Exact controllability results for a class of abstract nonlocal Cauchy problem with impulsive conditions

This paper deals with exact controllability of a class of abstract nonlocal Cauchy problem with impulsive conditions in Banach spaces. By using Sadovskii fixed point theorem and Monch fixed point theorem, exact controllability results are obtained without assuming the compactness and Lipschitz conditions for nonlocal functions. An example is given to illustrate the main results.


1.
Introduction. Impulsive dynamical systems reveal the various evolutionary processes, including those in engineering, biology and population dynamics which undergo abrupt changes in their state between intervals of continuous evolutions. In many evolution processes, such as optimal control models in economics, stimulated neural networks, frequency-modulated systems and some motions of missiles or aircrafts are characterized by the behavior of impulsive dynamical systems. In recent years, the analysis of impulsive systems are increasing due to their impact both in the theory and applications. We refer the reader for more facts of impulsive systems to [1,2,3,4,10,28,29].
Control theory deals with the behavior of dynamical systems. It is one of the basic concepts in mathematical control theory. By using various fixed point theorems, controllability of differential systems in Banach spaces under the assumption of compactness and noncompactness of the operator in semigroups has been studied by many authors [6,11,15,16,17,22,25,30,32,33,34,35,36].
The concept of nonlocal conditions can be applied in physics with improved effect than the usual initial condition x(0) = x 0 . Nonlocal condition was first initiated by Byszewski [8] and he investigated the existence and uniqueness of mild and classical solution of nonlocal Cauchy problems. In [9], he studied the existence and uniqueness of solutions of abstract functional differential equations with nonlocal conditions of the form x(a(t))), t ∈ J, where J = [t 0 , t 0 + b], b > 0 is constant, t 0 < t 1 < · · · < t m < t 0 + b, f : J × X × X → X and a : J → J are given functions , X is is a Banach space, x 0 ∈ X, c k ∈ R, c k = 0, (k = 1, 2, · · · , m), m ∈ N. The author pointed out that if c k = 0, (k = 1, 2, · · · , m), then the results of [9] can be applied to kinematics to determine the local evolution t → x(t) of a physical object for which we do not know the positions x(t 1 ), · · · , x(t m ), but the nonlocal conditions in (1) hold. We refer the reader for more facts of nonlocal systems to [19,20,27].
Recently, Chen et al. [12] considered the existence and uniqueness of strong solutions for semilinear evolution equations with nonlocal conditions. Liang et al. [21] studied the controllability of fractional integrodifferential evolution equations with nonlocal conditions. Motivated by the above works, this paper establishes sufficient conditions for the exact controllability results for abstract neutral impulsive differential evolution equations with nonlocal initial conditions in Banach spaces of the form where A(t) : D(A(t)) ⊂ X → X generates C 0 semigroup T (t)(t ≥ 0) in a Banach space X; g, f : J × X → X and I i : X → X, i = 1, 2, · · · , n are appropriate functions; the points 0 < t 0 < t 1 < · · · < t m < b are given and the symbol ∆x(t i ) represents the jump of the function x at t i , which is defined by ∆x(t i ) = x(t − i ) represent right and left hand limit of x(t) at t = t i respectively; the control function u(·) is considered in the space L 2 (J, V ), where V is a Banach space of control and B : V → X is a bounded linear operator.
We divided our work as follows. In the following section, we first introduce some notations and preliminaries which are used throughout this paper. In Section 3, exact controllability of abstract nonlocal Cauchy problem with impulsive conditions is established. In the last section, an example is given to demonstrate the application of the main results.

