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1、 重庆理工大学 数学专业英语 学 院 学 号 姓 名 年 月 2012 年 12 月 17 日 CONTROLLABILITY OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY 可控的无穷时滞中立型泛函微分方 程 In this article, we establish a result about controllability to the following class of partial neutral functional dierential equations with innite delay:

2、 0,),()(0 txxttFtCuA D x tD x tt (1) 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： 0,),()(0 txxttFtCuA D x tD x tt (1) where the state variable (.)x takes values in a Banach space ).,(E and the control (.)u is given in 0),0(2 TUTL ,the Banach space of admissible control functions with U a Banach space. C

3、 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 by BDD ,)0( 0 状态变量 (.)x 在 ).,(E 空间值和控制用 (.)u 受理控制范围 0),0(2 TUTL 的 Banach 空间，Bana

4、ch 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 BDD ,)0( 0 0D is 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 by 0,(),()( txxt F is an E-valued nonline

5、ar continuous mapping on . 0D 是从 B 到 E 的线性算子有界，每个 x : (, T E, T 0, ，和 t 0， T， xt 表示为像往常一样，从（映射 - ， 0到由 E 定义为 0,(),()( txxt F 是一个 E 值非线性连续映射在 。 The problem of controllability of linear and nonlinear systems represented by ODE in nit dimensional space was extensively studied. Many authors extended the

6、 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 rec

7、ent years, the theory of neutral functional dierential equations with innite delay in innite 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 。 dimension was developed and it is still a

8、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 i

9、s 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 o

10、btained 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. 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

11、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。 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,

12、suppose that ).,(BB 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. 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

13、 16进行了讨论。 （ A） There exist a positive constant H and functions K(.), M(.): ,with K continuous and M locally bounded, such that, for any and 0a ,if x : (, + a E, Bx and (.)x is continuous on , +a, then, for every t in , +a, the following conditions hold: (i) Bxt , (ii) BtxHtx )(,which is equivalent t

14、o BH )0(or every B (iii) B xtMsxtKxt tsB )()(s u p)( (A) For the function (.)x in (A), t xt is a B -valued continuous function for t in , + a. （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, ， Bx 和 (.)x 是在 ， + A 连续的，那么，每一个在 T ， + A，下列条件成立： (i) Bxt , (ii) BtxHtx )(,等同与 BH )0(或者对伊 B (iii) B xtMsxtKxt tsB )()(s u p)( （ A） 对于函数 (.)x 在 A 中 ， t xt 是 B 值连续函数在 , + a.