Approximate Controllability of Neutral Measure Evolution Equations with Nonlocal Conditions

In this paper, we consider a kind of neutral measure evolution equations with nonlocal conditions. By using semigroup theory and fixed point theorem, we can obtain sufficient conditions for the controllability results of such equations. Finally, an example is given to verify the reliability of the results.


Introduction
In the past decades, the theory of impulsive differential equations has been fully developed. An impulsive differential equation is used to simulate the evolution process of a system under perturbation during continuous evolution [1][2][3]. However, this type of system allows only a limited number of discontinuities within a limited range. As a result, it cannot simulate some complex phenomena, such as Zeno's behavior. However, the dynamic system with discontinuous trajectory is modeled by a measure differential equation or measure-driven equation [4][5][6][7][8][9][10]. Measure differential equations (MDEs) were studied in the early days [11][12][13][14][15][16][17][18]. In 1971, Das first studied measure differential equations. For a complete introduction of measure differential systems, we can refer to [12,13].
On the other hand, the complete controllability of several nonlinear dynamic systems, such as stochastic systems of fractional order and dynamic systems of impulsive differential equations, has been extensively studied. Recently, some authors had discussed existence, stability, and nonlocal controllability of the measure evolution equation [9,15,[19][20][21][22]. In the past few years, the existence and controllability of fractional abstract functional differential development systems with nonlocal conditions have been fully studied [2,[23][24][25][26][27][28][29][30][31]. However, the controllability problem of neutral measure evolution equations with nonlocal conditions is seldom studied. erefore, whether the appropriate solution of the system exists for any given control u and whether the system is approximately controllable are studied.
In the paper, we will study the following neutral measure evolution equations with nonlocal conditions: where f, q: [0, a] × G([− r, 0]; X) ⟶ X, x t (θ) � x(t + θ), θ ∈ [− r, 0], r > 0, p: G([− r, a]; X) ⟶ X is a specified function. e state variable x(·) takes values in Banach space X with the norm ‖ · ‖. A: D(A) ⊂ X ⟶ X generates a uniformly bounded analytic semigroup T(t) { } t≥0 in Banach space X. g: [0, a] ⟶ R is a nondecreasing leftcontinuous function. e control function u(t) takes values in another Banach space U, where U is a control set. e set G([− r, 0]; X) and G([− r, a]; X) represent the regulated functions space on [− r, 0] and [− r, a], respectively. e rest of this paper is organized as follows. In Section 2, some notations and preparation are given about Kurzweil-Henstock-Stieltjes integrals and regulated functions. In Section 3, we obtain the existence results for measure evolution system (1) by using Schauder's fixed point theorem. In Section 4, based on Krasnoselskii's fixed point theorem, we establish a controllability result for mild solutions of system (1). An example is given to prove validity of the results we obtained in Section 5.

Preliminaries
In this section, we will review some concepts and main results regarding Kurzweil-Henstock-Stieltjes integrals and regulated functions.
Consider a function δ: Definition 1 (See [22] for every δ ε -fine partition be a space of all p-ordered K-H-Stieltjes integral regulated functions from J to X with respect to g, with the norm ‖ · ‖ K p g defined by Definition 2 (See [7]). Let X be a Banach space with a norm ‖ · ‖ and [a, b] be a closed interval of the real line. A function f: exist and are finite. e space of all regulated functions f: [a, b] ⟶ X is denoted by G([a, b]; X). It is well known that the set of discontinuities of a regulated function is at most countable and that the space G([a, b]; X) is a Banach space endowed with the norm ‖f‖ ∞ � sup t∈ [a,b] |f(t)|. Let r > 0, for any element z ∈ G([− r, 0]; X), and we define the norm ‖z‖ * � sup s∈[− r,0] |z(s)|.
Let Y be another separable reflexive Banach space where control function u takes values. Let E ⊂ Y be bounded, and admissible control set U � K p g (J; E)(p > 1).
Lemma 1 (See [7]). Consider the functions f: J ⟶ X and g: J ⟶ R such that g is regulated and b a fdg exists. en,

] is regulated and satisfies
where where g(t − ) and g(t + ) denote the left limit and the right limit of the function g at t, respectively. Lemma 4 (See [7]). Let x n ∞ n�1 be a sequence of functions from [a, b] to X. If x n converges pointwisely to x 0 as n ⟶ ∞ and the sequence x n ∞ n�1 is equiregulated, then x n converges uniformly to x 0 .
For 0 < α < 1, (− A) α can be defined as a closed linear invertible operator with its domain D(− A) α being dense in X. We denote by X α the Banach space D(− A) α endowed with norm ‖x‖ α � ‖(− A) α x‖ which is equivalent to the graph norm of (− A) α . We have X β ↪X α for 0 < α < β, and the embedding is continuous [32].
Lemma 5 (See [32]). e following properties hold: (1) If 0 < α < β, then X α ⊂ X β and the embedding is compact whenever the resolvent operator of A is compact Lemma 6 (See [31]). Let M be a closed convex nonempty subset of a Banach space (S, | · |). We suppose that P and Q map M into S such that

Existence Results
In this section, we will study the existence of the mild solution of system (1). We first give the definition of solutions for measure system (1). Using the methods same as in [33], we can obtain the following definition.

Definition 4.
e function x ∈ G(J; X) is called a mild solution of system (1) on J if it satisfies the following measure integral equation: We introduce the following assumptions: and the inequality for all ; X) ⟶ X is continuous and compact, and there exist positive constants e and f such that

Theorem 1. If assumptions (H1)-(H5) hold, then problem (1) has a mild solution provided that
Me Proof. We define the operator Q: Let l > 0 be a constant and B l � G(J; X): ‖x‖ ∞ ≤ l , where B l is a bounded, closed, and convex set, Step I: we prove that there exists a constant l > 0 such that Q(B l ) ⊂ B l . Assuming that this conclusion is not true, for each l > 0, there will exist x l ∈ B l , t l ∈ J such that ‖Qx l (t l )‖ > l. According to (H1)-(H5),
Step II: Q(B l ) is equiregulated.

