可控的无穷时滞中立型泛函微分方程

CONTROLLABILITY OF NEUTRAL FUNCTIONAL
DIFFERENTIAL EQUATIONS WITH INFINITE DELAY
Abstract In this article, we give su fficient conditions for controllability of some partial neutral functional di fferential equations with infinite delay. We suppose that the lin ear part is not necessarily densely defined but satisfies 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 Controllabili ty; integrated semigroup; integral solution; infinity delay
1 Introduction
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) 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 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 specified later,
D is a bounded linear operator from B into
E defined by
B D D ∈-=ϕϕϕϕ,)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 defined by
]0,(),()(-∞∈+=θθθt x xt
F is an E-valued nonlinear continuous mapping on B ⨯ℜ+.
The problem of controllability of linear and nonlinear systems represented by ODE in finit dimensional space was extensively studied. Many authors extended the controllability concept to infinite dimensional sy stems 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 rec ent years, the theory of neutral functional di fferential equations with infinite delay in infinite
dimension was developed and it is still a field 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 defined but satisfies the resolvent estimates of the Hille-Yosida 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 i ntegrated 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.
Treating 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 functions mapping (−∞, 0] into E, and
satisfies the following fundamental axioms that were first introduced in [13] and widely discussed in [16].
(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) B t
x H t x ≤)(,which is equivalent to B H ϕϕ≤)0(or every B ∈ϕ (iii) B σσσσx t M s x t K xt t s B )()(sup )(-+-≤≤≤
(A) For the function (.)x in (A), t → xt is a B -valued continuous function for t in [σ, σ + a].
(B) The space B is complete.
Throughout this article, we also assume that the operator A satisfies the Hille -Yosida condition :
(H1) There exist and ℜ∈ω，such that )(),(A ρω⊂+∞ and
{}
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 for Ax x A A D Ax A D x A D It is well known that )()(0A D A D =and the operator 0A generates a strongly continuous semigroup ))((00≥t t T on )(A D . Recall that [19] for all )(A D x ∈ and 0≥t ,one has )()(000A D xds s T f t ∈ and x t T x sds s T A t )(0)(00=+⎪⎭
⎫ ⎝⎛⎰. We also recall that 00))((≥t t T coincides on )(A D with the derivative of the locally Lipschitz integrated semigroup 0))((≥t t S generated by A on E, which is, according to [3, 17, 18], a family of bounded linear operators on E, that satisfies
(i) S(0) = 0,
(ii) for any y ∈ E, t → S(t)y is strongly continuous with values in E,
(iii) ⎰-+=s
dr r s r t S t S s S 0))()(()()(for all t, s ≥ 0, and for any τ > 0 there exists a constant l(τ) > 0, such that
s t l s S t S -≤-)()()(τ or all t, s ∈ [0, τ] .
The C0-semigroup 0))((≥'t t S is exponentially bounded, that is, there exist two constants M and ω,such that t e M t S ω≤')( for all t ≥ 0.
Notice that the controllability of a class of non-densely defined functional di fferential equations was studied in [12] in the finite delay case.
2 Main Results
We start with introducing the fol lowing definition.
Definition 1 Let T > 0 and ϕ ∈ B. We consider the following definition.
We say that a function x := x(., ϕ) : (−∞, T ) → E, 0 < T ≤ +∞, is an integral solution of Eq.
(1) if
(i) x is continuous on [0, T ) ,
(ii)
⎰∈t A D Dxsds 0)( for t ∈ [0, T ) , (iii)
