Nonlocal controllability of fractional measure evolution equation

In this paper, we consider the following kind of fractional evolution equation driven by measure with nonlocal conditions: $$ \textstyle\begin{cases} {}^{C} D_{0+}^{\alpha }x(t)=Ax(t)\,dt+ (f(t,x(t))+Bu(t) )\,dg(t), \quad t\in (0, b],\\ x(0)+p(x)=x_{0}. \end{cases} $${D0+αCx(t)=Ax(t)dt+(f(t,x(t))+Bu(t))dg(t),t∈(0,b],x(0)+p(x)=x0. The regulated proposition of fractional equation is obtained for the first time. By noncompact measure method and fixed point theorems, we obtain some sufficient conditions to ensure the existence and nonlocal controllability of mild solutions. Finally, an illustrative example is given to show practical usefulness of the analytical results.


Introduction
Measure driven equations were investigated firstly in [1,2], and they can permit an infinite number of discontinuous points in a finite time interval, so it is convenient to model discontinuous dynamical systems. Differential equations and difference equations are special cases of measure differential equations. One can refer to [3,4] for the applications of measure differential equations such as modeling the quantum. Some recent papers have investigated the existence of solutions for measure differential equations (see [5][6][7][8]).
(see [35]), Cao and Sun in [36] discussed the complete controllability of measure differential equations by using Monch fixed point theorem and noncompact measure. Measure differential equation has developed rapidly, but it mainly focuses on integer order, there are few results on fractional order. Inspired by the above discussion, in this paper, we investigate a kind of fractional evolution equations driven by measure with nonlocal conditions. If g is an absolutely continuous function, then Eq. (1.1) becomes a fractional evolution equation; if g is the sum of an absolutely continuous function with a step function, then Eq. (1.1) becomes an impulsive fractional evolution equation. So fractional evolution equations driven by measure are more general. Since fractional measure differential equations are not as continuous or smooth as ordinary differential equations, they are only right continuous and bounded, this brings some difficulties to the further study of them. In order to solve this problem, we prove the regulated proposition of fractional equation for the first time. By noncompact measure method and fixed point theorems, we obtain some sufficient conditions to ensure the existence and nonlocal controllability of mild solutions. If the infinitesimal generator operator is noncompact, similar results can also be derived.
In this paper, we consider the following fractional measure evolution differential equation with nonlocal conditions: is a space of regulated functions on [0, b], which will be given later. The control function u(·) takes values in U ad , where U ad is a control set. g : [0, b] → R is a left continuous nondecreasing function.
The rest of the paper is organized as follows. Section 2 introduces some fundamentals that will be used later. In Sect. 3, we derive the existence result for fractional measure evolution Eq. (1.1) by means of Darbo-Sadovskii's fixed point theorem and noncompact measure. In Sect. 4, by Krasnoselskii's fixed point theorem, we obtain the controllability results for fractional measure evolution Eq. (1.1); if the infinitesimal generator operator is noncompact, similar results can also be derived. An illustrative example is given to show the practical usefulness of the analytical results in Sect. 5.

Preliminaries
In this section, we recall some basic concepts which will be used in what follows.
Definition 2.1 (Regulated function, see [8]) If a function f : K → X satisfies the limits then the function f is called regulated function on K .
By G(K; X) we denote the Banach space of all regulated functions with the norm f ∞ = sup t∈K f (t) , and the set of discontinuous points of a regulated function is at most countable. Definition 2.2 (Henstock-Lebesgue-Stieltjes integration, see [8]) A function f : Let HLS p g (K; X) (p > 1) be a space of all p-ordered Henstock-Lebesgue-Stieltjes integral regulated functions from K to X with respect to g, with the norm · HLS p g defined by Let Y be another separable reflexive Banach space where control function u takes values. Let E ⊂ Y be bounded, and the admissible control set U ad = HLS p g (K; E), p > 1.

Proposition 2.3
Consider the functions f ∈ HLS p g (K; X) (p > 1) and g : K → R satisfying that g is regulated. Then the function where q > 1, 1 , g(t + ) and g(t -) denote the right and left limits of function g at point t.
Proof Claim I: We prove that the function j(t) is regulated, i.e., lim τ →t -j(τ ) = j(t -), t ∈ (0, b]. The other direction can be proved in a similar way. For this purpose, we consider By the definition of Henstock-Lebesgue-Stieltjes integration, we have In terms of the regulated proposition of g and dominated convergence theorem, one has Claim II: As the proof of Claim I, we can easily derive the following inequality: This completes the proof. Definition 2.4 (Equiregulated set, see [8]) A set D ⊂ G(K; X) is called equiregulated if there is ν > 0; for every t 0 ∈ K and > 0, we have Lemma 2.5 (Uniform convergence, see [8]) Let {x n } ∞ n=1 be a sequence of functions from K to X. If the sequence {x n } ∞ n=1 is equiregulated and x n converges pointwisely to x 0 as n → ∞, then x n converges uniformly to x 0 . Definition 2.6 (Riemann-Liouville integral and derivative) The Riemann-Liouville fractional integral and derivative are defined respectively by where * denotes the convolution,

