A study of second order impulsive neutral evolution control systems with an infinite delay

This article, we study sufficient conditions for the controllability of second-order impulsive partial neutral evolution systems with infinite delay in Banach spaces by using the theory of cosine families of bounded linear operators and fixed point theorem.


Introduction
The study of impulsive functional differential equations is related to their utility in simulating processes and phenomena subject to short-time perturbations during their evolution.The perturbations are executed discretely and their duration is negligible in comparison with the total duration of the processes.That is why the perturbations are considered to take place instantaneously in the form of impulses.The theory of impulsive systems provides a common frame work for mathematical modeling of many real world phenomena.Moreover, these impulsive phenomena can also be found in fields such as information science, electronics, fed-batch culture in fermentative production, robotics and telecommunications (see [1,16,5,14,12,17] and references therein).
In recent years, the study of impulsive control systems has received increasing interest.Due to its importance several authors have investigated the controllability of impulsive systems (see [2,6]).
Motivated by the effort of the after mentioned papers [2,9], the primary inspiration driving this manuscript is mainly concerned with the study of controllability of second order impulsive partial neutral system of the form d dt ϑ (t) − g(t, ϑ t ) = A(t)ϑ(t) + Bu(t) + f (t, ϑ t ), t ∈ J = [0, T ], t = t k , k = 1, 2, ..., n, (1) where φ ∈ B and ζ ∈ X.The control function u(•) is given in L 2 (J, U ), a Banach space of admissible control functions with U as a Banach space and B : U → X as a bounded linear operator; For t ∈ J, x t represents the function ϑ t : (−∞, 0] → X defined by ϑ t (θ) = ϑ(t + θ), −∞ < θ ≤ 0 which belongs to some abstract phase space B defined axiomatically, f : J × B → X, I k : B → X, J k : B → X are appropriate functions and will be specified later.0 < t 1 < . . .< t n < a are fixed numbers and the symbol ∆ξ(t) represents the jump of a functionξ at t , which is defined by ∆ξ(t) = ξ(t + ) − ξ(t − ).Throughout the text we will assume that A(•) generates an evolution operator S(t, s).

Preliminaries
This section we review some basic concepts, notation, and properties required to find our main results.Nowadays there has been an increasing interest in studying the theoretical non-autonomous second order initial value problem where is a closed densely defined operator and f : J → X is an suitable function.Equations of this form have been considered in several papers.We refer the reader to [13,15] and the references therein.In the majority of works, the existence of results to the problem ( 5)-( 6) is related to the existence of an evolution operator S(t, s) for the homogeneous equation Let as assume that the domain of A(t) is a subspace D dense in X and not dependent of t, and for each ϑ ∈ D the function t −→ A(t)ϑ is continuous.The fundamental solution for the second-order evolution equation (7) has been developed by Kozak [11], and we will use the following concept of evolution operator.Definition 2.1 A family S of bounded linear operators S(t, s) : J × J → L(ϑ) is called an evolution operator for (7) if the following conditions are satisfied: (i) For each t ∈ [0, a], S(t, t) = 0, (ii) For all t, s ∈ [0, a], and for each ϑ ∈ X, Throughout this problem we assume that there exists an evolution operator S(t, s) associated to the operator A(t).To abbreviate the teϑ, we introduce the operator C(t, s) = − ∂S(t,s)  ∂s .In addition, we set N and N for positive constants such that sup 0<s,t<a S(t, s) ≤ N and sup 0<s,t<a C(t, s) ≤ N .In addition, we denote by N 1 a positive constant such that for all s, t, t+h ∈ [0, a].Assuming that f : J → X is an integrable function, the mild solution ϑ : [0, a] → X of the problem ( 5)-( 6) is given by In the literature a number of methods have been discussed to establish the existence of the evolution operator S(., .).In particular, a very studied situation is that A(t) is that perturbation of an operator A that generates a cosine operator function.In this reason, below we briefly analysis some essential properties of the theory of cosine functions.Let A : D(A) ⊆ ϑ → X be the infinitesimal generator of a strongly continuous cosine family of bounded linear operators (C(t)) t∈R on Banach space ϑ.We denote by (S(t)) t∈R the sine function associated with (C(t)) t∈R which is defined by S(t)ϑ = t 0 C(s)ϑds, for ϑ ∈ X and t ∈ R. We refer them to [3,19] for the necessary concepts about cosine functions.After that we only mention a few results and notations about this matter needed to establish our results.It is immediate that for all ϑ ∈ X.The notation D(A) stands for the domain of the operator A endowed with the graph norm ϑ A = ϑ + Aϑ , ϑ ∈ D(A).Moreover, in this work, E is the space formed by the vectors ϑ ∈ ϑ for which C(.)ϑ is of class C 1 on R. It was proved by Kisynski [10] that E endowed with the norm ϑ E = ϑ + sup 0≤t≤1 AS(t)ϑ , ϑ ∈ E, is a Banach space.The operator-valued function is a strongly continuous group of bounded linear operators on the space E × ϑ generated by the operator is a bounded linear map such that the operator valued maps S(.) is strongly continuous and where h : J → X is an integrable function, has been discussed in [20].Similarly the existence of solutions of semilinear second order abstract Cauchy problems has been treated in [21].
