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ORIGINAL RESEARCH article

Front. Appl. Math. Stat., 18 May 2023
Sec. Dynamical Systems
This article is part of the Research Topic Future Challenges in the Fractional-Order Dynamical Systems: From Mathematics to Applications View all 6 articles

Controllability of Hilfer fractional Langevin evolution equations

\r\nHaihua Wang
Haihua Wang*Junhua KuJunhua Ku
  • College of Science, Qiongtai Normal University, Haikou, China

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 [110]. 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 D0+α,β. Especially, D0+α,0 became the famous Riemman–Liouville fractional derivative whereas D0+α,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 [1216]. 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 [1821].

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

{ cDβ( cDα+λ)x(t)=f(t,x(t)),0<t<1,0<α1,1<β2,x(0)=0,x(η)=0,x(1)=0,

where cDα 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:

{D0+α,βx(t)+Ax(t)=f(t,x(t))+(Bu)(t),t(0,b],limt0+(I0+(1-β)(2-α)x)(t)=0,limt0+ddt(I0+(1-β)(2-α)x)(t)=b1,

where D0+α,β denotes the Hilfer fractional derivative, ASect(θ), where θρ(A)[0,(1-α2)π)], and b1 is an element in Banach space X. The control term uLp(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:

{D0+α1,β1(D0+α2,β2+A)x(t)=f(t,x(t))+Bu(t), tJ=[0,b],b>0,limt0+(I0+(1-α2)(1-β2)x)(t)=0,limt0+ddt(I0+(1-α1)(1-β1)+(1-α2)x)(t)+h(x)=x0,    (1.1)

where D0+αi,βi, i = 1, 2 denotes the Hilfer fractional derivative, respectively. 0 < αi ≤ 1, 0 ≤ βi ≤ 1, satisfies 1 < α1 + α2 ≤ 2. A generates a strongly continuous (α2, δ)-resolvent family Sα2, δ(t) (t ≥ 0), where 0 < δ ≤ α1 + α2. The function f:J × EE, let U be a Banach space, the control term uL2(J, U), B:UE 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 xC=maxtJx(t), for xC(J, E). We consider the Lp(J, R+) of Lebesgue p-integrable functions with 1 < p < ∞ on J, and let fLp denote the norm of Lp(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 AB(E), ρ(A) is defined by the set of {λ:(λIA)−1 exists in B(E)}.

Let gγ(γ > 0) denote the function

gγ(t)={tγ-1Γ(γ),t>0,0,t0.

For two given functions f1 and f2, the convolution of them is expressed in the form (f1*f2)(t)=0tf1(t-s)f2(s)ds.

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

S(t)Meωt,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 I0+γf(t)=(gγ*f)(t), t > 0.

Definition 2.3. Hilfer et al. [25] The Hilfer fractional derivative D0+α1,β1f(t) of order α1 ∈ (n − 1, n] and type β1 ∈ [0, 1] is defined by

D0+α1,β1f(t)=(I0+β1(n-α1)dndtn(I0+(1-β1)(n-α1)f))(t).

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 fL(0, b), n − 1 < α1n, 0 ≤ β1 ≤ 1, and I0+(1-β1)(n-α1)fACk[0,b], then

(I0+α1D0+α1,β1f)(t)=f(t)-k=0n-1(t-s)k-(n-α1)(1-β1)Γ(k-(n-α1)(1-β1)+1)limt0+dkdtk(I0+(1-β1)(n-α1)f)(t).

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 Sα, β is strongly continuous from [0, ∞) to B(E), such that Sα, β(t) is EB, {λα:(λαIA)−1 exists in B(E), Reλ > ω},

λα-β(λαI-A)-1x=0e-λtSα,β(t)x dt,Reλ>ω,xE.    (2.1)

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

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

(i) For t ≥ 0 and xD(A), we have Sα, β(t)xD(A). Moreover, Sα, β(t)Ax = ASα, β(t)x;

(ii) For xE, t ≥ 0, we have 0tgα(t-s)Sα,β(s)xdsD(A), and

Sα,β(t)x=gβ(t)x+A0tgα(t-s)Sα,β(s)x ds;

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

0tgα(t-s)ASα,β(s)x ds.

