# Approximate Controllability for a Kind of Fractional Neutral Differential Equations with Damping

Authors: Jun Du; Dongling Cui; Yeguo Sun; Jin Xu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper gains several meaningful results on the mild solutions and approximate controllability for a kind of fractional neutral differential equations with damping (FNDED) and order belonging to1,2 in Banach spaces. At first, a new expression for the mild solutions of FNDED via the (p, q)-regularized operator family and the technique of Laplace transform is acquired. Then, we consider the approximate controllability of FNDED by means of the approximate sequence method, and simultaneously, some applicable sufficient conditions are obtained.


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## Body

## 1. Introduction

The primary description of the fractional-order derivative was proposed by Riemann and Liouville toward the end of the nineteenth century, but the notion of the arbitrary derivative and integral which generalized the classical integer-order derivative and integral was presented by Leibniz and Liouville in 1695. However, until the late 1960s when many phenomena on physics, engineering technology, and economics were more accurately described by fractional differential equations (FDEs), scientists began to show great interest on fractional order calculus. For example, they are widely adopted for nonlinear oscillations of earthquakes and in the fluid-dynamic traffic model. In practice, FDEs are deemed to optimize the traditional differential equation model. About the elementary theory of fractional differential and evolution systems, one can refer to Podlubny [1], Kilbas et al. [2], Zhou [3, 4], and [5–10] and the references cited therein.In the particles’ realistic movement, the resistance to motion is unavoidable, so it is suitable to add the damping character in mathematical models and control systems. Recently, a great deal of meaningful conclusions for the mathematical models with damping influence have been presented by the researchers [11–17].As we all know, the controllability exerts a momentous effect on control theory and engineering technology. It lies in the fact that it is bound up with quadratic optimal control, observer design, and pole assignment. For this reason, the controllability has been actively investigated by many investigators, and an impressive progress has been made in recent years [7–12, 14–20]. Controllability of the deterministic systems in infinite dimensional spaces has been broadly investigated. In some results of the controllability for systems described by fractional differential models, the fixed-point and the approximate sequence method are felicitously used. Nonetheless, as demonstrated by Triggiani [21], for many parabolic partial differential systems, the conditions of complete controllability are very finite. The research to approximate controllability is more proper for the practical systems than to complete controllability. For the past few years, as regards differential dynamical systems in Banach spaces, several results are achieved about the approximate controllability [8, 19]. However, as far as we know, the approximate controllability of the fractional neutral differential equations with damping and order belonging to [1, 2] is still relatively infrequent, so it is more interesting and necessary to study it.