2.
Preliminaries. In this section, we recall some definitions, notations and results that we need in this paper. Throughout this paper, (X, · ) is a Banach and u(t + i ) exist, for all i = 1, 2, · · · , n. It is easy to see that PC is a Banach space with the norm Let {A(t) : t ∈ J} generates an evolution operator and let us assume the following hypotheses:
Under the assumptions (A1) − (A4), the family {A(t) : t ∈ J} generates an unique evolution system {U (t, s) : 0 ≤ s ≤ t ≤ b} satisfying: (a) There exists a positive constant M such that A two parameter family of bounded linear operators U (t, s), 0 ≤ s ≤ t ≤ b on X is called an evolution system if the following two conditions are satisfied: More details about evolution system can be found in Pazy [26] Lemma 2.2. [5] Let E + be the positive cone of an order Banach space (E, ≤). A function Φ defined on the set of all bounded subsets of the Banach space X with values in E + is called a measure of noncompactness (MNC) on X if β(coΩ) = Φ(Ω) for all bounded subsets Ω ⊆ X, where coΩ stands for the closed convex hull of Ω.
The MNC Φ is said to be, (1) Monotone if for all bounded subsets Ω 1 , Ω 2 of X we have,

if and only if Ω is relatively compact in X;
One of the examples of M N C is the noncompactness measure of Hausdorff β defined on each bounded subset Ω of X by β(Ω) = inf{ > 0; Ω has a finite -net in X}. It is well known that MNC β enjoys the above properties and other properties (see [5,18]), for all bounded subset Ω, Ω 1 , Ω 2 of X.
Let B ⊂ PC be bounded and equicontinuous. Then β(B(t)) is continuous on J and β PC (B) = max t∈J β(B(t)).
To prove the main results, for h ∈ C(J, X), we first consider the evolution equa- For the problem (3), let us assume: Lemma 2.5. Assume that the condition (H 0 ) holds. Then (3) has a unique mild solution x ∈ C(J, X) given by By operator spectrum theorem, the operator P : Definition 2.6. A function x := x(·; u) ∈ PC is said to be a mild solution of the system (2) if for any u ∈ L 2 (J, U ), x satisfies the following integral equation Definition 2.7. (Exact controllability [7]) The system (2) is said to be controllable on the interval J if and only if for every x 0 , x 1 ∈ X, there exists a control u ∈ L 2 (J, U ) such that the mild solution x(t) of (2) satisfies There is a difficulty in considering exact controllable for functional differential equations since the value x(b) = x T is often taken for that.
Consider the following linear control system We introduce the controllability operator associated with linear control system (5) by where B * and U * (t, s) denote the adjoint of B and U (t, s) respectively. It is clear that Γ b 0 is a linear bounded operator.

2])
Let Ω be a closed convex subset of a Banach space X and 0 ∈ Ω. Assume that Q : Ω → Ω is a continuous map, which satisfies Mönch's condition, i.e., for D ⊂ Ω is countable and D ⊂ co({0} ∪ Q(D)) ⇒ D is compact. Then Q has at least one fixed point in Ω.
3. Exact controllability. In order to establish the result, we need the following hypothesis: is a family of linear operators, A(t) : D(A) → X not depending on t and dense subset of X, generating an equicontinuous evolution system {U (t, s) : The function g : J ×X → X is continuous and there exist constants L, L 1 > 0 such that A(0)g(t, x(t)) ≤ L(||x|| + 1), for every (t, x) ∈ [0, b] × X, and the inequality for every 0 ≤ s 1 , s 2 ≤ b, x 1 , x 2 ∈ X.
(H 4 ) The function f : J × X → X satisfies the following conditions: (i) For each t ∈ J, the function f : (t, ·) : X → X is continuous, and for each x ∈ X, the function f (·, x) : J → X is strongly measurable. (ii) For any r > 0, there exists function f r ∈ L 1 (J, R + ) such that (ii) There exist a positive constant γ i such that (H 6 ) The linear control system (5) is exactly controllable.
, t ∈ J and any bounded subset D ⊂ X.
Proof. Let B r := {x ∈ PC : x PC ≤ r} for any r > 0. The B r is is a bounded, closed and convex subset in PC. For any x(·) ∈ PC, we introduce a control u(t) := u(t, x) by

