Controllability of Hilfer fractional Langevin evolution equations

The existence of fractional evolution equations has attracted a growing interest in recent years. The mild solution of fractional evolution equations constructed by a probability density function was first introduced by El-Borai. Inspired by El-Borai, Zhou and Jiao gave a definition of mild solution for fractional evolution equations with Caputo fractional derivative. Exact controllability is one of the fundamental issues in control theory: under some admissible control input, a system can be steered from an arbitrary given initial state to an arbitrary desired final state. In this article, using the (α, β) resolvent operator and three different fixed point theorems, we discuss the control problem for a class of Hilfer fractional Langevin evolution equations. The exact controllability of Hilfer fractional Langevin systems is established. An example is also discussed to illustrate the results.

1. Introduction

The application of fractional differential equations to many engineering and scientific disciplines is very important, as numerous fractional-order derivatives are used in the mathematical modeling in the fields of physics, chemistry, electrodynamics of complex media, and polymer rheology, see [1–10]. Currently, fractional differential equations are used extensively in every branch of science, for example, the electrical closed loops can be expressed as fractional equations by Kirchhoff's law [11]. In 2000, Hilfer introduced the definition of Hilfer fractional derivative $D_{0+}^{\alpha ,\beta }$. Especially, $D_{0+}^{\alpha ,0}$ became the famous Riemman–Liouville fractional derivative whereas $D_{0+}^{\alpha ,1}$ coincided with another fractional derivative, namely, the Caputo fractional derivative.

The study of fractional differential equations in infinite dimensional spaces includes the theoretical aspects, such as the existence and uniqueness of solutions, the numerical solutions, and so on. In general, it is interesting to find the existence of mild solutions, to arrive at the fact that, some technical tools, such as the method of lower and upper solutions and various fixed point theorems, are usually applied to the proof of existence.

The exact or approximate controllability is important in control theory. With some control input, a system can be guided from an initial state to any desired ultimate state. There are various articles with respect to the exact or approximate controllability of fractional differential equations [12–16]. However, a few articles have been written about the exact controllability of Hilfer fractional evolution equations.

Langevin first proposed a Brownian motion equation in 1908 and Langevin's equation was named so from then on. There have been a remarkably large number of frequently used theories to explain how physical phenomena evolve in fluctuating environments with respect to the Langevin equation. For example, if white noise is taken to be the random fluctuation force, Brownian motion can be well-described by the Langevin equation. More generally, if white noise is not taken to be the random fluctuation force, the generalized Langevin equation can be used to describe the particle's motion [17]. The formulation of Langevin equation is not unique. Currently, several versions of the conventional Langevin equation have been used in complex media to describe dynamical processes in a fractal medium, the reader can consult [18–21].

In 2012, Ahmad et al. [18] investigated the following fractional Langevin equation:

$\begin{array}{l}\{\begin{array}{l}{}^c{D}^{\beta }({}^c{D}^{\alpha }+\lambda )x(t)=f(t,x(t)),0<t<1,0<\alpha \le 1,1<\beta \le 2,\\ x(0)=0,x(\eta )=0,x(1)=0,\end{array}\end{array}$

where ^cD^α denotes the Caputo fractional derivative, and the authors obtained the existence of solutions by Krasnoselskii's fixed point theorem and the Banach contraction mapping theory, respectively.

In 2018, Lv et al. [22] considered approximative controllability of Hilfer fractional differential equations:

$\begin{array}{l}\{\begin{array}{l}{D}_{0+}^{\alpha ,\beta }x\left(t\right)+Ax\left(t\right)=f\left(t,x\left(t\right)\right)+\left(Bu\right)\left(t\right),t\in \left(0,b\right],\\ {\displaystyle \underset{t\to 0+}{lim}}\left(I_{0+}^{\left(1-\beta \right)\left(2-\alpha \right)}x\right)\left(t\right)=0,{\displaystyle \underset{t\to 0+}{lim}}\frac{\text{d}}{\text{d}t}\left(I_{0+}^{\left(1-\beta \right)\left(2-\alpha \right)}x\right)\left(t\right)=b_1,\end{array}\end{array}$

where $D_{0+}^{\alpha ,\beta }$ denotes the Hilfer fractional derivative, A ∈ Sect(θ), where $\theta \in \rho (A)\cap [0,(1-\frac{\alpha }{2})\pi )]$, and b₁ is an element in Banach space X. The control term u ∈ L^p(J, U), the approximate controllability of the above system, was discussed.

Recently, Gou et al. [23] discussed the controllability of an impulsive evolution equation. They proved that the system is controllable on J under the Mönch fixed point theorem.

However, controllability of the Hilfer Langevin evolution equation has received little attention. For the above-mentioned aspects, we discuss the controllability for a class of Hilfer Langevin evolution equations of the form:

$\begin{array}{ll}\{\begin{array}{l}{D}_{0+}^{{\alpha }_1,{\beta }_1}\left(D_{0+}^{{\alpha }_2,{\beta }_2}+A\right)x\left(t\right)=f\left(t,x\left(t\right)\right)+Bu\left(t\right),t\in J=\left[0,b\right],b>0,\\ {\displaystyle \underset{t\to 0+}{lim}}\left(I_{0+}^{\left(1-{\alpha }_2\right)\left(1-{\beta }_2\right)}x\right)\left(t\right)=0,{\displaystyle \underset{t\to 0+}{lim}}\frac{\text{d}}{\text{d}t}\left(I_{0+}^{\left(1-{\alpha }_1\right)\left(1-{\beta }_1\right)+\left(1-{\alpha }_2\right)}x\right)\\ \left(t\right)+h\left(x\right)=x_0,\end{array}& (1.1)\end{array}$

where $D_{0+}^{{\alpha }_i,{\beta }_i}$, i = 1, 2 denotes the Hilfer fractional derivative, respectively. 0 < α_i ≤ 1, 0 ≤ β_i ≤ 1, satisfies 1 < α₁ + α₂ ≤ 2. A generates a strongly continuous (α₂, δ)-resolvent family S_{α2, δ}(t) (t ≥ 0), where 0 < δ ≤ α₁ + α₂. The function f:J × E → E, let U be a Banach space, the control term u ∈ L²(J, U), B:U → E is linear and bounded.

This article aimed to study the controllability of system 1.1. The main approach is based on three different fixed point theorems and the properties of (α₂, δ)-resolvent operators. The structure of this article is given as follows: In Section 2, we list some notations, definitions, and preliminaries, which will be used in the next section. In Section 3, Theorem 3.1 is obtained without the compactness of the resolvent family, and Theorems 3.2 and 3.3 are obtained via compactness. Section 4 is devoted to illustrating the application of the results by an example.

2. Preliminaries and Lemmas

Throughout we let E be a Banach space with norm ‖·‖. The space C(J, E) denotes the space of continuous functions on J and taking values in E, with the norm $\Vert x{\Vert }_C=\underset{t\in J}{max}\Vert x(t)\Vert$, for x ∈ C(J, E). We consider the L^p(J, R⁺) of Lebesgue p-integrable functions with 1 < p < ∞ on J, and let $\Vert f{\Vert }_{L^p}$ denote the norm of L^p(J, R⁺). Let B(Y, X) denote the space of bounded linear operators from Y to X, B(X) = B(X, X) for short. Let A ∈ B(E), ρ(A) is defined by the set of {λ:(λI − A)⁻¹ exists in B(E)}.

Let g_γ(γ > 0) denote the function

$\begin{array}{l}{g}_{\gamma }(t)=\{\begin{array}{ll}\frac{t^{\gamma -1}}{\Gamma (\gamma )},& t>0,\\ 0,& t\le 0.\end{array}\end{array}$

For two given functions f₁ and f₂, the convolution of them is expressed in the form $(f_1*f_2)(t)={\int }_0^t{f}_1(t-s)f_2(s)\text{d}s$.

Definition 2.1. Li et al. [24] {S(t)}_{t ≥ 0} ⊂ B(E) is called exponentially bounded (EB) if there are constants ω ∈ R and M > 0, such that

$\Vert S(t)\Vert \le M{\text{e}}^{\omega t},\text{for all}t>0.$

ω or more precisely (M, ω) is called a type of S(t).

