可控地无穷时滞中立型泛函微分方程.doc

wordCONTROLLABILITY OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYAbstract In this article, we give sufficient conditions for controllability of some partial neutral functional differential 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 delay1 IntroductionIn this article, we establish a result about controllability to the following class of partial neutral functional differential equations with infinite delay: (1)where 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 will be specified later, D is a bounded linear operator from B into E defined 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 defined byF 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 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 understand the interesting properties of the phase space, suppose that 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(.):,with K continuous and M locally bounded, such that, for any and ,if x : (−∞, σ + a] → E, and is continuous on [σ, σ+a], then, for every t in [σ, σ+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 [σ, σ + a].(B) The space B is plete. Throughout this article, we also assume that the operator A satisfies the Hille-Yosida condition :(H1) There exist and ，such that and 〔2〕Let A0 be the part of operator A in defined byIt is well known that and the operator generates a strongly continuous semigroup on .Recall that [19] for all and ,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 satisfies(i) S(0) = 0,(ii) for any y ∈ E, t → S(t)y is strongly continuous with values in E,(iii) for all t, s ≥ 0, and for any τ > 0 there exists a constant l(τ) > 0, such that or all t, s ∈ [0, τ] .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 defined functional differential equations was studied in [12] in the finite delay case.2 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 < T ≤ +∞, is an integral solution of Eq. (1) if(i) x is continuous on [0, T ) ,(ii) for t ∈ [0, T ) ,(iii) for t ∈ [0, T ) ,(iv) for all t ∈ (−∞, 0].We deduce from [1] and [22] that inte。