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where e combination of the compactness of semigroup T(t), (t > 0) and its strong continuity shows the continuity of T(t) in the sense of uniform operator topology and p(x), q(t, x t ), f(t, x t ) are bounded, and applying dominated convergence theorem, I 1 , I 3 , as t ⟶ t + 0 , I 6 ⟶ 0. Moreover, let H(t) � t 0 m(s)dg(s), according to Lemma 1, and H(t) is a regulated function; then, as t ⟶ t + 0 . Also, we can follow the similar procedure to Step III: Q: B l ⟶ B l is a continuous operator. Let x n be a convergent sequence in B l with x n ⟶ x ∈ B l as n ⟶ ∞. According to hypothesis (H2)-(H5) and the boundedness of T(t), we have, for each s ∈ J, f s, x n s ⟶ f s, x s , q s, x n s ⟶ q s, x s , Journal of Mathematics as n ⟶ ∞, and then, by the dominated convergence theorem, we get In addition, the analysis same as in Step II demonstrates that Qx n { } ∞ n�1 is equiregulated on J. is property and the abovementioned verification together with Lemma 4 imply that Qx n converges uniformly to Qx(t) as n ⟶ ∞, namely, erefore, QB l is a continuous operator on J.
Step IV: be fixed, and let η be a given real number satisfying 0 < η < t. For every x(·) ∈ B l , we define

(23)
Since T(η) is compact, the set V η (t) � (Q η x)(t): x(·) ∈ B l } is relatively compact in X for every η, 0 < η < t. On the other hand, for every x(·) ∈ B l in view of assumption (H2)-(H3), we have By the left continuity of g and Lemma 1, as η ⟶ 0 + , ‖(Qx) − (Q η x)‖ ⟶ 0. erefore there are relatively compact sets arbitrarily close to the set V(t). Hence, for each t ∈ J, V(t) is a relatively compact set in X.
Step II and Step IV together with Lemma 2 imply that the set Q(B l ) is relatively compact in G(J; X). Hence, Q is a completely continuous operator on B l . By Schauder's fixed point theorem (Lemma 3), Q has a fixed point in B l , which is a mild solution of measure control system (1). e proof is completed.

Controllability Result
In this section, to investigate the controllability of system (1), we first give the definition of controllability.

Definition 5.
e system (1) is said to be controllable on the interval J if for every initial function x 0 ∈ G([− r, 0]; X) and x 1 ∈ X, there exists a control u ∈ K p g (J; E)(p > 1) such that the mild solution x(t) of (1) satisfies x(a) � x 1 .
We assume the following conditions: and for all φ 1 , φ 2 ∈ G([− r, a]; X), function p satisfies

Theorem 2. If hypotheses (H1)-(H9) are satisfied, system (1) is controllable on J.
Proof. Using assumption (H7) for an arbitrary function x(·), we define the control In what follows, it suffices to show that when using this control, the operator F defined by has a fixed point x(·) from which it follows that this fixed point is a mild solution of system (1). Clearly, x(a) � (Fx)(a) � x 1 , from which we conclude that the system is controllable. Let where ϕ(t) is taken as We define the operators F 1 and F 2 by

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Obviously, the operator F has a fixed point if only if operator F 1 + F 2 has a fixed point. us, we shall employ Krasnoselskii's fixed point theorem.
Step I: we have to show (F 1 + F 2 )B l ⊂ B l , i.e., any ϕ 1 , ϕ 2 ∈ B l implies that F 1 ϕ 1 + F 2 ϕ 2 ∈ B l . It is easy from hypotheses (H1-H9) and Lemma 1 to see that and thus, condition (i) in Lemma 6 is verified.
Step II: next, we need to show that operator F 1 is continuous.
Step III: we shall show that F 1 maps B l into an equicontinuous family. For y ∈ B l , θ 1 , θ 2 ∈ J, 0 < θ 2 < θ 1 ≤ a, we have 8 Journal of Mathematics where Hence, F 1 B l is equicontinuous. Since the case θ 2 < θ 1 < 0 or θ 2 < 0 < θ 1 is very simple, the proof of the equicontinuities for the two cases is omitted. Subsequentially, we shall show that F 1 B l is precompact. Let 0 < t ≤ a be fixed and ε be a real number satisfying 0 < ε < t. For y ∈ B l , we define T(t) is a compact operator, and Y ε (t) � (F ε 1 y)(t): y ∈ B l is relatively compact in X.
For every ε, 0 < ε < t. Moreover, for every y ∈ B l , we have ε ⟶ 0 + , (F 1 y)(t): Y ∈ B l is precompact in X. F 1 B l is uniformly bounded. By the Arzela-Ascoli theorem, it is concluded from the uniform boundedness, equicontinuity, and precompactness of the set F 1 B l that F 1 B l is compact.
Step IV: F(B l ) is equiregulated on J.

Conclusions
In this paper, the issue on approximate controllability of neutral measure evolution equations has been addressed, which can model a large class of hybrid systems without any restriction on their Zeno behavior. Firstly, by adopting Schauder's fixed point theorem, the existence results of mild solutions for this type of measure control system corresponding to some control function are obtained. en, the approximate controllability results are provided. Finally, we also use an example to illustrate the main result. Furthermore, we will investigate measure functional evolution equations of Sobolev type in the next work.

Data Availability
ere are no underlying data in the results.

Conflicts of Interest
e authors declare that they have no conflicts of interest.