## 2. Preliminaries and Notations

Inspired by the aforementioned analysis, the approximate controllability for a kind of fractional neutral differential equations with damping of order belonging to1,2 in the Banach space is studied in our work. We acquire several sufficient conditions to pledge the approximate controllability of the FNDED via the contraction mapping theory and approximate sequence method. The FNDED and order in [1, 2] is debated as follows:(1)DtpCx′t+qCDtpxt−ht,xt=Axt+ft,xt+But,t∈0,b,xt=φt,t∈−τ,0,x′0=x1,where p:0≤p≤1, DtpC denotes the Caputo fractional derivative, qq≥0, and the linear densely unbounded closed operator A:DA⊆E⟶E is the p,q−regularized family defined on the Banach space E. Here, the state x⋅ is evaluated in Banach space E. Let U be a Banach space of admissible control functions. The variable u⋅ takes a value in L20,b;U, B:U⟶E, which is linear and bounded. In addition, f:0,b×C−τ,0;E⟶E is nonlinear, which will be explained in detail later; xt∈L2−τ,0,E and is denoted by xtϑ=xt+ϑ|−τ≤ϑ≤0; φ=φϑ|−τ≤ϑ≤0∈L2−τ,0,E. The relevant linear neutral system of FNDED (1) is(2)DtpCy′t+qCDtpyt−ht,yt=Ayt+Bvt,t∈0,b,xt=φt,t∈−τ,0,x′0=x1.The structure of this paper is given as follows. In Section2, we review several fundamental concepts and provide a new form of the mild solution for FNDED (1). Then, in Sections 3 and 4, we acquire several meaningful results for the existence and uniqueness of the mild solution and, furthermore, the approximate controllability for FNDED (1).LetE be a Banach space with norm ⋅E and LE be the space of all linear bounded operators on E. Let Lm0,b;E,1≤m<∞, be the space of E-valued Bochner integrable function f:0,b⟶E, and(3)fLm=∫0bfEmdt1/m.In the following, we recall some definitions to be adopted in the entire work.Definition 1.
(see [1, 2]). The Riemann–Liouville fractional integral of a function z and order α and from lower limit 0 can be denoted by(4)I0+αzt=1Γα∫0tzst−s1−αds,t>0,α>0.Definition 2.
(see [1, 2]). The Caputo derivative of order α for a function z∈C0,∞,R can be defined as(5)D0+αCzt=1Γ1−αddt∫0tt−ν−αz′νdν,t>0,0<α<1.Remark 1 (see [1, 2]).
Ifz∈C0,∞,R, then(6)D0+αCzt=1Γ1−α∫0tt−ν−αz′νdν=I0+1−αz′t,t>0,0<α<1.Lemma 1.
(see [2]). Let α∈n−1,n. If z⋅∈ACna,b,E or z⋅∈Cna,b,E, then the Caputo derivative(7)I0+αD0+αzt=zt−∑i=0n−1zi0i!ti,t>0.Definition 3.
(see [15, 16]). Let q≥0 and 0≤p≤1. A is a linear closed operator, and its domain DA is in Banach space E. We say that A is the generator of a p,q-regularized operator family Sptt≥0⊂LE if the three formulas are established in the following:(a)
Spt is strongly continuous on R+, and Sp0=I.(b)