We only mention here that the function ϑ(•) given by is called a mild solution of ( 7)- (8) and that when υ ∈ E, ϑ(.) is continuously differentiable and In addition ,if υ ∈ D(A), ω ∈ E and f is a continuously differentiable function , then the function ϑ(•) is a solution of the initial value problem ( 11)- (12).Assume now that A(t) = A + B(t) where B(•) : R → L(E, ϑ) is a map such that the function t → B(t)ϑ is continuously differentiable in ϑ for each ϑ ∈ E. It has been established by serizawa [18] that for each (υ, ω) ∈ D(A) × E the non-autonomous abstract Cauchy problem has a unique solution ϑ(•) such that the function t → ϑ(t) is continuously differentiable in E. It is clear that the same argument allows us to conclude that Eq.( 14) with the initial condition ( 12) has a unique solution ϑ(•, s) such that the function t → ϑ(t, s) is continuously differentiable in E. It follows from ( 13) that In particular ,for υ = 0 we have Consequently, ϑ(τ, s 1 dτ and, applying the Gronwall -Bellman lemma we infer that We define the operator S(t, s)ω = ϑ(t, s).It follows from the previous estimate that S(t, s) is a bounded linear map on E. Since E is dense in X, we can extend S(t, s) to X.We keep the notation S(t, s) for this extension.It is well known that, exception the case dim(X) < ∞, the cosine function C(t) cannot be compact for all t ∈ R. By contrast, for the cosine functions that arise in specific applications, the sine function S(t) is very often a compact operator for all t∈ R.This motivates the result [[7], Theorem 1.2].We now consider some notations and definitions concerning impulsive differential equations.A function ϑ : [σ, τ ] → X is said to be a normalized piece wise continuous function on [σ, τ ] if ϑ is piece wise continuous and left continuous on (σ, τ ].We denote by PC ([σ, τ ], X) the space of normalized piecewise continuous functions from[σ, τ ] into X.In particular, we introduce the space PC formed by all normalized piece wise continuous func- ) and ϑ(t + k ) exists, for k = 1, 2, . . ., m.In this paper, we always assume that PC is endowed with the norm In what follows, we put t 0 = 0, t n+1 = a and, for ϑ ∈ PC, we denote by θk , for k = 0, 1, . . ., m, the function θk ∈ C ([t k , t k+1 ]; ϑ) given by θk (t) = ϑ(t) for t ∈ (t k , t k+1 ) and θk (t k ) = lim t→t + ϑ(t).Moreover, for a set E ⊆ PC, we denote by Ẽk , for k = 0, 1, . . ., m, the set Ẽk = θk : ϑ ∈ E.
In this work we will employ an axiomatic definition of the phase space B, similar to the one used in [8] and suitably modify to treat retarded impulsive differential equations.More precisely, B will denote the vector space of functions defined from (−∞, 0] into ϑ endowed with a seminorm denoted • B and such that the following axioms hold: then, for every t ∈ [µ, µ + b), the following conditions hold:and where The terminology and notations are those generally used in functional analysis.In particular, for Banach spaces (Z, • ), (W, • w ), the notation L(Z, W ) stands for the Banach space of bounded linear operators from Z into W and we abbreviate to L(Z) whenever Z = W .By σ(A) (respectively ,ρ(A)) we denote the spectrum (respectively ,the resolvent set)of a linear operator A. MoreoverB r (ϑ, Z) denotes the closed ball with center at ϑ and radius r > 0 in the space Z.
In order to obtain the controllability result, we introduce the following assumptions: (H1) The function f : J × B → XX is continuous and there exists constants and L f = sup t∈J f (t, 0) .
(H2) B is a continuous operator from U to X and the linear operator W : L 2 (J, U ) → X, defined by has a bounded invertible operator W −1 which takes values in L 2 (J, U )/kerW and there exist positive constant M such that BW −1 ≤ M 1 .
(H3) The impulsive functions satisfy the following conditions: (i) The maps I k : B → X, k = 1, 2, . . ., m is continuous and there exist constants and L I = I k (0) .
(H4) The function g : J × B → X is completely continuous and there exists L g > 0 such that From (H4), we have g(t, ψ) ≤ L g ψ B + L 1 where L 1 = sup t∈J g(t, 0) .
Definition 3. 3 The system (14-17) is said to be controllable on the interval J, if for every ϑ 0 = φ ∈ B, ϑ (0) = ζ and z 1 ∈ X, there exists a control u ∈ L 2 (J, U ) such that the mild solution ϑ(•) of ( 1)-( 4) satisfies ϑ(a) = z 1 The following results is an immediate application of the contraction principle of Banach.To simplify the text, we denote K a = sup 0≤t≤a K(t).