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

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

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

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

(i) For t > 0, Sα, β(t) is compact.

(ii) For μ > ω1/α, (μIA)−1 is compact.

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

(i) For t > 0, Sα, β(t) is a compact operator.

(ii) For μ > ω1/α, (μIA)−1 is compact.

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

λα-β(λαI-A)-1x=0e-λtSα,β(t)x dt,

from Definition 2.4. However, note that {Sα, β(t)}t>0 is uniform continuous by our hypothesis, where we can see that (λαIA)−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 gβ2Lloc1[0,) and therefore, by proposition in Haase [28], we obtain

limN12πiω-iNω+iNeλt(L(gβ2*Sα,β2))(λ)dλ=gβ2*Sα,β2=Sα,β(t),

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

12πiΓeλtλα-β(λαI-A)-1dλ=Sα,β(t),

where Γ is a vertical path lying in Re(z) = ω. By Lemma 2.4 and hypothesis, we observe for t > 0, Sα, 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, {Sα, β(t)}t≥0 is generated by operator A and {Sα, β(t)}t≥0 is strongly continuous. Then we have

 d dtSα,β(t)x=Sα,β-1(t)x,for tJ,xE.    (2.2)

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

λα-β(λαI-A)-1x=0e-λtSα,β(t)x dt,Reλ>ω,    (2.3)
λα-β+1(λαI-A)-1x0e-λtSα,β-1(t)x dt,Reλ>ω.    (2.4)

(2.3) and (2.4) together imply

0e-λtSα,β(t)x dt=λα-β(λαI-A)-1x                                             =1λλα-β+1(λαI-A)-1x                                             =1λ0e-λtSα,β-1(t)x dt                                             =0e-λt(g1*Sα,β-1)(t)x dt                                             =0e-λt(0tSα,β-1(s)ds)x dt,

It is easy to see that Sα,β(t)=0tSα,β-1(s)ds, 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, {Sα2, δ}t≥0 is generated by operator −A. Suppose that xC(J, E), if for tJ, x(t) ∈ D(−A) satisfies problem (1.1) and AxL1((0, b), E), then we have

x(t)=((gα1+α2-δ*Sα2,δ)*(f¯+Bu))(t)+(gγ1+α2-δ*Sα2,δ)(t)(x0-h(x)),    (2.5)

where f¯(t)=f(t,x(t)).

Proof. Using Liouville operators with I0+α1 on both sides of the equation

D0+α1,β1(D0+α2,β2+A)x(t)=f(t,x(t))+Bu(t),

in view of Lemma 2.1, we obtain

(D0+α2,β2+A)x(t)=I0+α1(f¯+Bu)(t)+c0Γ(γ1)tγ1-1,    (2.6)

where γ1 = α1 + β1 − α1β1. Using Liouville operators with I0+α2 on both sides of equation (2.6) again, we obtain

x(t)=I0+α1+α2(f¯+Bu)(t)-I0+α2(Ax)(t)+c0Γ(γ1+α2)tγ1+α2-1+c1Γ(γ2)tγ2-1,    (2.7)

where γ2 = α2 + β2 − α2β2. In view of the condition, we obtain c0 = x0h(x) and c1 = 0. Then we rewrite the representation of (2.7) as

x(t)=(gα1+α2*f¯)(t)+(gα1+α2*Bu)(t)-(gα2*Ax)(t)+x0-h(x)Γ(γ1+α2)tγ1+α2-1.    (2.8)

Applying the Laplace transform to (2.8), we obtain

(Lx)(λ)=1λα1+α2(Lf¯)(λ)+1λα1+α2(LBu)(λ)-1λα2A(Lx)(λ)+(x0-h(x))1λγ1+α2.