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Define an operator Φ : To prove the exact controllability of the control system in (2), we shall use the Lemma 2.9 to show that the operator Φ has a fixed point in B r . For this, we divide the proof into two steps.
Step 1. Φ : B r → B r is continuous. First we prove that Φ(B r ) ⊆ B r for some r > 0. Suppose that, for each positive integer r, there exists x ∈ B r such that (Φx)(t) > r for some t ∈ J. Dividing on both sides by r and taking limit as r → ∞ and by the hypotheses (H 4 ) − (H 5 ) and the equation (8) Lb+γ +λ) which is a contradicts to (6). Hence for some positive r, Φ(B r ) ⊆ B r .
Further more by (H 3 ) − (H 5 ), it is easy to prove that Φ is continuous.
Step 2. Φ : B r → B r is a condensing mapping. We decompose Φ into Φ = Φ 1 +Φ 2 , where the operators Φ 1 , Φ 2 are defined on B r as , we will verify that Φ 1 is contraction and Φ 2 is compact operator.
To prove Φ 1 is contraction, let x 1 , x 2 ∈ B r . For each t ∈ [0, b] and by (H 2 ) and (7), we have which shows that Φ 1 is contraction.
To prove Φ 2 is compact, first we prove that Φ 2 is continuous. By (H 1 ) − (H 4 ) it is easy to prove that Φ 2 is continuous. To prove that Φ 2 is equicontinuous function on J for any 0 ≤ τ 1 ≤ τ 2 ≤ b and x ∈ B r , denote Then we have By (H 1 ), we can easily verify that T j → 0(j = 1, 2, 3.) as τ 2 → τ 1 .
Therefore Φ 2 is equicontinuous. It remains to prove that Then from compactness of U (t, s), (t, s) > 0, we obtain that is relatively compact in X for every 0 < < t. Moreover x ∈ B r , we have Therefore, there are relatively compact sets arbitrarily close to the set V (t) and hence V (t) is also relatively compact in X. Thus by Arzela-Ascoli theorem Φ 2 is compact operator. These arguments above enable us to conclude that Φ = Φ 1 + Φ 2 is condense mapping on B r , and by Lemma 2.9 there exists a fixed point for Φ on B r . Thus the system (2) is controllable on J. Proof. In order to apply Lemma 2.10, we define the control u and the function Φ as in Theorem 3.1. By step 1 of Theorem 3.1, Φ is continuous. Next to prove that Φ satisfies Mönch condition. Let D ⊂ B r be countable and D ⊂ co({0} ∪ Φ(D)).
We will show that D is relatively compact. From the properties of MNC β, it is enough to prove that β(D) = 0.
To estimate β(Φ(D)), we decompose Φ as in Theorem 3.1. Without loss of generality, we may suppose that D = {x n } ∞ n=1 . By (H 6 ), we have Since Φ 2 D is equicontinuous on every J i , by proposition 7.3 of [14], we have Then, according to inequality (10), we have that
Hence, by the property (7) of Lemma 2.2 Since (K 1 + K 2 ) < 1, we obtain β(D) = 0. That is, D is relatively compact. Hence by Lemma 2.10, Φ has atlaest one fixed point which is the mild solution of the system (2). Thus the system is controllable on J.
Then A generates a strongly continuous semigroup T (·) which is compact, analytic and self-adjoint.
(a) Also A has a discrete spectrum representation (−n 2 ) w, w n w n , w ∈ D(A), n ∈ N ; where w n (x) = 2 π sin(nx); n = 1, 2, · · · is the orthogonal set of eigenvector of A. The eigen values are −n 2 , n ∈ N .
(b) The operator A 1/2 is given by A 1/2 w = ∞ n=0 n w, w n w n on the space n < w, w n < w n ∈ X .
This shows that F and A 1/2 F both take values in X 1/2 in terms of properties (a) and (b) and therefore the function g. Since, for any x 1 , x 2 ∈ X 1/2