Definition 2.2. Kilbas et al. [8] Let γ > 0, the γ-order Riemann–Liouville fractional integral of function f:[0, ∞) → R is given by $I_{0+}^{\gamma }f(t)=(g_{\gamma }*f)(t)$, t > 0.

Definition 2.3. Hilfer et al. [25] The Hilfer fractional derivative $D_{0+}^{{\alpha }_1,{\beta }_1}f(t)$ of order α₁ ∈ (n − 1, n] and type β₁ ∈ [0, 1] is defined by

$D_{0+}^{{\alpha }_1,{\beta }_1}f\left(t\right)=\left(I_{0+}^{{\beta }_1\left(n-{\alpha }_1\right)}\frac{{\text{d}}^n}{\text{d}{t}^n}\left(I_{0+}^{\left(1-{\beta }_1\right)\left(n-{\alpha }_1\right)}f\right)\right)\left(t\right).$

If f is taking values in E, then the corresponding integrals of the above two definitions are given in the sense of Bochner.

Lemma 2.1. Hilfer [6] Let f ∈ L(0, b), n − 1 < α₁ ≤ n, 0 ≤ β₁ ≤ 1, and $I_{0+}^{(1-{\beta }_1)(n-{\alpha }_1)}f\in A{C}^k\left[0,b\right]$, then

$\begin{array}{l}\begin{array}{l}\left(I_{0+}^{{\alpha }_1}{D}_{0+}^{{\alpha }_1,{\beta }_1}f\right)\left(t\right)=f\left(t\right)\\ -{\displaystyle \sum _{k=0}^{n-1}}\frac{{\left(t-s\right)}^{k-\left(n-{\alpha }_1\right)\left(1-{\beta }_1\right)}}{\Gamma \left(k-\left(n-{\alpha }_1\right)\left(1-{\beta }_1\right)+1\right)}{\displaystyle \underset{t\to 0+}{lim}}\frac{{\text{d}}^k}{\text{d}{t}^k}\left(I_{0+}^{\left(1-{\beta }_1\right)\left(n-{\alpha }_1\right)}f\right)\left(t\right).\end{array}\end{array}$

Definition 2.4. Chang et al. [26] Let A be a closed linear operator in Banach space E with domain D(A) ⊂ E. Assume that α, β > 0, A is called the generator of the resolvent family (α, β), if there exists an ω ≥ 0 and S_{α, β} is strongly continuous from [0, ∞) to B(E), such that S_{α, β}(t) is EB, {λ^α:(λ^αI − A)⁻¹ exists in B(E), Reλ > ω},

$\begin{array}{ll}{\lambda }^{\alpha -\beta }{({\lambda }^{\alpha }I-A)}^{-1}x={\displaystyle {\int }_0^{\infty }}{\text{e}}^{-\lambda t}{S}_{\alpha ,\beta }(t)x\text{ d}t,\text{Re}\lambda >\omega ,x\in E.& (2.1)\end{array}$

Then {S_{α, β}(t)}_t≥0 is called the resolvent family (α, β) generated by operator A. It is simply said that {S_{α, β}(t)}_t≥0 is generated by operator A.

Lemma 2.2. Li et al. [24] Let α, β > 0 and {S_{α, β}(t)}_{t ≥ 0} ⊂ B(E) is generated by operator A. Then, the main properties of S_{α, β}(t) are as per the following:

(i) For t ≥ 0 and x ∈ D(A), we have S_{α, β}(t)x ∈ D(A). Moreover, S_{α, β}(t)Ax = AS_{α, β}(t)x;

(ii) For x ∈ E, t ≥ 0, we have ${\int }_0^t{g}_{\alpha }(t-s)S_{\alpha ,\beta }(s)x\text{d}s\in D(A)$, and

$\begin{array}{l}{S}_{\alpha ,\beta }(t)x=g_{\beta }(t)x+A{\displaystyle {\int }_0^t}{g}_{\alpha }(t-s)S_{\alpha ,\beta }(s)x\text{ d}s;\end{array}$

moreover, if x ∈ D(A), then the second term on the right-hand side of the above equality can be replaced by

$\begin{array}{l}{\displaystyle {\int }_0^t}{g}_{\alpha }(t-s)A{S}_{\alpha ,\beta }(s)x\text{ d}s.\end{array}$

Theorem 2.1. Ponce [27] Let α > 0, 1 < β ≤ 2. Assume that {S_{α, β}(t)}_t≥0 is generated by operator A. Then for t > 0, S_{α, β}(t) is continuous in B(E).

Lemma 2.3. Ponce [27] {S_{α, β}(t)}_t≥0 is generated by operator A and (M, ω) is a type of S_{α, β}(t). Then for γ > 0, {S_{α, β + γ}(t)}_t≥0 is generated by operator A and (M/ω^γ, ω) is a type of S_{α, β+γ}(t).

Definition 2.5. Ponce [27] If S_{α, β}(t) is a compact operator for all t > 0, then we call the resolvent family {S_{α, β}(t)}_t≥0 as compact.

Theorem 2.2. Ponce [27] Let α > 0, 1 < β ≤ 2, {S_{α, β}(t)}_t≥0 is generated by operator A and (M, ω) is a type of S_{α, β}(t), and the following two conclusions are equivalent:

(i) For t > 0, S_{α, β}(t) is compact.

(ii) For μ > ω^1/α, (μI − A)⁻¹ is compact.

Lemma 2.4. Let α > 0, 0 < β ≤ 1, {S_{α, β}(t)}_t≥0 is generated by operator A and (M, ω) is a type of S_{α, β}(t). For t > 0, S_{α, β}(t) is uniform continuous. Then the following two conclusions are equivalent:

(i) For t > 0, S_{α, β}(t) is a compact operator.

(ii) For μ > ω^1/α, (μI − A)⁻¹ is compact.

Proof. If (i) is true, for λ > ω. Then we obtain

$\begin{array}{l}{\lambda }^{\alpha -\beta }{({\lambda }^{\alpha }I-A)}^{-1}x={\displaystyle {\int }_0^{\infty }}{\text{e}}^{-\lambda t}{S}_{\alpha ,\beta }(t)x\text{ d}t,\end{array}$

from Definition 2.4. However, note that {S_{α, β}(t)}_t>0 is uniform continuous by our hypothesis, where we can see that (λ^αI − A)⁻¹ is compact using Lemma 2.1 in Chang et al. [26].

On the contrary, for every fixed t > 0, let 0 ≤ β ≤ 1. For $g_{\frac{\beta }{2}}\in L_{loc}^1[0,\infty )$ and therefore, by proposition in Haase [28], we obtain

$\begin{array}{l}{\displaystyle \underset{N\to \infty }{lim}}\frac{1}{2\pi i}{\displaystyle {\int }_{\omega -iN}^{\omega +iN}}{e}^{\lambda t}(L(g_{\frac{\beta }{2}}*S_{\alpha ,\frac{\beta }{2}}))(\lambda )d\lambda =g_{\frac{\beta }{2}}*S_{\alpha ,\frac{\beta }{2}}=S_{\alpha ,\beta }(t),\end{array}$

in B(E). Hence, for t > 0,

$\begin{array}{l}\frac{1}{2\pi i}{\displaystyle {\int }_{\Gamma }}{e}^{\lambda t}{\lambda }^{\alpha -\beta }{({\lambda }^{\alpha }I-A)}^{-1}d\lambda =S_{\alpha ,\beta }(t),\end{array}$

where Γ is a vertical path lying in Re(z) = ω. By Lemma 2.4 and hypothesis, we observe for t > 0, S_{α, β}(t) is compact.

The definition and some Lemmas of Hausdorff measure of non-compactness can be found in Banas and Goebel [29], Deimling [30], Guo and Sun [31], and Lakshmikantham and Leela [32], so we omit their details here.

