微分方程英文论文和翻译.doc
《微分方程英文论文和翻译.doc》由会员分享,可在线阅读,更多相关《微分方程英文论文和翻译.doc(8页珍藏版)》请在三一办公上搜索。
1、Differential CalculusNewton 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 prima
2、rily 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 class of partial neutral functional dierential equations with innite delay: (1)w
3、here the state variabletakes values in a Banach spaceand the control is given in ,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
4、 will be specied later, D is a bounded linear operator from B into E dened byis 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 dened byF is an E-valued nonlinear continuous mapping on.The problem of controllabi
5、lity of linear and nonlinear systems represented by ODE in nit dimensional space was extensively studied. Many authors extended the controllability concept to innite dimensional systems in Banach space with unbounded operators. Up to now, there are a lot of works on this topic, see, for example, 4,
6、7, 10, 21. There are many systems that can be written as abstract neutral evolution equations with innite delay to study 23. In recent years, the theory of neutral functional dierential equations with innite delay in innite.dimension was developed and it is still a eld of research (see, for instance
7、, 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 dened
8、 but satises the resolvent estimates of the Hille-Yosida theorem. We shall assume conditions that assure global existence and give the sucient conditions for controllability of some partial neutral functional dierential equations with innite delay. The results are obtained using the integrated semig
9、roups theory and Banach xed point theorem. Besides, we make use of the notion of integral solution and we do not use the analytic semigroups theory.Treating equations with innite delay such as Eq. (1), we need to introduce the phase space B. To avoid repetitions and understand the interesting proper
10、ties of the phase space, suppose that is a (semi)normed abstract linear space of functions mapping (, 0 into E, and satises the following fundamental axioms that were rst introduced in 13 and widely discussed in 16.(A) There exist a positive constant H and functions K(.), M(.):,with K continuous and
11、 M locally bounded, such that, for any and ,if x : (, + a E, and is continuous on , +a, then, for every t in , +a, the following conditions hold:(i) .(ii) ,which is equivalent to or every.(iii) (A) For the function in (A), t xt is a B-valued continuous function for t in , + a.(B) 1.The space B is co
12、mplete. Throughout this article, we also assume that the operator A satises the Hille-Yosida condition :(H1) There exist and ,such that and (2)Let A0 be the part of operator A in dened byIt is well known that and the operator generates a strongly continuous semigroup on .Recall that 19 for all and ,
13、one has and .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 bounded linear operators on E, that satises(i) S(0) = 0,(ii) for any y E, t S(t)y is strongly continuous with values in E
14、,(iii) for all t, s 0, and for any 0 there exists a constant l() 0, such that or all t, s 0, .The C0-semigroup is exponentially bounded, that is, there exist two constants and ,such that for all t 0. Notice that the controllability of a class of non-densely dened functional dierential equations was
15、studied in 12 in the nite delay case.、2 Main Results We start with introducing the following denition.Denition 1 Let T 0 and B. We consider the following denition.We say that a function x := x(., ) : (, T ) E, 0 0, such that for 1, 2 B and t 0. (4)Using Theorem 7 in 1, we obtain the following result
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 微分方程 英文 论文 翻译
链接地址:https://www.31ppt.com/p-4194999.html