Proof.Using the assumption (H2), we define the control function we shall now show that when using this control the operator Γ on the space g(a) defined by(Γy) 0 = 0 and has a fixed point ϑ(•).This fixed point is then a mild solution of the system (1)-( 4).Clearly (Γϑ)(a) = z 1 which means that the control u steers the system from the initial state φ to z 1 in time a, provided we can obtain a fixed point of the operator Γ which implies that the system is controllable.From the assumptions, it is easy to see that Γ is well defined and continuous.For convenience let us take, First we show that Γ maps B r (0, g(a)) into B r (0, g(a)).To this end, from the definition of the operator Γ in (18) and our hypotheses, we obtain for y ∈ g(a) and t ∈ J. Hence Γy a ≤ r.Therefore, Γ maps B r (0, g(a)) into itself.Now for y, z ∈ B r (0, g(a)), we have Which implies that Γ is a contraction on B r (0, g(a)).Hence by the Banach fixed point theorem, Γ has a unique fixed point y in g(a).Defining ϑ(t) = y(t) + φ(t), −∞ < t ≤ a, we obtain that ϑ(•) is a mild solution of the problem (1)-( 4) and the proof is complete.We use the below condition instead of (H1) to avoid the Lipschitz continuity of f used in Theorem 3.1.
(A1) The function f : J × B → X satisfies the following conditions: (i) For each t ∈ J, the function f (t, •) : B → X is continuous and the function t → f (t, ϑ t ) is strongly measurable.
(b) The maps I k , J k : B → X are completely continuous and there exist positive constants Then the (1)-( 4) is controllable on J.
Proof.we define the map Γ on the space g(a) as in eq (18).To prove the controllability of the problem (1)-( 4), we must show that the operator Γ has a fixed point.This fixed point is then a mild solution of the system (1)-( 4).From the assumptions, it is easy to see that Γ is well defined and continuous.In order to apply Lemma 2.2, we need to obtain a priori bound for the solutions of the integral equation y = λΓ(y), λ ∈ (0, 1).To this end, let y λ be a solution of λΓ(y) = y, λ ∈ (0, 1).Using the notation ν λ = sup 0≤s≤t y λ s + φs we observe that Hence follows that Denoting by ω λ (t) the right-hand side of the previous inequality, we see that and subsequently, upon integrating over [0, t], we obtain This estimate permits us to conclude that the set of functions {ω λ : λ ∈ (0, 1)} is bounded and, in turn, that {y λ : λ ∈ (0, 1)} is bounded in g(a).Next we show that Γ is completely continuous.To clarify this proof, we decompose Γ in the form Γ = Γ 1 + Γ Using the hypotheses, condition (b) and Lemma 2.1, we obtain that Γ 1 is continuous and that Γ 2 is completely continuous.In order to use the Ascoli-Arzela theorem we prove that Γ 1 takes bounded sets into relatively compact ones.As above, B r = B r (0, g(a)) and y t + φt B ≤ K a r + c 1 = ρ f or t ∈ J.And also, (Bu)(s) ≤ B 0 .From the mean value theorem , we see that Γ 1 y(t) ∈ tco {S(t, s)f (s, ψ) : s ∈ [0, a], ψ B ≤ ρ} which implies that the set {Γ 1 y(t) : y ∈ B r (0, g(a))} is relatively compact for each t ∈ J.Moreover, from Γ 1 y(t + h) − Γ 1 y(t) = [p(s)Ω(ρ) + B 0 ]ds which implies that Γ 1 y(t + h) − Γ 1 y(t) → 0 as h → 0uniformly for y ∈ B r (0, g(a)) .From this we infer that Γ 1 y(t) : y ∈ B r (0, g(a) is relatively compact in G(a) and consequently that Γ 1 is completely continuous.This completes the proof of the assertion that the map Γ is completely continuous.
By an application of Lemma 2.1, we conclude that there exists a fixed point y of Γ.It is clear that the function ϑ = y + φ is a mild solution of the system (1)-( 4).This completes the proof.
a locally integrable function, then the function y(t) = t 0 S(t − s)ϑ(s)ds defines an E-valued continuous function.The existence of solutions for the second order abstract Cauchy problem

Remark 2 . 1
locally bounded and H, K, M are independent of ϑ(•).(B) The space B is complete.In impulsive functional differential systems, the map [µ, µ + b] → B, t → ϑ t , is in general discontinuous.For this reason, this property has been omitted from our description of the phase space B. Now we include that some of our proofs are based on the following well-known result [[4], Theroem 6.5.4].Lemma 2.2 (Leray-Schauder Alternative) Let D be a closed convex subsets of a normed lined space ϑ such that 0 ∈ D. Let F : D → D be a completely continuous map.Then the set {ϑ ∈ D : ϑ = λF (ϑ), f or some0 < λ < 1} is unbounded or the map F has a fixed point in D.