Lemma 2.5. Let α > 0, β > 1, {S_{α, β}(t)}_t≥0 is generated by operator A and {S_{α, β}(t)}_t≥0 is strongly continuous. Then we have

$\begin{array}{ll}\frac{\text{ d}}{\text{ d}t}{S}_{\alpha ,\beta }(t)x=S_{\alpha ,\beta -1}(t)x,\text{for}t\in J,x\in E.& (2.2)\end{array}$

Proof. Using (2.1), we have for t ≥ 0,

$\begin{array}{ll}{\lambda }^{\alpha -\beta }{({\lambda }^{\alpha }I-A)}^{-1}x={\displaystyle {\int }_0^{\infty }}{\text{e}}^{-\lambda t}{S}_{\alpha ,\beta }(t)x\text{ d}t,\text{Re}\lambda >\omega ,& (2.3)\end{array}$

$\begin{array}{ll}{\lambda }^{\alpha -\beta +1}{({\lambda }^{\alpha }I-A)}^{-1}x{\displaystyle {\int }_0^{\infty }}{\text{e}}^{-\lambda t}{S}_{\alpha ,\beta -1}(t)x\text{ d}t,\text{Re}\lambda >\omega .& (2.4)\end{array}$

(2.3) and (2.4) together imply

$\begin{array}{l}{\displaystyle {\int }_0^{\infty }}{\text{e}}^{-\lambda t}{S}_{\alpha ,\beta }(t)x\text{ d}t={\lambda }^{\alpha -\beta }{({\lambda }^{\alpha }I-A)}^{-1}x\\ =\frac{1}{\lambda }{\lambda }^{\alpha -\beta +1}{({\lambda }^{\alpha }I-A)}^{-1}x\\ =\frac{1}{\lambda }{\displaystyle {\int }_0^{\infty }}{\text{e}}^{-\lambda t}{S}_{\alpha ,\beta -1}(t)x\text{ d}t\\ ={\displaystyle {\int }_0^{\infty }}{\text{e}}^{-\lambda t}(g_1*S_{\alpha ,\beta -1})\left(t\right)x\text{ d}t\\ ={\displaystyle {\int }_0^{\infty }}{\text{e}}^{-\lambda t}\left({\displaystyle {\int }_0^t}{S}_{\alpha ,\beta -1}\left(s\right)ds\right)x\text{ d}t,\end{array}$

It is easy to see that $S_{\alpha ,\beta }(t)={\int }_0^t{S}_{\alpha ,\beta -1}(s)\text{d}s$, then we obtain (2.2) is true.

Remark 2.1. If β = 2 or β = α, the corresponding results can be found in Gou and Li [23].

Lemma 2.6. Let 0 < δ ≤ α₁ + α₂, {S_{α2, δ}}_t≥0 is generated by operator −A. Suppose that x ∈ C(J, E), if for t ∈ J, x(t) ∈ D(−A) satisfies problem (1.1) and Ax ∈ L¹((0, b), E), then we have

$\begin{array}{ll}x(t)=((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*(\overline{f}+Bu))(t)+(g_{{\gamma }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t)(x_0-h(x)),& (2.5)\end{array}$

where $\overline{f}(t)=f(t,x(t))$.

Proof. Using Liouville operators with $I_{0+}^{{\alpha }_1}$ on both sides of the equation

$\begin{array}{l}{D}_{0+}^{{\alpha }_1,{\beta }_1}(D_{0+}^{{\alpha }_2,{\beta }_2}+A)x(t)=f(t,x(t))+Bu(t),\end{array}$

in view of Lemma 2.1, we obtain

$\begin{array}{ll}(D_{0+}^{{\alpha }_2,{\beta }_2}+A)x(t)=I_{0+}^{{\alpha }_1}(\overline{f}+Bu)(t)+\frac{c_0}{\Gamma ({\gamma }_1)}{t}^{{\gamma }_1-1},& (2.6)\end{array}$

where γ₁ = α₁ + β₁ − α₁β₁. Using Liouville operators with $I_{0+}^{{\alpha }_2}$ on both sides of equation (2.6) again, we obtain

$\begin{array}{c}x(t)=I_{0+}^{{\alpha }_1+{\alpha }_2}(\overline{f}+Bu)(t)-I_{0+}^{{\alpha }_2}(Ax)(t)+\frac{c_0}{\Gamma ({\gamma }_1+{\alpha }_2)}{t}^{{\gamma }_1+{\alpha }_2-1}\\ +\frac{c_1}{\Gamma ({\gamma }_2)}{t}^{{\gamma }_2-1},& (2.7)\end{array}$

where γ₂ = α₂ + β₂ − α₂β₂. In view of the condition, we obtain c₀ = x₀ − h(x) and c₁ = 0. Then we rewrite the representation of (2.7) as

$\begin{array}{c}x(t)=(g_{{\alpha }_1+{\alpha }_2}*\overline{f})(t)+(g_{{\alpha }_1+{\alpha }_2}*Bu)(t)-(g_{{\alpha }_2}*Ax)(t)\\ +\frac{x_0-h(x)}{\Gamma ({\gamma }_1+{\alpha }_2)}{t}^{{\gamma }_1+{\alpha }_2-1}.& (2.8)\end{array}$

Applying the Laplace transform to (2.8), we obtain

$\begin{array}{c}(Lx)(\lambda )=\frac{1}{{\lambda }^{{\alpha }_1+{\alpha }_2}}(L\overline{f})(\lambda )+\frac{1}{{\lambda }^{{\alpha }_1+{\alpha }_2}}(LBu)(\lambda )-\frac{1}{{\lambda }^{{\alpha }_2}}A(Lx)(\lambda )\\ +(x_0-h(x))\frac{1}{{\lambda }^{{\gamma }_1+{\alpha }_2}}.\end{array}$

Thus, we obtain

$\begin{array}{l}(Lx)(\lambda )=\frac{1}{{\lambda }^{{\alpha }_1}}{({\lambda }^{{\alpha }_2}I+A)}^{-1}(L\overline{f})(\lambda )+\frac{1}{{\lambda }^{{\alpha }_1}}{({\lambda }^{{\alpha }_2}I+A)}^{-1}(LBu)(\lambda )\\ +(x_0-h(x))\frac{1}{{\lambda }^{{\gamma }_1}}{({\lambda }^{{\alpha }_2}I+A)}^{-1}.\end{array}$

Currently, by Definition 2.4, we can apply the inverse Laplace transform to the above equation, therefore

$\begin{array}{l}x(t)=((S_{{\alpha }_2,{\alpha }_1+{\alpha }_2})*(\overline{f}+Bu))(t)+S_{{\alpha }_2,{\gamma }_1+{\alpha }_2}(t)(x_0-h(x))\\ =((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*(\overline{f}+Bu))(t)\\ +(g_{{\gamma }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t)(x_0-h(x)).\end{array}$

Definition 2.6. Let 0 < δ ≤ α₁ + α₂, {S_{α2, δ}(t)}_{t ≥ 0} is generated by −A. We say that x(t) is a mild solution of (1.1) if $\underset{t\to 0+}{lim}\frac{\text{d}}{\text{d}t}\left(I_{0+}^{(1-{\alpha }_1)(1-{\beta }_1)+(1-{\alpha }_2)}x\right)(t)+h(x)=x_0$, $\underset{t\to 0+}{lim}\left(I_{0+}^{(1-{\alpha }_2)(1-{\beta }_2)}x\right)(t)=0$, x(·) ∈ C(J, E) satisfies the equation

$\begin{array}{r}x(t)=((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*(\overline{f}+Bu))(t)+(g_{{\gamma }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t)\\ (x_0-h(x)).\end{array}$

3. Main results

Let x be an arbitrary function in C(J, E), which we denote by x_b = x(b) during the final stages at time b in E.

Definition 3.1. Let the initial condition x₀ ∈ E and final stages x_b ∈ E, if there exists a control term u ∈ L²(J, U), such that x(t) is the mild solution of (1.1) with respect to u, which satisfies

$\begin{array}{r}{\displaystyle \underset{t\to 0+}{lim}}\left(I_{0+}^{\left(1-{\alpha }_2\right)\left(1-{\beta }_2\right)}x\right)\left(t\right)=0,{\displaystyle \underset{t\to 0+}{lim}}\frac{\text{ d}}{\text{ d}t}\left(I_{0+}^{\left(1-{\alpha }_1\right)\left(1-{\beta }_1\right)+\left(1-{\alpha }_2\right)}x\right)\left(t\right)\\ +h\left(x\right)=x_0\end{array}$

and x(b) = x_b, then we say that system (1.1) can be controlled on J.

Theorem 3.1. Let 0 < δ ≤ α₁ + α₂, {S_{α2, δ}(t)}_{t ≥ 0} is generated by operator −A and (M, ω) is a type of S_{α2, δ}(t). Assume that the following conditions are satisfied:

(H1) f:J × E → E satisfies the Carathéodory conditions.

