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 DELA Y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 di fferential equations with infinite delay:0,),()(0≥⎪⎩⎪⎨⎧∈=++=∂∂t x xt t F t Cu ADxt Dxt t βφ (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≥⎪⎩⎪⎨⎧∈=++=∂∂t x xt t F t Cu ADxt Dxt tβφ (1) where the state variable (.)x takes values in a Banach space ).,(E and the control (.)u is given in[]0),,,0(2>T U T L ,the Banach space of admissible control functions with U a Banach space. C is abounded 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 specified later, D is a bounded linear operator from B into E defined byB D D ∈-=ϕϕϕϕ,)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>T U T L 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B D D ∈-=ϕϕϕϕ,)0(00D is a bounded linear operator from B into E and for each x : (−∞, T ] → E, T > 0, and t ∈ [0, T ], xtrepresents, as usual, the mapping from (−∞, 0] into E defined by]0,(),()(-∞∈+=θθθt x xtF is an E-valued nonlinear continuous mapping on B ⨯ℜ+.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,(),()(-∞∈+=θθθt x xtF 是一个E 值非线性连续映射在B ⨯ℜ+。

The probl em of controllability of linear and nonlinear systems represented by ODE in finitdimensional space was extensively studied. Many authors extended the controllability concept to infinite 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 infinite delay to study [23]. In recent years, the theory of neutral functional di fferential e quations with infinite delay in infiniteODE 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 field of rese arch (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 defined but satisfies the resolvent estimates of the Hille -Y osida theorem. We shall assume conditions that assure global existence and give the su fficient conditions for controllability of some partial neutral functional di fferential equations with infinite delay. The results are obtained using the integrated semigroups theory and Banach fixed 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，但满足的Hille- Y osida 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。

Treatin g equations with infinite 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 ).,(B B is a (semi)normed abstract linear space of funct ions mapping (−∞, 0] into E, and satisfies the following fundamental axioms that were first 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[16]进行了讨论。

（A ）There exist a positive constant H and functions K(.), M(.):++ℜ→ℜ,with K continuous and M locally bounded, such that, for any ℜ∈σand 0>a ,if x : (−∞, σ + a] → E, B x ∈σ and (.)x is continuous on [σ, σ+a], then, for every t in [σ, σ+a], the following conditions hold:(i) B xt ∈,(ii) Bt x H t x ≤)(,which is equivalent to BH ϕϕ≤)0(or every B ∈ϕ(iii) Bσσσσx t M s x t K xtts B)()(sup )(-+-≤≤≤(A) For the function (.)x in (A), t → xt is a B -valued continuous function for t in [σ, σ + a].（一） 存在一个正的常数H 和功能K ，M ：++ℜ→ℜ连续与K 和M ，局部有界，例如，对于任何ℜ∈σ，如果x : (−∞, σ + a] → E,，B x ∈σ和(.)x 是在 [σ，σ+ A] 连续的，那么，每一个在T[σ，σ+ A]，下列条件成立： (i) B xt ∈,(ii) Btx H t x ≤)(,等同与 BH ϕϕ≤)0(或者对伊B ∈ϕ(iii) Bσσσσx t M s x t K xtts B)()(sup )(-+-≤≤≤（A ）对于函数(.)x 在A 中，t → xt 是B 值连续函数在[σ, σ + a].（B ）The space B is complete.Throughout this article, we also assume that the operator A satisfies the Hille -Y osida condition :(H1) There exist and ℜ∈ω，such that )(),(A ρω⊂+∞ and{}M N n A I n n ≤≥∈---ωλλωλ,:)()(sup （2）（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- Y osida 条件：(1) 在和ℜ∈ω，)(),(A ρω⊂+∞和 {}M N n A I n n ≤≥∈---ωλλωλ,:)()(sup （2）Let A0 be the part of operator A in )(A D defined by {}⎩⎨⎧∈=∈∈=)(,,)(:)()(000A D x f o r Ax x A A D Ax A D x A DIt is well known that )()(0A D A D =and the operator 0A generates a strongly continuous semigroup))((00≥t t T on )(A D .设A0是算子的部分一个由)(A D 定义为 {}⎩⎨⎧∈=∈∈=)(,,)(:)()(000A D x for Ax x A A D Ax A D x A D这是众所周知的，)()(0A D A D =和算子0A 对于)(A D 具有连续半群))((00≥t t T 。