文档介绍：Differential Calculus
Newton and Leibniz,quite independently of one another,were largely responsible for developing theators. 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 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 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 understa