(H2) There exist q₁ ∈ [0, 1) and two functions $m\in L^{\frac{1}{q_1}}(J,R^{+})$, Φ ∈ C(R⁺, R⁺) which are non-decreasing that satisfy

$\begin{array}{l}\Vert f(t,x)\Vert \le m(t)\Phi (\Vert x\Vert ),\text{for}x\in E,\text{a.e.}t\in J.\end{array}$

(H3) There exist q₂ ∈ [0, 1) and a function $n\in L^{\frac{1}{q_2}}(J,R^{+})$, such that for every bounded set D in E,

$\begin{array}{l}\alpha (f(t,D))\le n(t)\alpha (D),\text{for a.e.}t\in J.\end{array}$

(H4) (i) The function h:C(J, E) → E and there exist c₁, c₂ ≥ 0, such that

$\begin{array}{l}\Vert h(x)\Vert \le c_1,\Vert h(x)-h(y)\Vert \le c_2\Vert x-y\Vert ,x\in C(J,E);\end{array}$

(ii) There exists l > 0, such that for every bounded subset D in E,

$\begin{array}{l}\alpha (h(D))\le l\alpha (D),\end{array}$

(H5) W:L²(J, U) → E is a linear operator, which is given by

$\begin{array}{l}Wu={\displaystyle {\int }_0^b}{S}_{{\alpha }_2,{\alpha }_1+{\alpha }_2}(b-s)Bu(s)ds,u=u^x,\end{array}$

where u^x is defined in (3.4).

(i) The inverse operator W⁻¹:E → L²(J, U)\kerW exists, if there exist M₁ > 0, M₂ > 0, such that ‖B‖ ≤ M₁, $\Vert W^{-1}\Vert \le M_2$;

(ii) There exist q₃ ∈ [0, 1) and a function $K\in L^{\frac{1}{q_3}}(J,R^{+})$, such that for every bounded subset D in E,

$\begin{array}{l}\alpha (W^{-1}(D)(t))\le K(t)\alpha (D),t\in J.\end{array}$

Assume that max{Λ₁, Λ₂} < 1, where

$\begin{array}{c}{\Lambda }_1=\frac{M}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\left(1+\frac{M{M}_1{M}_2}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\frac{1}{\sqrt{2\omega }}{e}^{\omega b}\right){\left(\frac{1-q_1}{\omega }\right)}^{1-q_1}\\ e^{\omega b}\Vert m{\Vert }_{L^{\frac{1}{q_1}}}{\displaystyle {lim inf}_{r\to +\infty }}\frac{\Phi \left(r\right)}{r},& (3.1)\end{array}$

$\begin{array}{l}{\Lambda }_2=M{e}^{\omega b}\left[1+\frac{2M{M}_1}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left(\frac{1-q_3}{\omega }\right)}^{1-q_3}{e}^{\omega b}\Vert K{\Vert }_{L^{\frac{1}{q_3}}}\right]\\ \times \left[\frac{1}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}l+\frac{2}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left(\frac{1-q_2}{\omega }\right)}^{1-q_2}\Vert n{\Vert }_{L^{\frac{1}{q_2}}}\right],& (3.2)\end{array}$

then system (1.1) can be controlled on J.

Proof. Let us consider operator T in C(J, E) as follows:

$\begin{array}{l}(Tx)(t)=((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*(\overline{f}+Bu))(t)\\ +(g_{{\alpha }_2+{\gamma }_2-\delta }*S_{{\alpha }_2,\delta })(t)(x_0-h(x)),t\in J,& (3.3)\end{array}$

where the control term u is given by u(t) = u^x(t), x ∈ C(J, E) is given by

$\begin{array}{l}{u}^x(t)=W^{-1}{\displaystyle [x_b-(g_{{\alpha }_2+{\gamma }_2-\delta }*S_{{\alpha }_2,\delta })(b)(x_0-h(x))}\\ {\displaystyle -((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*\overline{f})(b)](t).}& (3.4)\end{array}$

Taking the control (3.4) in (3.3), we obtain (Tx)(b) = x_b. Next, we illustrate that the non-linear operator T has a fixed point.

Step 1: T(B_r) ⊂ B_r for some positive number r

If not, then for every r > 0, there exist x_r ∈ B_r and t_r ∈ J, such that ‖(Tx_r)(t_r)‖ > r. First, we observe that

$\begin{array}{l}\Vert (Tx)(t)\Vert \le {\displaystyle {\int }_0^t}\Vert (g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t-s){\Vert }_B\text{‖}f(s,x(s))\Vert \text{ d}s\\ +\Vert (g_{{\gamma }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t){\Vert }_B\text{‖}(x_0-h(x))\Vert \\ +{\displaystyle {\int }_0^t}\Vert (g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t-s){\Vert }_B\text{‖}B{u}^x(s)\Vert \text{ d}s\\ \le {\displaystyle {\int }_0^t}\frac{M{e}^{\omega (t-s)}}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}m(s)\text{Φ}(\Vert x(s)\Vert )\text{ d}s\\ +\frac{M{e}^{\omega t}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}\Vert (x_0-h(x))\Vert \\ +{\displaystyle {\int }_0^t}\frac{M{e}^{\omega (t-s)}}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{M}_1\Vert u^x(s)\Vert \text{ d}s\\ \le \frac{M}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\displaystyle {\int }_0^t}{e}^{\omega (t-s)}m(s)\text{Φ}(\Vert x(s)\Vert )\text{ d}s\\ +\frac{M{e}^{\omega t}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}(\Vert x_0\Vert +c_1)\\ +\frac{M{M}_1}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left[\frac{1}{2\omega }(e^{2\omega t}-1)\right]}^{\frac{1}{2}}\Vert u^x{\Vert }_{L^2},& (3.5)\end{array}$

where

$\begin{array}{l}\Vert u^x{\Vert }_{L^2}\le M_2{\displaystyle [\Vert x_b\Vert +\frac{M{e}^{\omega b}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}(\Vert x_0\Vert +c_1)}\\ {\displaystyle +\frac{M}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\displaystyle {\int }_0^b}{e}^{\omega (b-s)}m(s)\text{Φ}(\Vert x(s)\Vert )\text{ d}s].}& (3.6)\end{array}$

From (3.5) and (3.6), we conclude that

$\begin{array}{l}\Vert \left(Tx\right)\left(t\right)\Vert \le M\left\{1+\frac{M{M}_1{M}_2}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left[\frac{1}{2\omega }\left(e^{2\omega b}-1\right)\right]}^{\frac{1}{2}}\right\}\\ {\displaystyle [\frac{e^{\omega b}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}\left(\Vert x_0\Vert +c_1\right)}\\ {\displaystyle +\frac{1}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\displaystyle {\int }_0^b}{e}^{\omega \left(b-s\right)}m\left(s\right)\text{Φ}\left(\Vert x\left(s\right)\Vert \right)\text{ d}s]}\\ +\frac{M{M}_1{M}_2}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left[\frac{1}{2\omega }\left(e^{2\omega b}-1\right)\right]}^{\frac{1}{2}}\Vert x_b\Vert .\end{array}$

Consequently,

$\begin{array}{l}{\displaystyle r<\Vert \left(T{x}_r\right)\left(t_r\right)\Vert \le M\{1+\frac{M{M}_1{M}_2}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}}\\ {\displaystyle {\left[\frac{1}{2\omega }\left(e^{2\omega b}-1\right)\right]}^{\frac{1}{2}}\left\}\right\{\frac{e^{\omega b}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}\left(\Vert x_0\Vert +c_1\right)}\\ {\displaystyle +\frac{\text{Φ}\left(r\right)}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left[\frac{1-q_1}{\omega }\left(e^{\frac{\omega }{1-q_1}b}-1\right)\right]}^{1-q_1}\Vert m{\Vert }_{L^{\frac{1}{q_1}}}\}}\\ {\displaystyle +\frac{M{M}_1{M}_2}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left[\frac{1}{2\omega }\left(e^{2\omega b}-1\right)\right]}^{\frac{1}{2}}\Vert x_b\Vert }\\ {\displaystyle \le M\left(1+\frac{M{M}_1{M}_2}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\frac{1}{\sqrt{2\omega }}{e}^{\omega b}\right)[\frac{e^{\omega b}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}\left(\Vert x_0\Vert +c_1\right)}\\ {\displaystyle +\frac{\text{Φ}\left(r\right)}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left(\frac{1-q_1}{\omega }\right)}^{1-q_1}{e}^{\omega b}\Vert m{\Vert }_{L^{\frac{1}{q_1}}}]}\\ {\displaystyle +\frac{M{M}_1{M}_2}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\frac{1}{\sqrt{2\omega }}{e}^{\omega b}\Vert x_b\Vert .}& (3.7)\end{array}$

Dividing (3.7) by r and passing to the lower limit as r → +∞ yield

$\begin{array}{c}\frac{M}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\left(1+\frac{M{M}_1{M}_2}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\frac{1}{\sqrt{2\omega }}{e}^{\omega b}\right){\left(\frac{1-q_1}{\omega }\right)}^{1-q_1}\\ e^{\omega b}\Vert m{\Vert }_{L^{\frac{1}{q_1}}}{\displaystyle {lim inf}_{r\to +\infty }}\frac{\text{Φ}\left(r\right)}{r}\ge 1,\end{array}$

which contradicts Λ₁ < 1. Hence, T(B_r) ⊂ B_r for some r > 0.