SptDA⊂DA and ASptx=SptAx for all x∈DA,t≥0.(c)
For everyx∈DA, the following equation holds:(8)Sptx=e−qt+1Γp∫0t∫0νμp−1eqμ−νASpt−νxdμdν,t≥0,where Spt is the p,q-regularized operator family generated by A. On the basis of the operator Sp, the two operators Pp, Qp: R+⟶LE are reminded:(9)Ppt=∫0tSpνdν,t≥0,(10)Qpt=1Γp∫0tt−νp−1Spνdν,t≥0.
The following basic statement is deduced from [22].Lemma 2.
(see [15, 16]). Let A be a densely linear closed operator in Banach space E. Then, A is the generator of a p,q-regularized operator family if and only if there is a strong continuous operator Sp:R+⟶LE and constant α≥0 satisfying ηpη+q:η>δ⊂ρA and(11)Hηx≔ηpηpη+qI−A−1x=∫0∞e−ηtSptxdt,η>δ,x∈E.
In view of Lemma2, from (9) and (11), we have(12)ηp−1ηpη+qI−A−1x=∫0∞e−ηtPptxdt,η>δ,x∈E.
Combining (9) and (10), we obtain(13)ηpη+qI−A−1x=∫0∞e−ηtQptxdt,η>δ,x∈E.
At first, we put forward a new form of the mild solution for the system in the following.Lemma 3.
Let0≤p≤1 and g∈L1J,E; if x satisfies the equation(14)DtpCx′t+qCDtpxt−ht,xt=Axt+gt,t∈0,b,xt=φt,t∈−τ,0,x′0=x1,then x is denoted by the integral equation(15)xt=Sptφ0+Pptx1+qφ0−qh0,x0+q∫0tSpt−νhν,xνds+∫0tQpt−νgνdν.Proof.
Taking the Riemann–Liouville integral to both sides of the first equality of (14), we have(16)ItpDtpx′t+qItpDtpxt−ht,xt=ItpAxt+gt.
From Lemma1, we can get that(17)x′t−x′0+qxt−ht,xt−x0+h0,x0=ItpAxt+gt.
That is,(18)x′t+qxt=x′0+qφ0−qh0,x0+qht,xt+ItpAxt+gt.
Letη>0. Taking the Laplace transform for (18), we yield(19)−φ0+ηXη+qXt=1ηx1+qφ0−qh0,x0+qHη+1ηpAXη+Gη.
Hence, we obtain(20)Xη=ηpηpη+qI−A−1φ0+ηp−1ηpη+qI−A−1x1+qφ0−qh0,x0+ηpηpη+qI−A−1qHη+ηpη+qI−A−1Gη.
Then, applying Laplace inverse transform on (20) and combining with the result of Lemma 2 and the Laplace transform of the convolution, we achieve(21)xt=Sptφ0+Pptx1+qφ0−qh0,x0+q∫0tSpt−νhν,xνdν+∫0tQpt−νgνdν.
This completes the proof.
Because of Lemma3, we naturally deduce a new representation of the mild solution for (14).Definition 4.
A functionx⋅∈C−τ,b,E is reputedly a mild solution for the fractional neutral linear system with damping (14) if there exists g∈L1J,E satisfying(22)xt=Sptφ0+Pptx1+qφ0−qh0,x0+q∫0tSpt−νhν,xνdν+∫0tQpt−νgνdν,t∈0,b.
To gain the global existence of the mild solution for FNDED (1), we give hypothesis S(0).
S0: A is the infinitesimal generator of an exponentially bounded p,q−regularized operator family Spt on E, namely, there exist two real numbers M≥1 and ω≥0 satisfying(23)Spt≤Meωt,t≥0.Remark 2.
From hypothesis S(0) and formulas (9) and (10), we have(i)
Spt≤Mewb≜MS,t∈0,b.(ii)
Ppt=∫0tSpsds≤Mewbb≜MP,t∈0,b.(iii)
Qpt=∫0tgpt−sSpsds≤Mewbbp/Γp+1≜MQ,t∈0,b.