⎰⎰+++=t s t t ds x s F s Cu Dxsds A D Dx 00),()(ϕfor t ∈ [0, T ) ,
(iv) )()(t t x ϕ= for all t ∈ (−∞, 0].
We deduce from [1] and [22] that integral solutions of Eq. (1) are given for ϕ ∈ B, such that )(A D D ∈ϕ by the following system
⎪⎩⎪⎨⎧-∞∈=∈+-'+'=⎰+∞→],0,(),()(),,0[,)),()(()(lim )(0t t t x t t ds x s F s Cu B s t S D t S Dxt t s ϕλϕλ
(3)
Where 1)(--=A I B λλλ.
To obtain global existence and uniqueness, we supposed as in [1] that (H2) 1)0(0<D K .
(H3) E F →B ⨯+∞],0[:is continuous and there exists 0β> 0, such that
B -≤-21021),(),(ϕϕβϕϕt F t F for ϕ1, ϕ2 ∈ 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 ϕ ∈ B such that D ϕ ∈ D(A). Then, there exists a unique integral solution x(., ϕ) of Eq. (1), defined on (−∞,+∞) .
Definition 2 Under the above conditions, Eq. (1) is said to be controllable on the interval J =
[0, δ], δ > 0, if for every initial function ϕ ∈ B with D ϕ ∈ D(A) and for any e1 ∈ D(A), there exists a control u ∈ L2(J,U), such that the solution x(.) of Eq. (1) satisfies 1)(e x =δ.
Theorem 2 Suppose that(H1), (H2), and (H3) hold. Let x(.) be the integral solution of Eq. (1) on (−∞, δ) , δ > 0, and assume that (see [20]) the linear operator W from U into D(A) defined by
ds s Cu B s S Wu )()(lim
0λδλδ⎰-'=+∞→， （5） nduces an invertible operator W ~on KerW U J L /),(2，such that there exist positive constants
1N and 2N satisfying 1N C ≤and 21~N W ≤-，then, Eq. (1) is controllable on J provided
that
1))(2221000<++δδωδωδβδβK e M N N e M D ， （6） Where )(max :0t K K t δδ≤≤=.
Proof Following [1], when the integral solution x(.) of Eq. (1) exists on (−∞, δ) , δ > 0, it is given for all t ∈ [0, δ] by
ds s Cu s t S dt d ds x s F s t S dt d D t S x D t x t t s t ⎰⎰-+-+'+=0
00)()(),()()()(ϕ Or
ds x s B s t S D t S x D t x t s t ⎰-'+'+=+∞→00),()(lim )()(λλϕ ds s Cu B s t S t ⎰-'++∞→0)()(lim λλ
Then, an arbitrary integral solution x(.) of Eq. (1) on (−∞, δ) , δ > 0, satisfies x(δ) = e1 if and only if ds s Cu B s t S ds x s F s S d d
D S x D e t s ⎰⎰-'+-+
'+=+∞→0001)()(lim ),()()(λλδδδδϕδThis implies that, by use of (5), it su ffices to take, for all t ∈ J, {})()()(lim
~)(01t ds s Cu B s t S W t u t ⎰-'=+∞→-λλ {})(),()(lim )(~0011t ds x s B s t S D S x D e W t
s ⎰-'-'--=+∞→-λλϕδδ
i n order to have x(δ) = 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, δ] by
⎰-+'+=t s t ds z s F s t S dt d D t S z D t Pz 0
0),()()(:))((ϕ {ϕδδδD S z D z W C s t S dt d t )()(~)(00
1'---=⎰- ds s d z F B S )}(),()(lim
0τττδτλδλ⎰-'-+∞→ Without loss of generality, suppose that ω ≥ 0. Using similar arguments as in [1], we can see
hat, for every 1z ,)(2ϕδZ z ∈and t ∈ [0, δ] ,
∞-+≤-2
10021)())(())((z z K e M D t Pz t Pz δδωβ As K is continuous and 1)0(0<K D ,we can choose δ > 0 small enough, such that
1)2221000<++δδωδωδββK e M N N e M D .
Then, P is a strict contraction in )(ϕδZ ,and t he fixed point of P gives the unique integral olution
x(., ϕ) on (−∞, δ] that verifies x(δ) = e1. Remark 1 Suppose that all linear operators W from U into D(A) defined by ds s Cu B s b S Wu )()(lim 0λδλ⎰
-'=+∞→
0 ≤ a < b ≤ T, T > 0, induce invertible operators W ~ on KerW U b a L /)],,([2,such that there exist positive constants N1 and N2 satisfying 1N C ≤ and 21~N W ≤-,taking N T =δ,N large enough and following [1]. A similar argument as the above proof can be used inductively in 11],)1(,[-≤≤+N n n n δδ,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
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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：
0,),()(0≥⎪⎩
⎪⎨⎧∈=++=∂∂t x xt t F t Cu ADxt Dxt t βφ (1) 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(0
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 xt
F 是一个E 值非线性连续映射在B ⨯ℜ+。