Step 2: T:B_r → B_r is continuous.

Assume that {x_n} ⊂ B_r satisfying x_n → x. Let us show that ‖Tx_n − Tx‖_C → 0. For this, we consider the inequality

$\begin{array}{l}\Vert (T{x}_n)(t)-(Tx)(t)\Vert \le {\displaystyle {\int }_0^t}\Vert (g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t-s){\Vert }_B\text{‖}{\overline{f}}_n(s)\\ -\overline{f}(s)\text{‖ d}s+{\displaystyle {\int }_0^t}\Vert (g_{{\alpha }_1+{\alpha }_2-\delta }\\ *S_{{\alpha }_2,\delta })(t-s){\Vert }_B\text{‖}B{u}^{x_n}(s)-B{u}^x(s)\Vert \text{ d}s\\ +\Vert (g_{{\gamma }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t){\Vert }_B\text{‖}h(x_n)-h(x)\Vert \\ \le \frac{M}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\displaystyle {\int }_0^t}{e}^{\omega (t-s)}\Vert {\overline{f}}_n(s)-\overline{f}(s)\Vert \text{ d}s\\ +\frac{M{M}_1}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left[\frac{1}{2\omega }(e^{2\omega t}-1)\right]}^{\frac{1}{2}}\Vert u^{x_n}-u^x{\Vert }_{L^2}\\ +\frac{M{e}^{\omega t}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}{c}_2\Vert x_n-x\Vert ,& (3.8)\end{array}$

where ${\overline{f}}_n(t)=f(t,x_n(t))$ and

$\begin{array}{l}{\displaystyle \Vert u^{x_n}-u^x{\Vert }_{L^2}\le M_2[\frac{M{e}^{\omega b}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}{c}_2\Vert x_n-x{\Vert }_C}\\ {\displaystyle +\frac{M}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\displaystyle {\int }_0^b}{e}^{\omega \left(b-s\right)}\Vert {\overline{f}}_n\left(s\right)-\overline{f}\left(s\right)\Vert \text{ d}s].}& (3.9)\end{array}$

By means of the Lebesgue dominated convergence theorem and condition (H1), together with (3.8) and (3.9), proves that ‖Tx_n − Tx‖_C → 0 as n → ∞.

Step 3: T satisfies conditions of the Mönch fixed point theorem.

Let D be a countable subset in B_r satisfying D is a subset in the closed convex hull of {0} ∪ T(D), and we will later prove α(D) = 0. Assume, without loss of generality, that $D={\left\{x_n\right\}}_{n=1}^{\infty }\subset B_r$, let 0 ≤ t₁ < t₂ ≤ b, then

$\begin{array}{l}\begin{array}{l}\Vert (T{x}_n)(t_1)-(T{x}_n)(t_2)\Vert \\ \le {\displaystyle {\int }_0^{t_1}}\Vert (g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t_2-s)-(g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t_1-s){\Vert }_B\\ \times \Vert \overline{f}(s)+B{u}^{x_n}(s)\Vert \text{ d}s\\ +{\displaystyle {\int }_{t_1}^{t_2}}\Vert (g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t_2-s){\Vert }_B\text{‖}\overline{f}(s)+B{u}^{x_n}(s)\Vert \text{ d}s\\ +\Vert (g_{{\gamma }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t_2)-(g_{{\gamma }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t_1){\Vert }_B\text{‖}(x_0-h(x_n))\Vert .\end{array}\end{array}$

By Lemma 2.5, (g_{α1 + α2 − δ}*S_{α2, δ})(t) = S_{α2, α1 + α2}(t) and (g_{γ1 + α2 − δ}*S_{α2, δ})(t) = S_{α2, γ1 + α2}(t) for t ≥ 0. Furthermore, by Theorem 2.1, we obtain S_{α2, α2 + α1}(t) and S_{α2, γ1 + α2}(t) which are norm continuous. Since the right-hand side of the inequality approaches zero as t₂ → t₁, T(D) is equicontinuous on J.

Using the properties of the measure of non-compactness in Deimling [30], Lakshmikantham and Leela [32],

$\begin{array}{l}\alpha \left(T{x}_n\left(t\right)\right)\le \alpha \left(\left\{((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*\overline{f}(s))\left(t\right)\right\}\right)\\ +\alpha \left(\left\{((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*B{u}^{x_n})\left(t\right)\right\}\right)\\ +\alpha \left(\left\{(g_{{\gamma }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t)(h(x_n))\right\}\right)\\ \le \frac{2M}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\displaystyle {\int }_0^t}{e}^{\omega \left(t-s\right)}n\left(s\right)\text{ d}s\alpha \left(\left\{x_n\right\}\right)\\ +\frac{2M{M}_1}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\displaystyle {\int }_0^t}{e}^{\omega \left(t-s\right)}\alpha \left(\left\{u^{x_n}\left(s\right)\right\}\right)\text{ d}s\\ +\frac{M{e}^{\omega t}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}l\alpha \left(\left\{x_n\right\}\right),& (3.10)\end{array}$

where

$\begin{array}{l}\alpha \left({\left\{u^{x_n}\left(s\right)\right\}}_{n=1}^{\infty }\right)\le K\left(s\right){\displaystyle [\alpha \left(\left\{(g_{{\gamma }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(b)h(x_n)\right\}\right)}\\ {\displaystyle +\alpha \left(\left\{((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*{\overline{f}}_n)(b)\right\}\right)]}\\ \le K\left(s\right)\alpha \left({\left\{x_n\right\}}_{n=1}^{\infty }\right)(\frac{M{e}^{\omega b}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}l\\ +\frac{2M}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\displaystyle {\int }_0^b}{e}^{\omega \left(b-s\right)}n\left(s\right)\text{ d}s.& (3.11)\end{array}$

By (3.10) and (3.11), we obtain

$\begin{array}{l}\alpha \left(T{x}_n\left(t\right)\right)\le \frac{2M}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\alpha \left(D\right){\left[\frac{1-q_2}{\omega }\left(e^{\frac{\omega }{1-q_2}t}-1\right)\right]}^{1-q_2}\Vert n{\Vert }_{L^{\frac{1}{q_2}}}\\ +\frac{2M{M}_1}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\alpha \left(D\right)(\frac{M{e}^{\omega b}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}l\\ +\frac{2M}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\displaystyle {\int }_0^b}{e}^{\omega \left(b-s\right)}n\left(s\right)ds\\ \times {\displaystyle {\int }_0^t}{e}^{\omega \left(t-s\right)}K\left(s\right)ds+\frac{M{e}^{\omega t}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}l\alpha \left(D\right)\\ \le \frac{2M}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\alpha \left(D\right){\left(\frac{1-q_2}{\omega }\right)}^{1-q_2}{e}^{\omega t}\Vert n{\Vert }_{L^{\frac{1}{q_2}}}\\ +\frac{2M^2{M}_1}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\alpha \left(D\right)[\frac{e^{\omega b}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}l\\ +\frac{2}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left(\frac{1-q_2}{\omega }\right)}^{1-q_2}{e}^{\omega b}\Vert n{\Vert }_{L^{\frac{1}{q_2}}}\\ \times {\left(\frac{1-q_3}{\omega }\right)}^{1-q_3}{e}^{\omega t}\Vert K{\Vert }_{L^{\frac{1}{q_3}}}+\frac{M{e}^{\omega t}}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}l\alpha \left(D\right)\\ \le M{e}^{\omega b}\left[1+\frac{2M{M}_1}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left(\frac{1-q_3}{\omega }\right)}^{1-q_3}{e}^{\omega b}\Vert K{\Vert }_{L^{\frac{1}{q_3}}}\right]\\ \times \left[\frac{1}{{\omega }^{{\gamma }_1+{\alpha }_2-\delta }}l+\frac{2}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\left(\frac{1-q_2}{\omega }\right)}^{1-q_2}\Vert n{\Vert }_{L^{\frac{1}{q_2}}}\right]\\ \alpha \left(D\right),\end{array}$

we have

$\begin{array}{l}\alpha (TD)\le {\Lambda }_2\alpha (D).\end{array}$

Thus, by condition of the Mönch fixed point theorem, we obtain

$\begin{array}{l}\alpha (D)\le \alpha (\overline{co}(\left\{0\right\}\cup T(D))).\end{array}$