Thus, we obtain

(Lx)(λ)=1λα1(λα2I+A)-1(Lf¯)(λ)+1λα1(λα2I+A)-1(LBu)(λ)                  +(x0-h(x))1λγ1(λα2I+A)-1.

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

x(t)=((Sα2,α1+α2)*(f¯+Bu))(t)+Sα2,γ1+α2(t)(x0-h(x))        =((gα1+α2-δ*Sα2,δ)*(f¯+Bu))(t)        +(gγ1+α2-δ*Sα2,δ)(t)(x0-h(x)).

Definition 2.6. Let 0 < δ ≤ α1 + α2, {Sα2, δ(t)}t ≥ 0 is generated by −A. We say that x(t) is a mild solution of (1.1) if limt0+ddt(I0+(1-α1)(1-β1)+(1-α2)x)(t)+h(x)=x0, limt0+(I0+(1-α2)(1-β2)x)(t)=0, x(·) ∈ C(J, E) satisfies the equation

x(t)=((gα1+α2-δ*Sα2,δ)*(f¯+Bu))(t)+(gγ1+α2-δ*Sα2,δ)(t)(x0-h(x)).

3. Main results

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

Definition 3.1. Let the initial condition x0E and final stages xbE, if there exists a control term uL2(J, U), such that x(t) is the mild solution of (1.1) with respect to u, which satisfies

limt0+(I0+(1-α2)(1-β2)x)(t)=0,limt0+ d dt(I0+(1-α1)(1-β1)+(1-α2)x)(t)+h(x)=x0

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

Theorem 3.1. Let 0 < δ ≤ α1 + α2, {Sα2, δ(t)}t ≥ 0 is generated by operator −A and (M, ω) is a type of Sα2, δ(t). Assume that the following conditions are satisfied:

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

(H2) There exist q1 ∈ [0, 1) and two functions mL1q1(J,R+), Φ ∈ C(R+, R+) which are non-decreasing that satisfy

f(t,x)m(t)Φ(x),for xE,a.e. tJ.

(H3) There exist q2 ∈ [0, 1) and a function nL1q2(J,R+), such that for every bounded set D in E,

α(f(t,D))n(t)α(D),for a.e. tJ.

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

h(x)c1,h(x)-h(y)c2x-y,xC(J,E);

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

α(h(D))lα(D),

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

Wu=0bSα2,α1+α2(b-s)Bu(s)ds,u=ux,

where ux is defined in (3.4).

(i) The inverse operator W−1:EL2(J, U)\kerW exists, if there exist M1 > 0, M2 > 0, such that ‖B‖ ≤ M1, W-1M2;

(ii) There exist q3 ∈ [0, 1) and a function KL1q3(J,R+), such that for every bounded subset D in E,

α(W-1(D)(t))K(t)α(D),tJ.

Assume that max{Λ1, Λ2} < 1, where

Λ1=Mωα1+α2-δ(1+MM1M2ωα1+α2-δ12ωeωb)(1-q1ω)1-q1eωbmL1q1lim infr+Φ(r)r,    (3.1)
Λ2=Meωb[1+2MM1ωα1+α2-δ(1-q3ω)1-q3eωbKL1q3]        ×[1ωγ1+α2-δl+2ωα1+α2-δ(1-q2ω)1-q2nL1q2],    (3.2)

then system (1.1) can be controlled on J.

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

(Tx)(t)=((gα1+α2-δ*Sα2,δ)*(f¯+Bu))(t)               +(gα2+γ2-δ*Sα2,δ)(t)(x0-h(x)),tJ,    (3.3)

where the control term u is given by u(t) = ux(t), xC(J, E) is given by

ux(t)=W-1[xb-(gα2+γ2-δ*Sα2,δ)(b)(x0-h(x))           -((gα1+α2-δ*Sα2,δ)*f¯)(b)](t).    (3.4)

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

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

If not, then for every r > 0, there exist xrBr and trJ, such that ‖(Txr)(tr)‖ > r. First, we observe that

