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Differential Calculus

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 le routine methods.The succeful 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 establish a result about controllability to the following cla of partial neutral functional differential equations with infinite delay:

DxtADxtCu(t)F(t,xt),t0

(1)tx0where the state variablex(.)takes values in a Banach space(E,.)and the control u(.)is given in L2(0,T,U),T0,the Banach space of admiible 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

D(0)D0,B

D0is 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

xt()x(t),(,0]

F is an E-valued nonlinear continuous mapping on.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 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 differential 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 discued by many mathematicians, see, for example, [5, 8].The objective of this article is to discu 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 aume conditions that aure global existence and give the sufficient conditions for controllability of some partial neutral functional differential 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.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 discued 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 a0,if x :(−∞, σ + a] → E, xB and x(.)is continuous on [σ, σ+a], then, for every t in [σ, σ+a], the following conditions hold:(i)xtB.(ii)x(t)HxtB,which is equivalent to

(0)HBor everyB.(iii)xtBK(t)supx(s)M(t)xstB

(A)For the function x(.)in(A), t → xt is a B-valued continuous function for t in [σ, σ + a].(B)1.The space B is complete.Throughout this article, we also aume that the operator A satisfies the Hille-Yosida condition :(H1)There exist and ，such that(,)(A)and

nn:nN,M

（2）

sup()(IA)

Let A0 be the part of operator A in D(A)defined by

D(A0)xD(A):AxD(A)A0xAx,for,xD(A0)It is well known that D(A0)D(A)and the operator A0 generates a strongly continuous semigroup(T0(t)t0)on D(A).Recall that [19] for allxD(A)and t0,one has f0tT0(s)xdsD(A0)and

tA0T0(s)sdsxT0(t)x.We also recall that(T0(t))t0coincides on D(A)with the derivative of the locally Lipschitz integrated semigroup(S(t))t0 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(s)S(t)(S(tr)s(r))drfor all t, s ≥ 0, and for any τ > 0 there exists a

0sconstant l(τ)> 0, such that

S(t)S(s)l()ts or all t, s ∈ [0, τ].The C0-semigroup(S(t))t0 is exponentially bounded, that is, there exist two constants Mand ,such that S(t)Met for all t ≥ 0.Notice that the controllability of a cla of non-densely defined functional differential equations was studied in [12] in the finite delay case.、Main Results

We start with introducing the following 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

tt00DxtDADxsdsCu(s)F(s,xs)dsfor t ∈ [0, T), x(t)(t)for all t ∈(−∞, 0].We deduce from [1] and [22] that integral solutions of Eq.(1)are given for ϕ ∈ B, such that DD(A)by the following system

tDxtS(t)DlimS(ts)B(Cu(s)F(s,xs))ds,t[0,t),、(0x(t)(t),t(,0],Where B(IA)1.To obtain global existence and uniquene, we supposed as in [1] that(H2)K(0)D01.(H3)F:[0,]Eis continuous and there exists

F(t,1)F(t,2)00> 0, such that

12for ϕ1, ϕ2 ∈ B and t ≥ 0.(4)

Using Theorem 7 in [1], we obtain the following result.Theorem 1 Aume that(H1),(H2), and(H3)hold.Let ϕ ∈ B such that Dϕ ∈ 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 Dϕ ∈ 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 x()e1.Theorem 2 Suppose that(H1),(H2), and(H3)hold.Let x(.)be the integral solution of Eq.(1)on(−∞, δ), δ > 0, and aume that(see [20])the linear operator W from U into D(A)defined by

Wulim~0S(s)BCu(s)ds，（5）

nduces an invertible operator Won L2(J,U)/KerW，such that there exist positive constants

~N1and N2satisfying CN1and W1N2，then, Eq.(1)is controllable on J provided that

Where(D0)0Me0N1N2M2e2)K1，（6）

0tK:maxK(t)