We obtain α(D) = 0 for Λ₂ < 1. Applying the Mönch fixed point theorem, we know that there exists a fixed point x ∈ B_r of T, which, of course, is a mild solution of 1.1 and satisfies x(b) = x_b. Hence, system 1.1 can be controlled on J.

Theorem 3.2. Let 0 < δ ≤ α₁ + α₂, {S_{α2, δ}(t)}_{t ≥ 0} is generated by operator −A and (M, ω) is a type of S_{α2, δ}(t). In addition to assumptions (H1), (H2), (H4)(i), and (H5)(i) of Theorem 3.1, we suppose that the following assumptions hold:

(H6) ${({\lambda }^{{\alpha }_2}I+A)}^{-1}$ is compact for all $\lambda >{\omega }^{1/{\alpha }_2}$.

If max{Λ₁, Λ₃} < 1, where ${\Lambda }_3=\frac{M{e}^{\omega b}}{{\omega }^{{\gamma }_1+{\alpha }_1-\delta }}{c}_2$, then (1.1) can be controlled on J.

Proof. We define two operators T₁, T₂ in C(J, E) as follows:

$\begin{array}{ll}(T_{1}x)(t)={\displaystyle {\int }_0^t}((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*(\overline{f}+Bu)(t),t\in J,& (3.12)\end{array}$

$\begin{array}{ll}(T_{2}x)(t)=(g_{{\gamma }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t)(x_0-h(x)),t\in J.& (3.13)\end{array}$

As in Step 1 of Theorem 3.1, we can find r > 0, such that T₁x + T₂y ∈ B_r for x, y ∈ B_r. Moreover, with a similar method used in Step 2 of Theorem 3.1, it follows that T₁ is continuous on B_r and T₂ is a contraction on B_r. Currently, we are going to illustrate that {T₁x:x ∈ B_r} is precompact. The uniformly bounded nature of {T₁x:x ∈ B_r} is obvious.

Step 1: {T₁x:x ∈ B_r} is an equicontinuous family.

For x ∈ B_r, without loss of generality, we assume that 0 ≤ t₁ < t₂ ≤ b, then

$\begin{array}{l}\begin{array}{l}\Vert (T_{1}x)(t_1)-(T_{1}x)(t_2)\Vert \\ \le {\displaystyle {\int }_0^{t_1}}\Vert (g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t_2-s)-(g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t_1-s){\Vert }_B\\ \times \Vert \overline{f}(s)+B{u}^x(s)\Vert \text{ d}s\\ +{\displaystyle {\int }_{t_1}^{t_2}}\Vert (g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t_2-s){\Vert }_B\text{‖}\overline{f}(s)+B{u}^x(s)\Vert \text{ d}s:=I_1+I_2.\end{array}\end{array}$

For I₁, we have

$\begin{array}{l}{\displaystyle I_1\le \left(\text{Φ}\left(r\right)\Vert m{\Vert }_{L^{\frac{1}{q_1}}}++M_1\Vert u^x{\Vert }_{L^2}\right)({\int }_0^{t_1}\Vert (g_{{\alpha }_1+{\alpha }_2-\delta }}\\ {\displaystyle *S_{{\alpha }_2,\delta })(t_2-s)-(g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t_1-s){\Vert }_B^{\frac{1}{1-q_1}}\text{ d}s{)}^{1-q_1}.}& (3.14)\end{array}$

By Theorem 2.1, we have the norm continuity of S_{α2, α2 + α1}(t) and therefore if t₂ → t₁, then S_{α2, α2 + α1}(t₂ − s) − S_{α2, α2 + α1}(t₁ − s) → 0 in B(E). We can have that $\underset{t_2\to t_1}{lim}{I}_1=0$ using Lebesgue's theorem.

For I₂, we have

$\begin{array}{l}{I}_2\le \frac{M\text{Φ}\left(r\right)}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\displaystyle {\int }_{t_1}^{t_2}}{e}^{\omega \left(t_2-s\right)}m\left(s\right)ds\\ +\frac{M{M}_1}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}{\displaystyle {\int }_{t_1}^{t_2}}{e}^{\omega \left(t_2-s\right)}{u}^x\left(s\right)\text{ d}s\\ \le \frac{M\text{Φ}\left(r\right)}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\Vert m{\Vert }_{L^{\frac{1}{q_1}}}{\left[\frac{1-q_1}{\omega }\left(e^{\frac{\omega \left(t_2-t_1\right)}{1-q_1}}-1\right)\right]}^{1-q_1}\\ +\frac{M{M}_1}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\Vert u^x{\Vert }_{L^2}{\left[\frac{1}{2\omega }\left(e^{2\omega \left(t_2-t_1\right)}-1\right)\right]}^{\frac{1}{2}},& (3.15)\end{array}$

and therefore $\underset{t_2\to t_1}{lim}{I}_2=0$. From the above two inequalities, we find that {T₁x:x ∈ B_r} is an equicontinuous family.

Step 2: For every t ∈ [0, b], it remains to show that H(t) = {(T₁x)(t):x ∈ B_r} is precompact.

First, it is obvious that H(0) is precompact. Finally, let 0 < t ≤ b be a fixed number. For ∀ϵ ∈ (0, t), we consider the operator $T_1^{ϵ}$ on B_r by the formula

$\begin{array}{l}(T_1^{ϵ}x)(t)=((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*(\overline{f}+Bu))(t-ϵ),x\in B_r.\end{array}$

From (H6) and Theorem 2.2, we know that the compactness of $\left\{(g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t-s)(\overline{f}(s)+Bu):0\le s\le t-ϵ\right\}$ for ϵ > 0. Using the Mazur theorem and the mean-value theorem with respect to the Bochner integral, we have that for ϵ > 0, $H_{ϵ}(t)=\left\{(T_1^{ϵ}x)(t):x\in B_r\right\}$ is precompact in E. In addition, for every x ∈ B_r, we obtain

$\begin{array}{l}\Vert (T_{1}x)(t)-(T_1^{ϵ}x)(t)\Vert \le {\displaystyle {\int }_{t-ϵ}^t}\Vert (g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t-s)[\overline{f}(s)\\ +Bu(s)]\Vert ds\le \frac{M\text{Φ}(r)}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\Vert m{\Vert }_{L^{\frac{1}{q_1}}}\\ {\left[\frac{1-q_1}{\omega }\left(e^{\frac{\omega ϵ}{1-q_1}}-1\right)\right]}^{1-q_1}\\ +\frac{M{M}_1}{{\omega }^{{\alpha }_1+{\alpha }_2-\delta }}\Vert u^x{\Vert }_{L^2}{\left[\frac{1}{2\omega }(e^{2\omega ϵ}-1)\right]}^{\frac{1}{2}}.\end{array}$

Therefore, H(t) = {(T₁x)(t):x ∈ B_r} is precompact in E.

According to Ascoli–Arzela's Theorem and above, we conclude that {T₁x:x ∈ B_r} is precompact. Thus, T₁ is a completely continuous operator by the continuity of T₁ and the relative compactness of {T₁x:x ∈ B_r}. According to Krasnoselskii's fixed point theorem, it is natural to obtain that T₁ + T₂ has a fixed point on B_r. Hence, (1.1) can be controlled on J, and the proof is complete.