(Tx)(t)0t(gα1+α2-δ*Sα2,δ)(t-s)Bf(s,x(s)) ds                      +(gγ1+α2-δ*Sα2,δ)(t)B(x0-h(x))                      +0t(gα1+α2-δ*Sα2,δ)(t-s)BBux(s) ds                      0tMeω(t-s)ωα1+α2-δm(s)Φ(x(s)) ds                      +Meωtωγ1+α2-δ(x0-h(x))                      +0tMeω(t-s)ωα1+α2-δM1ux(s) ds                      Mωα1+α2-δ0teω(t-s)m(s)Φ(x(s)) ds                      +Meωtωγ1+α2-δ(x0+c1)                      +MM1ωα1+α2-δ[12ω(e2ωt-1)]12uxL2,    (3.5)

where

uxL2M2[xb+Meωbωγ1+α2-δ(x0+c1)                  +Mωα1+α2-δ0beω(b-s)m(s)Φ(x(s)) ds].    (3.6)

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

(Tx)(t)M{1+MM1M2ωα1+α2-δ[12ω(e2ωb-1)]12}                     [eωbωγ1+α2-δ(x0+c1)                     +1ωα1+α2-δ0beω(b-s)m(s)Φ(x(s)) ds]                     +MM1M2ωα1+α2-δ[12ω(e2ωb-1)]12xb.

Consequently,

r<(Txr)(tr)M{1+MM1M2ωα1+α2-δ       [12ω(e2ωb-1)]12} {eωbωγ1+α2-δ(x0+c1)    +Φ(r)ωα1+α2-δ[1-q1ω(eω1-q1b-1)]1-q1mL1q1}    +MM1M2ωα1+α2-δ[12ω(e2ωb-1)]12xb    M(1+MM1M2ωα1+α2-δ12ωeωb)[eωbωγ1+α2-δ(x0+c1)    +Φ(r)ωα1+α2-δ(1-q1ω)1-q1eωbmL1q1]    +MM1M2ωα1+α2-δ12ωeωbxb.    (3.7)

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

Mωα1+α2-δ(1+MM1M2ωα1+α2-δ12ωeωb)(1-q1ω)1-q1eωbmL1q1lim infr+Φ(r)r1,

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

Step 2: T:BrBr is continuous.

Assume that {xn} ⊂ Br satisfying xnx. Let us show that ‖TxnTxC → 0. For this, we consider the inequality

(Txn)(t)-(Tx)(t)0t(gα1+α2-δ*Sα2,δ)(t-s)Bf¯n(s)                                             -f¯(s)‖ ds+0t(gα1+α2-δ                                             *Sα2,δ)(t-s)BBuxn(s)-Bux(s) ds                                             +(gγ1+α2-δ*Sα2,δ)(t)Bh(xn)-h(x)                                             Mωα1+α2-δ0teω(t-s)f¯n(s)-f¯(s) ds                                             +MM1ωα1+α2-δ[12ω(e2ωt-1)]12uxn-uxL2                                             +Meωtωγ1+α2-δc2xn-x,    (3.8)

where f¯n(t)=f(t,xn(t)) and

uxn-uxL2M2[Meωbωγ1+α2-δc2xn-xC                                 +Mωα1+α2-δ0beω(b-s)f¯n(s)-f¯(s) ds].    (3.9)

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

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

Let D be a countable subset in Br 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={xn}n=1Br, let 0 ≤ t1 < t2b, then

(Txn)(t1)-(Txn)(t2)0t1(gα1+α2-δ*Sα2,δ)(t2-s)-(gα1+α2-δ*Sα2,δ)(t1-s)B×f¯(s)+Buxn(s) ds+t1t2(gα1+α2-δ*Sα2,δ)(t2-s)Bf¯(s)+Buxn(s) ds+(gγ1+α2-δ*Sα2,δ)(t2)-(gγ1+α2-δ*Sα2,δ)(t1)B(x0-h(xn)).