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 ：+
+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) B t
x H t x ≤)(,等同与 B H ϕϕ≤)0(或者对伊B ∈ϕ (iii) B σσσσx t M s x t K xt t s 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 ）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ℜ∈ω，)(),(A ρω⊂+∞和
{}
M N n A I n n ≤≥∈---ωλλωλ,:)()(sup （2） 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 。

回想一下，[19]所有)(A D x ∈和)()(000A D xds s T f t ∈。

x t T x sds s T A t )(0)(00=+⎪⎭
⎫ ⎝⎛⎰. 我们还知道)()(000A D xds s T f t ∈在)(A D ，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) ⎰-+=s
dr r s r t S t S s S 0))()(()()(for all t, s ≥ 0, 对于 τ > 0这里存在一个常数l(τ) > 0, s 所以
s t l s S t S -≤-)()()(τ 或者 t, s ∈ [0, τ] .
C0 -半群指数0))((≥'t t S 有界，即存在两个常数M 和 ω，例如t e M t S ω≤')(对所有的t ≥0。

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φ∈B.我们认为以下的定义。

我们说一个函数X ：= X ：（ - ∞，T ）→E ，0<T ≤+∞，是一个方程的整体解决方程Eq.
1. x 在[0, T )是连续的。

2. ⎰∈t
A D Dxsds 0)( 对于 t ∈ [0, T )
3. ⎰⎰+++=t
s t t ds x s F s Cu Dxsds A D Dx 00),()(ϕ对于t ∈ [0, T ) 4. )()(t t x ϕ= 对于所有t ∈ (−∞, 0].
我们推断[1]和[22]式的整体解决方法。

（1）给出了ϕ ∈ B ，)(A D D ∈ϕ如以下结论
⎪⎩⎪⎨⎧-∞∈=∈+-'+'=⎰+∞→],0,(),()(),,0[,)),()(()(lim )(0t t t x t t ds x s F s Cu B s t S D t S Dxt t s ϕλϕλ
(3)
当1)(--=A I B λλλ。

为了获得全局的存在性和唯一，我们应该在[1]中 (H2) 1)0(0<D K .
(H3) E F →B ⨯+∞],0[:i 是连续的，存在 0β> 0, 所以
B -≤-21021),(),(ϕϕβϕϕt F t F for ϕ1, ϕ2 ∈ 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ϕ ∈ B ，这样D ϕ ∈ D(A).。

则，存在一个独特的整数解x(., ϕ) 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, δ], δ > 0，如果为每一个初始函数ϕ ∈ B ，ϕ ∈ 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1)(e x =δ。

定理2假设(H1), (H2), (H3).x(.)式为整体解决方法在Eq. (1)中(−∞, δ) , δ > 0。

并假设（见[20]） 的线性算子从W 到U 在D(A)定义为
ds s Cu B s S Wu )()(lim
0λδλδ⎰-'=+∞→， （5） 诱导可逆的算子，
W ~存在KerW U J L /),(2正数1N 和2N 满足1N C ≤ 和 21~N W ≤-那 么，Eq. (1)是可控的前提是在J
1))(2221000<++δδωδωδβδβK e M N N e M D ， （6） 当)(max :0t K K t δδ
≤≤=. 证明 以下[1]，当整体解决方案x(.)式。

Eq. (1)存在于(−∞, δ) , δ > 0，这是对所有的t ∈
[0, δ] ds s Cu s t S dt d ds x s F s t S dt d D t S x D t x t t s t ⎰⎰-+-+'+=0
00)()(),()()()(ϕ 或者
ds x s B s t S D t S x D t x t s t ⎰-'+'+=+∞→00),()(lim
)()(λλϕ ds s Cu B s t S t ⎰-'++∞→0)()(lim λλ
然后，一个任意整数解x(.)式。

（1）在(−∞, δ) , δ > 0，满足x(δ) = e1，当且仅当 ds s Cu B s t S ds x s F s S d d
D S x D e t s ⎰⎰-'+-+
'+=+∞→0001)()(lim ),()()(λλδδδδϕδ这意味着，使用（5），它足以采取对所有的t ∈ J ， {})()()(lim
~)(01t ds s Cu B s t S W t u t ⎰-'=+∞→-λλ {})(),()(lim )(~0011t ds x s B s t S D S x D e W t
s
⎰-'-'--=+∞→-λλϕδδ 以x(δ) = 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, δ]的整体解的存在性
⎰-+'+=t s t ds z s F s t S dt d D t S z D t Pz 0
0),()()(:))((ϕ {ϕδδδD S z D z W C s t S dt d t )()(~)(00
1'---=⎰-
ds s d z F B S )}(),()(lim
0τττδτλδλ⎰-'-+∞→ 为了不失一般性，假设ω ≥ 0。

[1]类似的论点，我们可以看到的1z ,)(2ϕδZ z ∈和t
∈[0，δ]
∞-+≤-2
10021)())(())((z z K e M D t Pz t Pz δδωβ 为K 是1)0(0<K D 连续的，δ > 0足够小，这样我们可以选择
1)2221000<++δδωδωδββK e M N N e M D .
然后，P 是一个严格的收缩在)(ϕδZ ，和固定的P 点给出了独特的不可分割的线上的x(., ϕ) on (−∞, δ]，验证x(δ) = e1。

注1 假设所有D(A)从U W 时的线性算子定义
ds s Cu B s b S Wu )()(lim 0λδλ⎰-'=+∞→
0 ≤ a < b ≤ T, T > 0，诱发可逆的算子W ~在KerW U b a L /)],,([2，如存在正常数N1
和N2满足1N C ≤，同时21~N W ≤-，N
T =δ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11],)1(,[-≤≤+N n n n δδ，看到Eq. (1)在[0，T]的所有T>0是可控的。

友情提示：方案范本是经验性极强的领域，本范文无法思考和涵盖全面，供参考!最好找专业人士起草或审核后使用。