Theorem 3.3. Let 0 < δ ≤ α₁ + α₂, {S_{α2, δ}(t)}_{t ≥ 0} is generated by operator −A and (M, ω) is a type of S_{α2, δ}(t). In addition to assumptions (H1), (H2), (H4)(i), (H5)(i), and (H6) of Theorem 3.1, suppose that

(H7) For 0 < δ ≤ 1, {S_{α2, δ}(t)}_t>0 is uniform continuous.

Then (1.1) can be controlled on J for Λ₁ < 1.

Proof. We consider the operator T in C(J, E), which is the same as (3.3). Similarly, there exists r > 0, such that T:B_r → B_r is continuous. We shall now examine the precompact nature of {Tx:x ∈ B_r}. Furthermore, we can see that {Tx:x ∈ B_r} is not only uniformly bounded, but also equicontinuous.

Next, we verify that for all t ∈ [0, b], {Tx(t):x ∈ B_r} is precompact. Obviously, {(Tx)(0):x ∈ B_r} is precompact. Let 0 < t ≤ b be a number, ∀ϵ ∈ (0, t), we consider operator T^ϵ on B_r as follows:

$\begin{array}{l}(T^{ϵ}x)(t)=S_{{\alpha }_2,\delta }(ϵ)((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*(\overline{f}+Bu))(t-ϵ),x\in B_r.\end{array}$

If 0 < δ ≤ 1, then (H6), (H7), and Lemma 2.4 show that for t > 0, S_{α2, δ}(t) is compact, if 1 < δ ≤ α₁ + α₂, then (H6) and Theorem 2.2 also illustrate that S_{α2, δ}(t) is compact for t > 0, Finally, we obtain that $\left\{(T^{ϵ}x)(t):x\in B_r\right\}$ is precompact in E for ∀ϵ ∈ (0, t). Furthermore, for every x ∈ B_r, we have

$\begin{array}{l}{\displaystyle \Vert S_{{\alpha }_2,\delta }(ϵ)((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*(\overline{f}+Bu))(t-ϵ)}\\ {\displaystyle -{\displaystyle {\int }_0^{t-ϵ}}(g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t-s)\left[\overline{f}(s)+Bu(s)\right]\text{ d}s\Vert }\\ {\displaystyle \le \left(\text{Φ}\left(r\right)\Vert m{\Vert }_{L^{\frac{1}{q_1}}}+M_1\Vert u^x{\Vert }_{L^2}\right)({\displaystyle {\int }_0^{t-ϵ}}\Vert S_{{\alpha }_2,\delta }(ϵ)(g_{{\alpha }_1+{\alpha }_2-\delta }}\\ *S_{{\alpha }_2,\delta })(t-s-ϵ)\\ {\displaystyle -(g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })(t-s){\Vert }_B^{\frac{1}{1-q_1}}\text{ d}s{)}^{1-q_1}}\end{array}$

By Theorem 2.1, (g_{α1 + α2 − δ}*S_{α2, δ})(t) is norm continuous for all t > 0, using Lebesgue's theorem, we have

$\begin{array}{l}{\displaystyle \Vert S_{{\alpha }_2,\delta }(ϵ)((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*(\overline{f}+Bu))(t-ϵ)-{\displaystyle {\int }_0^{t-ϵ}}(g_{{\alpha }_1+{\alpha }_2-\delta }}\\ {\displaystyle *S_{{\alpha }_2,\delta })(t-s)\left[\overline{f}(s)+Bu(s)\right]\text{ d}s\Vert \to 0,ϵ\to 0.}\end{array}$

Hence, the set $((g_{{\alpha }_1+{\alpha }_2-\delta }*S_{{\alpha }_2,\delta })*(\overline{f}+Bu))(t):x\in B_r\}$, t > 0 is precompact. The compactness of (g_{α2 + γ2 − δ} * S_{α2, δ})(t) is obtained by Theorem 2.2. Hence, we have proved that for t ∈ (0, b], {Tx(t):x ∈ B_r} is relatively compact in E. Consequently, by Ascoli–Arzela's Theorem, the set {Tx:x ∈ B_r} is precompact. This further leads to T being compact on B_r. We therefore have, by applying Schauder's fixed point theorem, a fixed point on B_r of T, which implies that 1.1 can be controlled on J.

4. An example

Example 4.1. Set E = U = L²([0, π], R), α_i ∈ (0, 1], β_i ∈ [0, 1], and i = 0, 1. We consider the fractional control system

$\begin{array}{ll}\{\begin{array}{l}{D}_{0+}^{{\alpha }_1,{\beta }_1}\left(D_{0+}^{{\alpha }_2,{\beta }_2}+A\right)x\left(t,\xi \right)=f\left(t,x\left(t,\xi \right)\right)+Bu\left(t,\xi \right),t\in \left(0,1\right),\\ \xi \in \left[0,\pi \right],\\ {\displaystyle \underset{t\to 0+}{lim}}\left(I_{0+}^{\left(1-{\alpha }_2\right)\left(1-{\beta }_2\right)}x\right)\left(t,\xi \right)=0,\\ {\displaystyle \underset{t\to 0+}{lim}}\frac{\text{ d}}{\text{ d}t}\left(I_{0+}^{\left(1-{\alpha }_1\right)\left(1-{\beta }_1\right)+\left(1-{\alpha }_2\right)}x\right)\left(t,\xi \right)=x_0\left(\xi \right),\end{array}& (4.1)\end{array}$

where t ∈ (0, 1), ξ ∈ [0, π], and $D_{0+}^{{\alpha }_i,{\beta }_i}$ are Hilfer fractional derivatives. Operator A is given by $Ax=x-\frac{{\partial }^{2}x}{\partial {\xi }^2}$, let D(A) = {x ∈ E:x, x′ absolutely continuous, x″ ∈ E, x(t, 0) = x(t, π) = 0} and E be the domain and the range of A, respectively. We can see that (1 + n²) and $x_n(\xi )=\sqrt{\frac{2}{\pi }}sin(n\xi )$ are the eigenvalues and the normalized eigenvectors of A, respectively.

For x ∈ E and 1 ≤ δ ≤ α₁ + α₂, we have

$\begin{array}{l}{\lambda }^{{\alpha }_2-\delta }{({\lambda }^{{\alpha }_2}I+A)}^{-1}x={\displaystyle \sum _{n=1}^{\infty }}\langle x,x_n\rangle x_n\frac{{\lambda }^{{\alpha }_2-\delta }}{{\lambda }^{{\alpha }_2}+(1+n^2)}\\ ={\displaystyle \sum _{n=1}^{\infty }}\langle x,x_n\rangle x_n{\displaystyle {\int }_0^{\infty }}{\text{e}}^{-\lambda t}{t}^{\delta -1}{E}_{{\alpha }_2,\delta }\\ (-(1+n^2)t^{{\alpha }_2})\text{ d}t\\ ={\displaystyle {\int }_0^{\infty }}\langle x,x_n\rangle x_n{\text{e}}^{-\lambda t}{\displaystyle \sum _{n=1}^{\infty }}{t}^{\delta -1}{E}_{{\alpha }_2,\delta }\\ (-(1+n^2)t^{{\alpha }_2})\text{ d}t.\end{array}$

Hence, {S_{α2, δ}(t)}_t≥0 is generated by operator −A,

$\begin{array}{l}{S}_{{\alpha }_2,\delta }(t)x={\displaystyle \sum _{n=1}^{\infty }}\langle x,x_n\rangle x_n{t}^{\delta -1}{E}_{{\alpha }_2,\delta }(-(1+n^2)t^{{\alpha }_2}),x\in E,\end{array}$

which is norm continuous by the continuity of E_{α2, δ}(·). Moreover, for λ > 0, we have $\underset{n\to \infty }{lim}\frac{{\lambda }^{{\alpha }_2-\delta }}{{\lambda }^{{\alpha }_2}+(1+n^2)}=0$, which implies that ${\lambda }^{{\alpha }_2-\delta }{({\lambda }^{{\alpha }_2}I+A)}^{-1}$ is compact on the Hilbert space E, then ${({\lambda }^{{\alpha }_2}I+A)}^{-1}$ is compact for λ > 0.

Otherwise, for each x ∈ E, we obtain $\Vert S_{{\alpha }_2,\delta }(t)x\Vert \le \frac{b^{\delta -1}}{\Gamma (\delta )}\Vert x\Vert$. Therefore, S_{α2, δ}(t) is of type (b^δ−1/Γ(δ), 1).

