数学专业英语论文(含中文版)

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 less routine methods.The successful 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 give su fficient conditions for controllability of some partial neutral functional di fferential equations with infinite delay. We suppose that the linear 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 Controllability; 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 repr esented 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 di fferential equations with infinite delay in infinite

dimension was deve loped 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 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 discussed