Definition 2.7 (Caputo derivative) The Caputo derivative of order
Now, we introduce the Hausdorff noncompact measure σ (·) defined on each bounded subset Ω of Banach space X by is a metric space. For more details, see [37]. Some basic properties of σ (·) are given in the following lemmas. Lemma 2.8 (see [37]) The noncompact measure σ (·) satisfies: Since the Lebesgue-Stieltjes measure is a regular Borel measure, then we refer to Theorem 3.1 in [38], the following result can be derived.
Theorem 2.10 (Darbo-Sadovskii fixed point, see [37]) If D ⊂ X is a convex bounded and closed set, the continuous mapping Z : D → D is a σ -contraction, then Z has at least one fixed point in D.
Theorem 2.11 (Krasnoselskii fixed point, see [39]) If B ⊂ X is bounded closed and convex, where X is a Banach space, operators P : D → X and Q : D → X satisfy the following conditions: (ii) P is a compact and continuous mapping; (iii) Q is a contraction operator. Then P + Q has a fixed point in D.
Referring to the definition of mild solution given in [21,40], we define the mild solution for fractional measure evolution Eq. (1.1) in the space of regulated function G(K; X) as follows.

Definition 2.12
A function x ∈ G(K; X) is said to be a mild solution of problem (1.1) if it satisfies Lemma 2.13 (see [40]) T α , S α have the following properties: (i) For every fixed t ≥ 0, the operators T α and S α are all linear and bounded, i.e., for each x ∈ X, The operators T α and S α are all compact if T(t) (t > 0) is compact for any t ≥ 0. (ii) T α and S α are all strongly continuous operators.

Existence of solution
In this section, by using the measure of noncompactness and fixed point theorem, we obtain a sufficient condition in order to ensure the existence of a mild solution.
The following hypotheses will be used: is measurable for all x ∈ X, and f (t, ·) is continuous for a.e. t ∈ K ; (ii) There is a function h ∈ HLS p g (K; R + ) and a nondecreasing continuous function Φ : (H5) The control function u is given in U ad , a Banach space of admissible control functions.
Define the operators F, F 1 , F 2 as follows: The proof process is divided into four steps.
Step I. We can find a positive number r such that F(B r ) ⊆ B r . If this is not the case, there is a function x r satisfying F(x r )(t) > r for some t ∈ K . According to assumptions (H1)-(H4), we get We can divide both sides of this inequality by r and take the limit r → +∞ in both sides to get which contradicts (3.1), so we can find r satisfying F(B r ) ⊆ B r .
Step II. F is continuous on B r . Let x n → x ∈ B r as n → ∞, where {x n } ∞ n=1 is a sequence in B r . Then we have From the continuity of p(x) and (H2)(i), we derive that the operator F : B r → B r is continuous. Step In terms of conditions (H2) and (H3) we know that the sets {p(x) : x ∈ B r } and {f (s, x(s)) : s ∈ K, x ∈ B r } are bounded. Moreover, according to the compactness and strong continuity of T(t), we know T(t) satisfies uniform operator topology continuity, and applying dominated convergence theorem, we can derive that A 1 , A 2 , A 4 all tend to zero independently of

s) = (ts) α-1 h(s), referring to Proposition 2.3, j(t) : K → X is a regulated function, we can obtain
when t → t + 0 , we have A 3 → 0. In a similar way, we know that A 5 → 0 as t → t + 0 . According to the above discussion, we can also derive that Step IV. F is a contraction mapping. In fact, for x, y ∈ B r , 0 < t < b, we have In the following we prove that F 2 is a compact operator. In terms of Step II, we know that lim F 2 x n = F 2 x as n → ∞, F 2 is continuous on B r . Let t be fixed, where t ∈ K , φ is a positive constant and satisfies 0 < φ ≤ t for each

s, x(s) + Bu(s) dg(s).
It follows from the compactness of S α that the set F 2φ x(t) = {F 2φ x(t) : x(·) ∈ B r } is relatively compact in X for each 0 < φ < t. On the other hand, for any x(·) ∈ B r , under condition (H2), we have According to hypothesis (H2), we know that t t-φ (ts) α-1 h(s) dg(s) is regulated and continuous from the left, then the last inequality tends to zero as φ → 0 + . So, for each t ∈ K , F 2 is relatively compact on B r for any D ⊂ B r , we obtain that By (3.1) we know cM 1 < 1, so the operator F is condensed.
According to Darbo-Sadovskii's fixed point theorem, we know that the fractional measure evolution problem (1.1) has a solution on the interval K . This completes the proof.