Let $f(t,x)=\frac{e^{-t}}{1+t}x$, then we can choose $m(t)=\frac{e^{-t}}{1+t}$ and Φ = I.

$\begin{array}{l}\Vert f(t,x)\Vert \le \frac{e^{-t}}{1+t}\end{array}$

Assume that $Bu(t)={\displaystyle \sum _{n=1}^{\infty }}{\widehat{u}}_n(t)x_n$, where

$\begin{array}{l}{\widehat{u}}_n(t)=\{\begin{array}{l}0,t\in \left[0,1-\frac{1}{n}\right),\\ u_n(t),t\in \left[1-\frac{1}{n},1\right].\end{array}\end{array}$

Similar to Lv and Yang [22], we see that B is a bounded linear operator and W satisfies (H5). Then (4.1) can be controlled on J by Theorem 3.3.

5. Conclusion

In this article, we consider the exact controllability of a Hilfer fractional Langevin equation and the corresponding results are obtained using three fixed point theorems, respectively. One result is obtained without the compactness of proper {S_{α2, δ}(t)}, whereas the other two results rely on the compactness of {S_{α2, δ}(t)}.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

HW wrote and revised this manuscript. JK discussed with HW and gave some valuable suggestions in this paper. Both authors have participated in this research and approved the final manuscript.

Funding

The authors acknowledge the support from the Hainan Provincial Natural Science Foundation of China (122MS088) and from the Qiongtai Normal University (Grants QTjg2022-4 and QTjg2022-49).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

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References

1. Abu Arqub O, Maayah B. Adaptive the Dirichlet model of mobile/immobile advection/dispersion in a time-fractional sense with the reproducing kernel computational approach: formulations and approximations. Int J Modern Phys B. (2022) 1–17. doi: 10.1142/S0217979223501795. [Epub ahead of print].

CrossRef Full Text | Google Scholar

2. Abu Arqub O, Alsulami H, Alhodaly M. Numerical Hilbert space solution of fractional Sobolev equation in (1+1)-dimensional space. Math Sci.. (2022) 1–12. doi: 10.1007/s40096-022-00495-9. [Epub ahead of print].

CrossRef Full Text | Google Scholar

3. Diethelm K, Freed AD. On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity. In:Keil F, Mackens W, Vob H, Werther, J, , editors. Scientific Computing in Chemical Engineering II. Heidelberg: Springer-Verlag (1999). p. 217–24.

Google Scholar

4. Gaul L, Klein P, Kempfle S. Damping description involving fractional operators. Mech Systems Signal Process. (1991) 5:81–8. doi: 10.1016/0888-3270(91)90016-X

CrossRef Full Text | Google Scholar

5. Glockle WG, Nonnenmacher TF. A fractional calculus approach of self-similar protein dynamics. Biophys J. (1995) 68:46–53.

PubMed Abstract | Google Scholar

6. Hilfer R. Applications of Fractional Calculus in Physics. Singapore: World Scientific (2000).

Google Scholar

7. Maayah B, Moussaoui A, Bushnaq S, Abu Arqub O. The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach. Demonstratio Mathematica. (2022) 55:963–77. doi: 10.1515/dema-2022-0183

CrossRef Full Text | Google Scholar

8. Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier (2006).

Google Scholar

9. Sabatier J, Agrawal OP, Machado JAT. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Dordrecht: Springer (2007).

Google Scholar

10. Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives. Theory and Applications. Yverdon: Gordon and Breach Science Publishers (1993).

PubMed Abstract | Google Scholar

11. Sathiyaraj T, Wang JR, Balasubramaniam P. Ulam's stability of Hilfer fractional stochastic differential systems. Eur Phys J Plus (2019) 134:605. doi: 10.1140/epjp/i2019-12952-y

CrossRef Full Text | Google Scholar

12. Aimene D, Baleanu D, Seba D. Controllability of semilinear impulsive Atangana-Baleanu fractional differential equations with delay. Chaos Solitons Fractals. (2019) 128:51–7. doi: 10.1016/j.chaos.2019.07.027

CrossRef Full Text | Google Scholar

13. Baleanu D, Fedorov VE, Gordievskikh DM, Tas K. Approximate controllability of infinite-dimensional degenerate fractional order systems in the sectorial case. Mathematics. (2019) 7:735. doi: 10.3390/math7080735

CrossRef Full Text | Google Scholar

14. Debbouchea A, Baleanu D. Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput Math Appl. (2011) 62:1442–50. doi: 10.1186/1687-1847-2011-5

CrossRef Full Text | Google Scholar

15. Liang J, Yang H. Controllability of fractional integro-differential evolution equations with nonlocal conditions. Appl Math Comput. (2015) 254: 20–9. doi: 10.1016/j.amc.2014.12.145

CrossRef Full Text | Google Scholar

16. Sakthivel R, Anthoni SM, Kim JH. Existence and controllability result for semilinear evolution integrodifferential systems. Math Comput Model. (2005) 41:1005–11. doi: 10.1016/j.mcm.2004.03.007

CrossRef Full Text | Google Scholar

17. Zwanzig R. Nonequilibrium Statistical Mechanics. New York, NY: Oxford University Press (2001).

Google Scholar

18. Ahmada B, Nieto JJ, Alsaedi A, El-Shahed M. A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal. (2012) 13:599–606. doi: 10.1016/j.nonrwa.2011.07.052

CrossRef Full Text | Google Scholar

19. Torres C. Existence of solution for fractional Langevin equation: variational approach. Qual J Theory Differ Equ. (2014) 54:1–14. doi: 10.14232/ejqtde.2014.1.54

CrossRef Full Text | Google Scholar

20. Wang J, Fěckan M, Zhou Y. Presentation of solutions of impulsive fractional Langevin equations and existence results. Eur Phys J Spcl Top. (2013) 222:1857–74. doi: 10.1140/epjst/e2013-01969-9

CrossRef Full Text | Google Scholar

21. Zhou H, Alzabut J, Yang L. On fractional Langevin differential equations with anti-periodic boundary conditions. Eur Phys J Spcl Top. (2017) 226:3577–90. doi: 10.1140/epjst/e2018-00082-0

CrossRef Full Text | Google Scholar

22. Lv JY, Yang XY. Approximate controllability of Hilfer fractional differential equations. Math Methods Appl. Sci. (2019) 43:1–13. doi: 10.1002/mma.5862

PubMed Abstract | CrossRef Full Text | Google Scholar

23. Gou HD, Li YX. A study on controllability of impulsive fractional evolution equations via resolvent operators. Bound Value Probl. (2021) 25:1–22. doi: 10.1186/s13661-021-01499-5

CrossRef Full Text | Google Scholar

24. Li K, Peng J, Jia J. Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives. Funct J Anal. (2012) 263:476–510. doi: 10.1016/j.jfa.2012.04.011

CrossRef Full Text | Google Scholar

25. Hilfer R, Luchko Y, Tomovski Z. Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives. Fract Calc Appl Anal. (2009) 12: 299–318.

PubMed Abstract | Google Scholar

26. Chang YK, Pereira A, Ponce R. Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators. Fract Calc Appl Anal. (2017) 20:963–87. doi: 10.1515/fca-2017-0050

CrossRef Full Text | Google Scholar

27. Ponce R. Existence of mild solutions to nonlocal fractional cauchy problems via compactness. Abatr Appl Anal. (2016) 2016:4567092. doi: 10.1155/2016/4567092

CrossRef Full Text | Google Scholar

28. Haase M. The complex inversion formula revisited. Aust J Math Soc. (2008) 84:73–83. doi: 10.1017/S1446788708000050

CrossRef Full Text | Google Scholar

29. Banas J, Goebel K. Measure of Noncompactness in Banach Spaces. New York, NY: Marcel Dekker (1980).

Google Scholar

30. Deimling K. Nonlinear Functional Analysis. Berlin: Springer (1985).

Google Scholar

31. Guo DJ, Sun JX. Ordinary Differential Equations in Abstract Spaces. Shandong Science and Technology, Jinan (1989).

Google Scholar

32. Lakshmikantham V, Leela S. Nonlinear Differential Equations in Abstract Spaces. New York, NY: Pergamon Press (1969).

Google Scholar