Controllability of fractional evolution systems of Sobolev type via resolvent operators

In this paper, we consider the nonlocal controllability of α∈(1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha\in (1,2)$\end{document}-order fractional evolution systems of Sobolev type in abstract spaces. By utilizing fixed point theorems and the theory of resolvent operators we establish some sufficient conditions for the nonlocal controllability of Sobolev-type fractional evolution systems.


Introduction
The theory of fractional differential equations admits wide applications in the fields of biology, physics, chemistry, control theory, and so on, and hence it has been regarded as an active aspect of mathematics in recent years. Controllability of fractional differential systems of order 0 < α < 1 has been investigated by many authors; we refer the readers to [6,13,[15][16][17] for more detail. However, as far as we know, the works on the fractional order 1 < α < 2 are limited. In 2013, using the Sadovskii fixed point theorem and vectorvalued operator theory, Li et al. [10] proved the controllability of fractional differential systems of order α ∈ (1,2] of the form In the present work, we consider the controllability of the fractional control system of Sobolev type with nonlocal conditions in a Hilbert space X of the form C D α t (Ex(t)) = Ax(t) + f (t, x(t)) + Bu(t), t ∈ J, where 1 < α < 2, C D α t denotes the Caputo fractional derivative operator of order α, A and E are two closed linear operators defined in X with domains D(A) and D(E), respectively, the control function u is given in L 2 (J, U), U is a Hilbert space, B is a bounded linear operator from U to X, and f , g, and h are appropriate functions to be specified later.
To deal with the Sobolev-type differential equations, the common assumptions are: (1) E, A are linear operators, and A is closed; (2) D(E) ⊂ D(A), and E is bijective; (3) E -1 is a compact operator. In this case, -AE -1 is a bounded operator, which generates a uniformly continuous semigroup; see [2,7] for more detail. In this paper, without assuming the existence and compactness of E -1 , we define the solution operator of (1.1) by fractional resolvent family generated by the pair (A, E). More precisely, we assume that the pair (A, E) generates an (α, 1)-resolvent family {C E α,1 (t)} t≥0 . Then we prove some properties of {C E α,1 (t)} t≥0 . Applying these properties and the Laplace transform, we define the solution operator of the fractional control system (1.1). By utilizing fixed point theorems and resolvent operator theory we obtain some controllability results without any compactness conditions on the (α, 1)-resolvent family {C E α,1 (t)} t≥0 .

Preliminaries
Let X be a Hilbert space with norm · and inner product ·, · X . We denote by C(J, X) the set of all X-valued continuous functions on J. Then C(J, X) is a Banach space with norm x C = sup t∈J x(t) . For 1 ≤ p < +∞, L p (J, X) denotes the Banach space of all Bochner- Let B(X) := B(X, X) be the Banach space of all bounded linear operators from X to X with operator norm · .
We recall some definitions of fractional calculus,; see [1,3,5] and the reference therein for more detail. For simplicity, for every ν ≥ 0, let where Γ is the gamma function. As usual, we define Denote by n = α the smallest integer greater than or equal to α.
Definition 1 Let u ∈ L 1 (J). The Riemann-Liouville fractional integral of order α > 0 is defined by

Definition 2
The Riemann-Liouville fractional derivative of order α > 0 is defined for all u ∈ L 1 (J) satisfying g n-α * u ∈ W n,1 (J) by where D n t = d n dt n .

Definition 3
The Caputo fractional derivative of order α > 0 is defined for all u ∈ L 1 (J) by If u ∈ C n [0, ∞), then the Caputo fractional derivative of order α ∈ (n -1, n) is By (1.23) of [1] the Laplace transform of Caputo fractional derivative is given by where n = α . We further introduce some results on fractional resolvent family; see [4,14] for more detail. We assume that A is a closed linear densely defined operator in X. Denote , and for all x ∈ X, In this case, {C E α,β (t)} t≥0 is called the (α, β)-resolvent family generated by the pair (A, E).

Definition 6
A function x ∈ C(J, X) is called a mild solution of (1.1) if for each t ∈ J, x satisfies the integral equation Proof Since M := sup t∈J C E α,1 (t) < +∞, by (2.4) and (2.5), for any t ≥ 0, we have Thus the conclusion is proved.
Now we recall some definitions and lemmas on the Hausdorff measure of noncompactness (H-MNC). Let D ⊂ X be a nonempty bounded subset of X. Denote by γ (D) the H-MNC of D with respect to X, that is, γ (D) := inf{ε > 0 : D has a finite ε-net in X}.
We denote by γ (·) and γ C (·) the H-MNCs of a bounded subset of X and C(J, X), respectively. Let B ⊂ C(J, X) be a bounded subset, and let t ∈ J. Then B(t) := {u(t) : u ∈ B} is a bounded subset of X, and γ (B(t)) ≤ γ C (B). It is well known (see [9]) that the H-MNC γ (·) has the following properties: For any bounded subsets D 1 , D 2 , and D of X, we have , if a mapping F : X → X satisfies the Lipschitz condition for all nonempty bounded subsets D of X.

Lemma 3
Let B ⊂ C(J, X) be a bounded and equicontinuous subset. Then γ (B(t)) is continuous on J, and γ C (B) = max t∈J γ (B(t)).