## 3. Existence and Uniqueness of Mild Solutions

In this segment, firstly, we consider the existence and uniqueness of mild solutions for FNDED (1). To this end, we impose the following conditions.S1: h:0,b×E⟶E is continuous and satisfies(24)ht,xt−ht,ytE≤nh⋅xt−yt,for every t∈0,b, and xt,yt∈L2−τ,0;E, nh⋅∈L1J,R+, and nh⋅≤Nh∈R+.S2: for almost every t∈0,b, the nonlinear item ft,⋅:E⟶E is continuous, and there is nf⋅∈L1J,R+ satisfying(25)ft,xt−ft,ytE≤nf⋅xt−yt,for all xt,yt∈L2−τ,0;E and nft≤Nf∈R+.Theorem 1.
Under conditionsS0–S2, the fractional neutral differential equations with damping (1) has one and only one mild solution belonging to the space C−τ,b,E with xt=φt,t∈−τ,b, and x′t=x1 for every control item u⋅∈L20,b;U, provided that(26)bb+τqMSNh+MQNf<1.Proof.
Define the mappingΩ:C−τ,b,E⟶C−τ,b,E by(27)Ωxt=Sptφ0+Pptx1+qφ0−qh0,x0+q∫0tSpt−νhν,xνds+∫0tQpt−νBuν+fν,xνdν,t∈0,b,xt=φt,t∈−τ,0,x′0=x1.
LetMh=max0≤t≤bht,0E, Mf=max0≤t≤bft,0E, BE≤MB, r=1−qMSNh+MQNfbb+τ−1MS+qMP+qMPNhφ+MPx1+qMSb+MPMh+bMQMf+b1/2MQMBuL2, and Br=x⋅∈C−τ,b;E:x≤r,xt=φt,t∈−τ,0, which is a closed and bounded subset of C−τ,b;E. For each y∈Br, we prove Ω has a fixed point in C−τ,b;E. Firstly, we prove that Ω maps Br into itself. Obviously, Ωx0∈Br. For t∈0,b, we achieve(28)Ωxt≤Sptφ+Pptx1+qφ−qh0,x0+q∫0tSpt−νhν,xνdν+∫0tQpt−νBuν+fν,xνdν≤MSφ+MPx1+qφ+qh0,x0−h0,0+h0,0+qMS∫0thν,xν−hν,0+hν,0dν+MQ∫0tBuνdν+MQ∫0tfν,xν−fν,0+fν,0dν≤MS+qMP+qMPNhφ+MPx1+qMSb+MPMh+bMQMf+b1/2MQMBuL2+qMSNh+MQNfbb+τr=r.
This is because(29)∫0txνdν≤∫0b∫−τ0xν+ϑdϑdν=∫0b∫ν−τνxωdωdν≤bb+τsupt∈u−τ,bxω=bb+τx,which means Ωxt≤r,Ωx⊆Br.
Next, we demonstrate thatΩ is a contraction mapping on Br. In fact, for x⋅,y⋅∈Br,(30)Ωxt−Ωyt≤q∫0tSpt−νhν,xν−hν,yνdν+∫0tQpt−sfs,xs−fs,ysds≤qMS∫0tnhνxν−yνds+MQ∫0tnfνxν−yνdν≤bb+τqMSNh+MQNfx−y.
Since(31)∫0txν−yνdν≤∫0b∫−τ0xν+ϑ−yν+ϑdϑdν=∫0b∫ν−τνxω−yωdωdν≤∫0b∫−τbxω−yωdωdν≤bb+τsupt∈−τ,bxω−yω=bb+τx−y,due to bb+τqMSNh+MQNf<1, Ω is a contraction mapping. Consequently, Ω has the unique fixed point belonging to the space C−τ,b;E. (27) is the unique mild solution of FNDED (1).