By Lemma 2.5, (gα1 + α2 − δ*Sα2, δ)(t) = Sα2, α1 + α2(t) and (gγ1 + α2 − δ*Sα2, δ)(t) = Sα2, γ1 + α2(t) for t ≥ 0. Furthermore, by Theorem 2.1, we obtain Sα2, α2 + α1(t) and Sα2, γ1 + α2(t) which are norm continuous. Since the right-hand side of the inequality approaches zero as t2t1, T(D) is equicontinuous on J.

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

α(Txn(t))α({((gα1+α2-δ*Sα2,δ)*f¯(s))(t)})                    +α({((gα1+α2-δ*Sα2,δ)*Buxn)(t)})                    +α({(gγ1+α2-δ*Sα2,δ)(t)(h(xn))})                    2Mωα1+α2-δ0teω(t-s)n(s) dsα({xn})                    +2MM1ωα1+α2-δ0teω(t-s)α({uxn(s)}) ds                    +Meωtωγ1+α2-δlα({xn}),    (3.10)

where

α({uxn(s)}n=1)K(s)[α({(gγ1+α2-δ*Sα2,δ)(b)h(xn)})                                    +α({((gα1+α2-δ*Sα2,δ)*f¯n)(b)})]                                    K(s)α({xn}n=1)(Meωbωγ1+α2-δl                                    +2Mωα1+α2-δ0beω(b-s)n(s) ds.    (3.11)

By (3.10) and (3.11), we obtain

α(Txn(t))2Mωα1+α2-δα(D)[1-q2ω(eω1-q2t-1)]1-q2nL1q2                    +2MM1ωα1+α2-δα(D)(Meωbωγ1+α2-δl                    +2Mωα1+α2-δ0beω(b-s)n(s)ds                    ×0teω(t-s)K(s)ds+Meωtωγ1+α2-δlα(D)                    2Mωα1+α2-δα(D)(1-q2ω)1-q2eωtnL1q2                    +2M2M1ωα1+α2-δα(D)[eωbωγ1+α2-δl                    +2ωα1+α2-δ(1-q2ω)1-q2eωbnL1q2                    ×(1-q3ω)1-q3eωtKL1q3+Meωtωγ1+α2-δlα(D)                    Meωb[1+2MM1ωα1+α2-δ(1-q3ω)1-q3eωbKL1q3]                    ×[1ωγ1+α2-δl+2ωα1+α2-δ(1-q2ω)1-q2nL1q2]                        α(D),

we have

α(TD)Λ2α(D).

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

α(D)α(co¯({0}T(D))).

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

Theorem 3.2. Let 0 < δ ≤ α1 + α2, {Sα2, δ(t)}t ≥ 0 is generated by operator −A and (M, ω) is a type of Sα2, 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) (λα2I+A)-1 is compact for all λ>ω1/α2.

If max{Λ1, Λ3} < 1, where Λ3=Meωbωγ1+α1-δc2, then (1.1) can be controlled on J.

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

(T1x)(t)=0t((gα1+α2-δ*Sα2,δ)*(f¯+Bu)(t),tJ,    (3.12)
(T2x)(t)=(gγ1+α2-δ*Sα2,δ)(t)(x0-h(x)),tJ.    (3.13)

As in Step 1 of Theorem 3.1, we can find r > 0, such that T1x + T2yBr for x, yBr. Moreover, with a similar method used in Step 2 of Theorem 3.1, it follows that T1 is continuous on Br and T2 is a contraction on Br. Currently, we are going to illustrate that {T1x:xBr} is precompact. The uniformly bounded nature of {T1x:xBr} is obvious.

Step 1: {T1x:xBr} is an equicontinuous family.

For xBr, without loss of generality, we assume that 0 ≤ t1 < t2b, then

(T1x)(t1)-(T1x)(t2)0t1(gα1+α2-δ*Sα2,δ)(t2-s)-(gα1+α2-δ*Sα2,δ)(t1-s)B×f¯(s)+Bux(s) ds+t1t2(gα1+α2-δ*Sα2,δ)(t2-s)Bf¯(s)+Bux(s) ds:=I1+I2.

