數(shù)學專業(yè)英語外文翻譯

1、<p><b>　　重慶理工大學</b></p><p><b>　　數(shù)學專業(yè)英語</b></p><p>　　學 院 </p><p>　　學 號 </p><p>　　姓 名 </p>

2、<p>　　年 月 2012年12月17日 </p><p>　　CONTROLLABILITY OF NEUTRAL FUNCTIONAL DIFFERENTIAL </p><p>　　EQUATIONS WITH INFINITE DELAY</p><p>　　可控的無窮時滯中立型泛函微分方程</p>&

3、lt;p>　　In this article, we establish a result about controllability to the following class of partial neutral functional di?erential equations with in?nite delay:</p><p><b>　　(1)</b></p>

4、<p>　　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)</p><p>　　where the state variabletakes values in a Banach spaceand the control is given in ,the Banach space of admissible c

5、ontrol 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 speci?ed later, D

6、is a bounded linear operator from B into E de?ned by</p><p>　　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</p><p>　　is a

7、 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 de?ned by</p><p>　　F is an E-valued nonlinear continu

8、ous mapping on.</p><p>　　是從B到E的線性算子有界，每個x : (?∞, T ] → E, T > 0,，和t∈[0，T]，xt表示為像往常一樣，從（映射 - ∞，0]到由E定義為</p><p>　　F是一個E值非線性連續(xù)映射在。</p><p>　　The problem of controllability of linear

9、and nonlinear systems represented by ODE in ?nit dimensional space was extensively studied. Many authors extended the controllability concept to in?nite dimensional systems in Banach space with unbounded operators. Up to

10、 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 in?nite delay to study [23]. In recent years, the th

11、eory of neutral fun</p><p>　　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。</p><p&

12、gt;　　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 mathematic

13、ians, 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 de?ned but satis?es the resolvent estimates of the Hille-Yo

14、sida theorem. We shall assume conditions that assure global exist</p><p>　　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)該被非密集的定義，但滿足的Hille- Yosida定理

15、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。</p><p>　　Treating equations with in?nite delay such as Eq. (1), we need to introduce the phase space B. To

16、 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 satis?es the following fundamental axioms th

17、at were ?rst introduced in [13] and widely discussed in [16].</p><p>　　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。</p><p>　　（A）T

18、here exist a positive constant H and functions K(.), M(.):,with K continuous and M locally bounded, such that, for any and ,if x : (?∞, σ + a] → E, and is continuous on [σ, σ+a], then, for every t in [σ, σ+a], the follo

19、wing conditions hold:</p><p><b>　　(i) ,</b></p><p>　　(ii) ,which is equivalent to or every</p><p><b>　　(iii) </b></p><p>　　(A) For the function

20、in (A), t → xt is a B-valued continuous function for t in [σ, σ + a].</p><p>　?。ㄒ唬?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 : (?∞, σ + a] → E,，和是在 [σ，σ+ A] 連續(xù)的，那么，每一個在T[σ，σ+ A]，下列條件成立： (i) ,</p><p&

21、gt;　　(ii) ,等同與 或者對伊</p><p><b>　　(iii) </b></p><p>　　對于函數(shù)在A中，t → xt是B值連續(xù)函數(shù)在[σ, σ + a].</p><p>　?。˙）The space B is complete.</p><p>　　Throughout this article,

22、 we also assume that the operator A satis?es the Hille-Yosida condition :</p><p>　　(H1) There exist and ，such that and </p><p><b>　　（2）</b></p><p>　?。˙）空間B是封閉

23、的</p><p>　　整篇文章中，我們還假定算子A滿足的Hille- Yosida條件：</p><p>　　(1) 在和，和 （2）</p><p>　　Let A0 be the part of operator A in de?ned by </p><p>　　It is well known that and the

24、operator generates a strongly continuous semigroup on .</p><p>　　設(shè)A0是算子的部分一個由定義為 </p><p>　　這是眾所周知的，和算子對于具有連續(xù)半群。</p><p>　　Recall that [19] for alland ,one has and .</p><p

25、>　　回想一下，[19]所有和。 .</p><p>　　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 bounde

26、d linear operators on E, that satis?es</p><p>　　(i) S(0) = 0,</p><p>　　(ii) for any y ∈ E, t → S(t)y is strongly continuous with values in E,</p><p>　　(iii) for all t, s ≥ 0,

27、and for any τ > 0 there exists a constant l(τ) > 0, such that or all t, s ∈ [0, τ] .</p><p>　　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</p><p>　　(i) S(0) = 0,<

28、/p><p>　　(ii) for any y ∈ E, t → S(t)y判斷為E,</p><p>　　(iii) for all t, s ≥ 0, 對于 τ > 0這里存在一個常數(shù)l(τ) > 0, s所以 或者 t, s ∈ [0, τ] .</p><p>　　The C0-semigroup is exponentially boun