心思想是微分学的概念衍生。 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，即存在两个常数和 ，例如对所有的t0。一类非密集定义泛函微分方程的可控性12研究在有限的延误。2 Main Results我们开始引入以下定义。定义1设T 0和B.我们认为以下的定义。我们说一个函数X：= X：（ - ，T）E，0 0, 所以 for 1, 2 B 和 t 0. (4)使用1定理7中，我们得到以下结论。定理1 假设（H1），（H2）（H3），。设 B，这样D D(A).。则，存在一个独特的整数解x(., ) 对于Eq. (1),。 （1），定义在(,+) .。定义2 在上述条件下，方程Eq. (1)被说成是在区间J = 0, , 0，如果为每一个初始函数 B， D(A)和任何e1 D(A)，存在可控一个控制u L2(J,U)的，这样的解x(.)的Eq. (1)满足。定理2假设(H1), (H2), (H3).x(.)式为整体解决方法在Eq. (1)中(, ) , 0。并假设（见20）的线性算子从W到U在D(A)定义为， （5）诱导可逆的算子，存在正数和满足 和 那么，Eq. (1)是可控的前提是在J ， （6）当.证明 以下1，当整体解决方案x(.)式。Eq. (1)存在于(, ) , 0，这是对所有的t 0, 或者 然后，一个任意整数解x(.)式。 （1）在(, ) , 0，满足x() = e1，当且仅当这意味着，使用（5），它足以采取对所有的t J， 以x() = e1因此，我们必须采取上述控制，因此，证明是减少对所有的t 0, 的整体解的存在性为了不失一般性，假设 0。 1类似的论点，我们可以看到的,和t0，为K是连续的， 0足够小，这样我们可以选择.然后，P是一个严格的收缩在，和固定的P点给出了独特的不可分割的线上的x(., ) on (, ，验证x() = e1。注1 假设所有D(A)从U W时的线性算子定义0 a 0，诱发可逆的算子在，如存在正常数N1和N2满足，同时，中N足够大，下面的1。上述证明的一个类似的说法可以使用，看到Eq. (1)在0，T的所有T0是可控的。微分方程的研究是数学的一部分，也许比其他分支更多的直接受到力学，天文学和数学物理的推动。他的历史起源于17世纪，当时牛顿、莱布尼茨、伯努利家族解决了一些来自几何和力学的简单的微分方程。哲学开始于1690年的早期发现，逐渐引起了解某些特殊类型的微分方程的大量特殊技巧的发展。尽管这些特殊的技巧只是用于相对较少的几种情况，但是他们能够解决力学和几何中出现的许多微分方程，因此，他们的研究具有重要的实际应用。
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