For I1, we have

I1(Φ(r)mL1q1++M1uxL2)(0t1(gα1+α2-δ    *Sα2,δ)(t2-s)-(gα1+α2-δ*Sα2,δ)(t1-s)B11-q1 ds)1-q1.    (3.14)

By Theorem 2.1, we have the norm continuity of Sα2, α2 + α1(t) and therefore if t2t1, then Sα2, α2 + α1(t2s) − Sα2, α2 + α1(t1s) → 0 in B(E). We can have that limt2t1I1=0 using Lebesgue's theorem.

For I2, we have

I2MΦ(r)ωα1+α2-δt1t2eω(t2-s)m(s)ds    +MM1ωα1+α2-δt1t2eω(t2-s)ux(s) ds    MΦ(r)ωα1+α2-δmL1q1[1-q1ω(eω(t2-t1)1-q1-1)]1-q1    +MM1ωα1+α2-δuxL2[12ω(e2ω(t2-t1)-1)]12,    (3.15)

and therefore limt2t1I2=0. From the above two inequalities, we find that {T1x:xBr} is an equicontinuous family.

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

First, it is obvious that H(0) is precompact. Finally, let 0 < tb be a fixed number. For ∀ϵ ∈ (0, t), we consider the operator T1ϵ on Br by the formula

(T1ϵx)(t)=((gα1+α2-δ*Sα2,δ)*(f¯+Bu))(t-ϵ),xBr.

From (H6) and Theorem 2.2, we know that the compactness of {(gα1+α2-δ*Sα2,δ)(t-s)(f¯(s)+Bu):0st-ϵ} for ϵ > 0. Using the Mazur theorem and the mean-value theorem with respect to the Bochner integral, we have that for ϵ > 0, Hϵ(t)={(T1ϵx)(t):xBr} is precompact in E. In addition, for every xBr, we obtain

(T1x)(t)-(T1ϵx)(t)t-ϵt(gα1+α2-δ*Sα2,δ)(t-s)[f¯(s)                                                +Bu(s)]dsMΦ(r)ωα1+α2-δmL1q1                                                  [1-q1ω(eωϵ1-q1-1)]1-q1                                                +MM1ωα1+α2-δuxL2[12ω(e2ωϵ-1)]12.

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

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

Theorem 3.3. Let 0 < δ ≤ α1 + α2, {Sα2, δ(t)}t ≥ 0 is generated by operator −A and (M, ω) is a type of Sα2, 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, {Sα2, 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:BrBr is continuous. We shall now examine the precompact nature of {Tx:xBr}. Furthermore, we can see that {Tx:xBr} is not only uniformly bounded, but also equicontinuous.

Next, we verify that for all t ∈ [0, b], {Tx(t):xBr} is precompact. Obviously, {(Tx)(0):xBr} is precompact. Let 0 < tb be a number, ∀ϵ ∈ (0, t), we consider operator Tϵ on Br as follows:

(Tϵx)(t)=Sα2,δ(ϵ)((gα1+α2-δ*Sα2,δ)*(f¯+Bu))(t-ϵ),xBr.

If 0 < δ ≤ 1, then (H6), (H7), and Lemma 2.4 show that for t > 0, Sα2, δ(t) is compact, if 1 < δ ≤ α1 + α2, then (H6) and Theorem 2.2 also illustrate that Sα2, δ(t) is compact for t > 0, Finally, we obtain that {(Tϵx)(t):xBr} is precompact in E for ∀ϵ ∈ (0, t). Furthermore, for every xBr, we have

Sα2,δ(ϵ)((gα1+α2-δ*Sα2,δ)*(f¯+Bu))(t-ϵ)    -0t-ϵ(gα1+α2-δ*Sα2,δ)(t-s)[f¯(s)+Bu(s)] ds    (Φ(r)mL1q1+M1uxL2)(0t-ϵSα2,δ(ϵ)(gα1+α2-δ    *Sα2,δ)(t-s-ϵ)    -(gα1+α2-δ*Sα2,δ)(t-s)B11-q1 ds)1-q1

By Theorem 2.1, (gα1 + α2 − δ*Sα2, δ)(t) is norm continuous for all t > 0, using Lebesgue's theorem, we have

Sα2,δ(ϵ)((gα1+α2-δ*Sα2,δ)*(f¯+Bu))(t-ϵ)-0t-ϵ(gα1+α2-δ*Sα2,δ)(t-s)[f¯(s)+Bu(s)] ds0,ϵ0.