.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

dx(t)D0xtS(t)DdtOr

dS(ts)F(s,x)dss0dttS(ts)Cu(s)ds

0tx(t)D0xtS(t)DlimS(ts)B(s,xs)ds0t

limS(ts)BCu(s)ds

0tThen, an arbitrary integral solution x(.)of Eq.(1)on(−∞, δ), δ > 0, satisfies x(δ)= e1 if and only if e1D0xS()Ddd0S(s)F(s,xs)dslim0S(ts)BCu(s)dstThis implies that, by use of(5), it suffices to take, for all t ∈ J,t~1u(t)WlimS(ts)BCu(s)ds(t)0

t~1We1D0xS()DlimS(ts)B(s,xs)ds(t)

0in 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

dt(Pz)(t):D0ztS(t)DS(ts)F(s,zs)ds

dt0dt~S(ts)CW1z()D0zS()D dt0limS()BF(,z)d}(s)ds

0Without lo of generality, suppose that hat, for every

 ≥ 0.Using similar arguments as in [1], we can see z1,z2Z()and t ∈ [0, δ] ,(Pz1)(t)(Pz2)(t)(D00Me)Kz1z2As K is continuous and



D0K(0)1,we can choose δ > 0 small enough, such that

D00Me0N1N2M2e2)K1.Then, P is a strict contraction in Z(),and the 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

WulimS(bs)BCu(s)ds

0~20 ≤ a 0, induce invertible operators W on L([a,b],U)/KerW,such that there

T~1N2,taking ,N exist positive constants N1 and N2 satisfying CN1 and WNlarge enough and following [1].A similar argument as the above proof can be used inductively

1nN1,to see that Eq.(1)is controllable on [0, T ] for all T > 0.in [n,(n1)],The 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：

DxtADxtCu(t)F(t,xt),t0

(1)tx0状态变量x(.)在(E,.)空间值和控制用u(.)受理控制范围L(0,T,U),T0的Banach空

2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]到由E定义为

xt()x(t),(,0]

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（∞，T）→E，0

00tt4.x(t)(t)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，DD(A)如以下结论

tDxtS(t)DlimS(ts)B(Cu(s)F(s,xs))ds,t[0,t),(3)0x(t)(t),t(,0],当B(IA)1。

为了获得全局的存在性和唯一，我们应该在[1]中

(H2)K(0)D01.(H3)F:[0,]Ei是连续的，存在F(t,1)F(t,2)00> 0, 所以

12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Dϕ ∈ 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x()e1。

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

Wulim~么，Eq.(1)是可控的前提是在J

当K0t0S(s)BCu(s)ds，（5）

~1N2那 W存在L2(J,U)/KerW正数N1和N2满足CN1 和 W诱导可逆的算子，(D0)0Me0N1N2M2e2)K1，（6）

:maxK(t).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]

dx(t)D0xtS(t)Ddt或者

dS(ts)F(s,x)dss0dttS(ts)Cu(s)ds

0tx(t)D0xtS(t)DlimS(ts)B(s,xs)ds0t

limS(ts)BCu(s)ds

0t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

de1D0xS()Dd0S(s)F(s,xs)dslim0S(ts)BCu(s)dst这意味着，使用（5），它足以采取对所有的t ∈ J，t~1u(t)WlimS(ts)BCu(s)ds(t)

t~1We1D0xS()DlimS(ts)B(s,xs)ds(t)00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

(Pz)(t):D0ztS(t)DddtddtS(ts)F(s,z0ts)ds

t0~S(ts)CW1z()D0zS()D

limS()BF(,z)d}(s)ds

0为了不失一般性，假设 ≥ 0。[1]类似的论点，我们可以看到的z,z[0，δ]

12Z()和t∈

(Pz1)(t)(Pz2)(t)(D00Me)Kz1z2为K是

D0K(0)1连续的，δ > 0足够小，这样我们可以选择

D00Me0N1N2M2e2)K1.然后，P是一个严格的收缩在Z()，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

WulimS(bs)BCu(s)ds

0~20 ≤ a 0，诱发可逆的算子W在L([a,b],U)/KerW，如存在正常数N1和N

2T~1WN满足CN1，同时中N足够大，下面的[1]。上述证明的一个类2，N1nN1，看到Eq.(1)在[0，T]的所有T>0是可控似的说法可以使用[n,(n1)],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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