Nonlocal controllability
In this section, we obtain some sufficient conditions ensuring the nonlocal controllability of mild solutions by employing the measure of noncompactness and fixed point theorem.
Definition 4.1 (Nonlocally controllable, see [36]) If for each x 0 , x 1 ∈ X there is a control u ∈ U ad such that the mild solution x(·) of (1) satisfies that x(b) + q(x) = x 1 , then system (1.1) is said to be nonlocally controllable on K .
Furthermore, we suppose that (H6) There is a function W ∈ HLS p g (K; R + ) (p > 1) such that (H7) Define an operator Λ ∈ HLS p g (K; X) (p > 1) by and assume it satisfies the following: (i) Operator Λ -1 taking value in HLS p g (K; X)/ ker Λ exists and there is a positive constant M 3 such that (ii) There is a function J ∈ HLS p g (K; R + ) (p > 1) such that, for almost t ∈ K and any bounded set D ⊆ X, Proof Let x 0 ∈ X be fixed. Define the operators F, F 1 , F 2 as follows: From condition (i) of (H7), we can define the control for arbitrary function x(·) ∈ G(K; X) as follows: Let r ≥ 0 and B r = {G(K; X) : The proof process is divided into three steps.
Step I. Claim 1: we can find a positive number r such that F(B r ) ⊆ B r .
Claim 2: we show that, for any x, x ∈ B r , we can derive In order to prove Claim 1, we can prove by contradiction. We suppose that there is a function x r (·) ∈ B r satisfying F(x r )(t) > r for some t ∈ K . According to assumptions (H1)-(H4), we have where Due to F(x r )(t) > r, so we have We divide the above inequality on both sides by r, then passing to the lower limit as r → +∞, we have It is a contradiction to (4.1). So we can find some positive number r satisfying F(B r ) ⊆ B r . Based on Claim 1, it is easy to prove Claim 2 as follows: For t ∈ K and x, x ∈ B r , we get that Step II. F 1 is continuous on B r .
Let lim x n → x ∈ B r as n → ∞, where {x n } ∞ n=1 is a sequence in B r . Then we have from the continuity of p(x), we derive that the operator F 1 : B r → B r is continuous.
Step III. The operator F 1 is compact.
due to the strong continuity of T α , we can conclude that lim μ→ν T α (ν)-T α (μ) = 0, which implies that F 1 (B r ) is equicontinuous. According to Ascoli theorem, we get F 1 is a compact mapping.
Step IV. F(B r ) is equiregulated on K . For any t 0 ∈ [0, b), we get In terms of the conditions we know that the set {p(x) : x ∈ B r } and {f (s, x(s)) : s ∈ K, x ∈ B r } are bounded. Furthermore, according to the compactness and strong continuity of T(t), we know that T(t) satisfies uniform operator topology continuity, and applying dominated convergence theorem, we can derive that A 1 , A 2 , A 4 all tend to zero independently of x as where k α (s) = (ts) α-1 h(s) referring to Proposition 2.3, j(t) : K → X is a regulated function, we can obtain In a similar way, we know that A 5 → 0 as t → t + 0 . According to the above discussion, we can also derive that Step V. F 2 is a contraction mapping.
In terms of assumptions (H2)(iii) and (H4)(ii), we can obtain Hence we can derive that operator F 2 is condensed. According to Krasnoselskii's fixed point theorem, we see that the operator F has a fixed point in B r . Then Hence, the fractional measure evolution system (1.1) is nonlocally controllable on the interval K . The proof is completed.
(Ĥ1) The C 0 -semigroup T(t) generated by a linear operator A : D(A) ⊆ X → X is equicontinuous for t > 0. and Proof We can use the similar way in Theorem 4.2 to derive the result.

Conclusions
This paper is concerned with the existence and nonlocal controllability of mild solution for fractional evolution equation driven by measure with nonlocal conditions for the first time. We prove the regulated proposition of fractional equation firstly, then by constructing operator F, using the noncompact measure method and different fixed point theorems, we derive some sufficient conditions to guarantee the existence and nonlocal controllability of mild solutions. Finally, an illustrative example is given to show the practical usefulness of the analytical results. Furthermore, we will investigate the fractional measure evolution equations with Riemann-Liouville and Hilfer fractional derivatives in the next work.