Hence, the set ((gα1+α2-δ*Sα2,δ)*(f¯+Bu))(t):xBr}, t > 0 is precompact. The compactness of (gα2 + γ2 − δ * Sα2, δ)(t) is obtained by Theorem 2.2. Hence, we have proved that for t ∈ (0, b], {Tx(t):xBr} is relatively compact in E. Consequently, by Ascoli–Arzela's Theorem, the set {Tx:xBr} is precompact. This further leads to T being compact on Br. We therefore have, by applying Schauder's fixed point theorem, a fixed point on Br 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), αi ∈ (0, 1], βi ∈ [0, 1], and i = 0, 1. We consider the fractional control system

{D0+α1,β1(D0+α2,β2+A)x(t,ξ)=f(t,x(t,ξ))+Bu(t,ξ),t(0,1),ξ[0,π],limt0+(I0+(1-α2)(1-β2)x)(t,ξ)=0,limt0+ d dt(I0+(1-α1)(1-β1)+(1-α2)x)(t,ξ)=x0(ξ),    (4.1)

where t ∈ (0, 1), ξ ∈ [0, π], and D0+αi,βi are Hilfer fractional derivatives. Operator A is given by Ax=x-2xξ2, let D(A) = {xE: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 xn(ξ)=2πsin(nξ) are the eigenvalues and the normalized eigenvectors of A, respectively.

For xE and 1 ≤ δ ≤ α1 + α2, we have

λα2-δ(λα2I+A)-1x=n=1x,xnxnλα2-δλα2+(1+n2)                                                    =n=1x,xnxn0e-λttδ-1Eα2,δ                                                          (-(1+n2)tα2) dt                                                    =0x,xnxne-λtn=1tδ-1Eα2,δ                                                          (-(1+n2)tα2) dt.

Hence, {Sα2, δ(t)}t≥0 is generated by operator −A,

Sα2,δ(t)x=n=1x,xnxntδ-1Eα2,δ(-(1+n2)tα2),xE,

which is norm continuous by the continuity of Eα2, δ(·). Moreover, for λ > 0, we have limnλα2-δλα2+(1+n2)=0, which implies that λα2-δ(λα2I+A)-1 is compact on the Hilbert space E, then (λα2I+A)-1 is compact for λ > 0.

Otherwise, for each xE, we obtain Sα2,δ(t)xbδ-1Γ(δ)x. Therefore, Sα2, δ(t) is of type (bδ−1/Γ(δ), 1).

Let f(t,x)=e-t1+tx, then we can choose m(t)=e-t1+t and Φ = I.

f(t,x)e-t1+t

Assume that Bu(t)=n=1u^n(t)xn, where

u^n(t)={0,t[0,1-1n),un(t),t[1-1n,1].

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 {Sα2, δ(t)}, whereas the other two results rely on the compactness of {Sα2, 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

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: control, mild solution, existence, non-compactness, evolution

Citation: Wang H and Ku J (2023) Controllability of Hilfer fractional Langevin evolution equations. Front. Appl. Math. Stat. 9:1191661. doi: 10.3389/fams.2023.1191661

Received: 22 March 2023; Accepted: 14 April 2023;
Published: 18 May 2023.

Edited by:

Zhouchao Wei, China University of Geosciences Wuhan, China

Reviewed by:

Savin Treanta, Polytechnic University of Bucharest, Romania
Omar Abu Arqub, Al-Balqa Applied University, Jordan

Copyright © 2023 Wang and Ku. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Haihua Wang, d2FuZ2hvaXdhbiYjeDAwMDQwOzE2My5jb20=

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.