Lemma 5 Let X be a separable Hilbert space, and let B
To investigate the nonlocal controllability of system (1.1), we introduce the definition of controllability.
Definition 8 (Nonlocal controllability) System (1.1) is said to be nonlocally controllable on [0, b] if for all x 0 , y 0 , x 1 ∈ X, there exists a control u ∈ L 2 (J, U) such that the mild solution At the end of this section, we present a fixed point theorem, on which the proof of our main results are based.

Nonlocal controllability
In this section, we state and prove some results on the nonlocal controllability of system (1.1). The discussion is based on the theory of resolvent operators and fixed point theorems. For this purpose, we first make the following assumptions.
( J → X is strongly measurable; for each t ∈ J, f (t, ·) : X → X is continuous.
(H g ) g : C(J, X) → X, and there exists a constant L g > 0 such that (H B ) B : U → X is a linear bounded operator, and let M B := B . By assumption (H W ), for any x 1 ∈ X and x ∈ C(J, X), we define the control u x ∈ L 2 (J, U) as If x ∈ C(J, X) is a mild solution of system (1.1) corresponding to the control u x , then by (H W ) and (2.3) we have which implies x(b) + g(x) = x 1 , and system (1.1) is nonlocally controllable on J. Hence we will now prove that system (1.1) has mild solutions by using resolvent operator theory and fixed point theorems. Define the operator Q : C(J, X) → C(J, X) by By Definition 6 the mild solution of system (1.1) is equivalent to the fixed point of Q. We first apply the contraction mapping principle to prove that Q has a fixed point in C(J, X).

Lemma 7 Assume that conditions (H AE ), (H W ), (H f 2 ), (H g ), and (H h ) are satisfied. Then for all x, y ∈ C(J, X) and t ∈ J, we have
Proof For any x, y ∈ C(J, X) and t ∈ J, by the definition of u x and u y we have This completes the proof.
Proof For any x, y ∈ C(J, X) and t ∈ J, by (3.1) and Lemma 7 we have (Qx)(t) -(Qy)(t) From Then for any x ∈ Ω r and t ∈ J, we have Remark 2 If f satisfies the linear growth conditions, for example, f (t, x) ≤ a 1 (t)x + a 2 (t), t ∈ J, x ∈ X, where a 1 , a 2 ∈ L 1 (J, R), then condition (H f 2 ) holds when we choose ϕ r (t) = a 1 (t) r + a 2 (t) .
Proof It is obvious that the operator Q : C(J, X) → C(J, X) is continuous under these assumptions. Hence we just prove Q(Ω r ) ⊂ Ω r for some r > 0. If this were not true, then for any r > 0, there would be x ∈ Ω r such that r < (Qx)(t) for all t ∈ J. By Lemma 7 and (3.1) we have Dividing both sides by r and taking the lower limit as r → ∞, we obtain which is a contradiction to (3.4). Thus there is r > 0 such that Q(Ω r ) ⊂ Ω r .

Theorem 2 Let assumptions
Proof We define two operators Q 1 , Q 2 : C(J, X) → C(J, X) by and Then by (3.1) we know that Q = Q 1 + Q 2 . By (H AE ), (H g ), and (H h ) it is clear that Next, we prove that the set V := {Q 2 x : x ∈ Ω r } is relatively compact in C(J, X). To apply the Ascoli-Arzelà theorem, we prove that V := {Q 2 x : x ∈ Ω r } is equicontinuous in C(J, X) and V (t) := {(Q 2 x)(t) : x ∈ Ω r } is relatively compact in X. For any 0 ≤ t 1 < t 2 ≤ b and x ∈ Ω r , by Lemmas 2 and 8 we have By the assumption (H f 3 ), conv(V ε ) is also a compact set, where conv(V ε ) means the convex closure of V ε . By the mean value theorem for Bochner integrals, we deduce that x ∈ Ω r } is relatively compact in X. Moreover, for any x ∈ Ω r , we have x ∈ Ω r } is relatively compact in X. By the Ascoli-Arzelà theorem the set V is relatively compact. Hence γ C (V ) = γ C (Q 2 (Ω r )) = 0.
At last, by the properties of H-MNC and because of M(L g + bL h ) < 1, we obtain that which implies that Q : Ω r → Ω is a condensing mapping. By Sadovskii's fixed point theorem (see Lemma 6) Q has at least one fixed point x in Ω r , which is the mild solution of system (1.1) satisfying x(b) + g(x) = x 1 . Therefore system (1.1) is nonlocally controllable.
H-MNC condition is another important tool guaranteeing the compactness of the solution operator. In what follows, we assume that f satisfies the following H-MNC condition: (H f 4 ) There exists a constant L 1 > 0 such that for every countable subset D 0 ⊂ X.
where D 0 ⊂ Ω r is a countable subset of Ω r .
Proof By Lemma 5 we obtain that The proof is completed.

Conclusion
In this paper, we investigated the nonlocal controllability of α ∈ (1, 2)-ordered fractional evolution systems of Sobolev type of the form (1.1) in a Hilbert space X. We first define the (α, 1)-resolvent family {C E α,1 (t)} t≥0 generated by the pair (A, E). Without assuming the compactness of {C E α,1 (t)} t≥0 , we prove some nonlocal controllability results for the fractional evolution system (1.1) by using Banach's contraction mapping principle and Sadovskii's fixed point theorem. The discussion is based on fractional resolvent operator theory. Our results improve and extend some existing results.