## 4. Approximate Controllability

In this segment, we acquire several appropriate sufficient conditions of the approximate controllability for FNDED (1) by virtue of the approximate technique and the iterative approach.Define two continuous linear operatorsℒ and ℐ from L20,b,E to E by(32)ℒv=∫0bPpb−νvνdν,v⋅∈L20,b,E,ℐv=q∫0bSpb−νvνdν,v⋅∈L20,b,E.If we let the combinationx,u be the mild solution for FNDED (1) with u∈L20,b,U, then we represent it as xt=xt;x0,x1,u, and the terminal item xb can be written as(33)xb=xb;x0,x1,u=Spbφ0+Ppbx1+qφ0−qh0,x0+q∫0bSpb−ν⋅hν,xνBudν+∫0bQpb−νBuν+fν,xνBudν≜Hb,φ,x1+ℐhν,xνBu+ℒfν,xνBu+ℒBuν,where Hb;φ,x1=Spbφ0+Ppbx1+qφ0−qh0,x0.So, the reachable setKbf which is composed of all possible final states at time b is(34)Kbf=xb;x0,x1,u;u∈U.Thus, the approximate controllability for FNDED (1) means the set Kbf is dense on space E. That is to say, the definition of approximate controllability is acquired.Definition 5.
Letx0,x1∈E. We called the fractional neutral differential equations with damping (1) is approximately controllable on 0,b if, for any ε>0 and xb∈E, there is a control function uNε∈U satisfying(35)xb−Hb;φ,x1−ℐhν,xνBuNε−ℒfν,xνBuNε−ℒBuNε<ε.
And we add some of the postulated conditions.
S3: for each given ε>0 and v⋅,w⋅∈L20,b;E, there is u⋅∈U satisfying(36)ℐw−ℒv−ℒBuE<ε,where Bu⋅L2J,U≤Muv⋅L2J,U, and Mu is a positive real number irrelevant of v⋅ and holds(37)qMSNh+MQNf+MuMQNfbb+τ<1.
Because condition (37) implies (26), the existence is still met when inequality (26) is changed into (37) in Theorem 1. Next, to demonstrate the conclusion of the approximate controllability remains true for (1), we give the following two lemmas.Lemma 4.
Letx1,u1,x2,u2 be two pairs relevant to FNDED (1). Then, in terms of conditions S(0)–S(3), the following result holds:(38)xtBu1−xtBu2≤MQb1/21−qMSNh+MQNfbb+τBu1−Bu2L2.Proof.
The mild solutionxt=xt;x0,x1,u of FNDED (1) in E satisfies(39)xtBu=φt,t∈−τ,0,Sptφ0+Pptx1+qφ0−qh0,x0+q∫0tSpt−νhν,xνdν+∫0tQpt−νgνdν,t∈0,b.
We definey⋅,φ:−τ,b⟶E as(40)yt,φ=φt,t∈−τ,0,Sptφ0+Pptx1+qφ0−qh0,x0,t∈0,b.
Letxt=yt+zt, t∈−τ,b. Apparently, x⋅ meets (1) if and only if z0=0andt∈−τ,0, and for each t∈0,b, we acquire(41)zt=q∫0tSpt−νhν,xνdν+∫0tQpt−νBuν+fν,xνdν=q∫0tSpt−νhν,zνBu+yνdν+∫0tQpt−νBuν+fν,zνBu+yνdν.
Now, let us takextBu1,xtBu2∈E and u1⋅,u2⋅∈U; then,(42)xtBu1−xtBu2=ztBu1+yt−ztBu2+yt=ztBu1−ztBu2≤q∫0tSpt−ν⋅hν,xνBu1−hν,xνBu1dν+∫0tQpt−ν⋅Bu1ν−Bu2νdν+∫0tQpt−ν⋅fν,xνBu1−fν,xνBu2dν≤qMSNh∫0txνBu1−xνBu2dν+b1/2MQBu1ν−Bu2νL2+MQNf∫0txνBu1−xνBu2dν=MQb1/2Bu1ν−Bu2νL2+qMSNh+MQNf∫0tzνBu1−zνBu2dν≤MQb1/2Bu1ν−Bu2νL2+qMSNh+MQNfbb+τzνBu1−zνBu2.
Recombining the formula, we get(43)xtBu1−xtBu2=ztBu1−ztBu2≤MQb1/21−qMSNh+MQNfbb+τBu1−Bu2L2.
This completes the proof.Theorem 2.
If hypothesesS0)–S3 hold, the fractional neutral differential equations with damping (1) is approximately controllable on J.Proof.
A is the infinitesimal generator of a p,q-regularized operator family in a Banach space, and its domain is dense in Banach space E. So, in accordance with the definition of approximate controllability, it is only needed to demonstrate that DA⊂Kbφ,x1. Next, we should prove that, for every ε>0 and xb∈DA, there is uNε∈U with(44)xb−Hb,φ,x1−ℐhν,xνBuNεν−ℒfν,xνBuNεν−ℒBuNεν<ε,where xνt=xt;φ,x1,uNε satisfies(45)xνt=Ht;φ,x1+q∫0tSpt−νhν,xνBuNενdν+∫0tQpt−νBuNεν+fν,xνBuνdν,t∈0,b.
By the theory of the fractional resolvent, we deduce thatHb;φ,x1∈DA for φt,x1∈E, which implies that xb−Hb;φ,x1∈DA for xb∈DA. It is evident that there is one v∈L10,b,E such that(46)ℒv=xb−Hb;φ,x1.
Condition (S3) suggests for eachε>0 and u1∈U, there is u2∈U such that(47)xb−Hb;φ,x1−ℐhν,xνBu1−ℒfν,xνBu1−ℒBu2<ε32.
Furthermore, foru2∈U, we determine w2∈U again by condition S(3) and (36) with the following two properties:(48)ℐhν,xνBu2−hν,xνBu1+ℒfν,xνBu2−fν,xνBu1−ℒBw2<ε33,Bw2L2≤Mufν,xνBu2−fν,xνBu1L2≤MuNf∫0bxνBu2−xνBu1L12dν1/2≤MuNfb1/2b+τxνBu2−xνBu1≤MQMuNfbb+τ1−qMSNh+MQNfbb+τBu1−Bu2L2,where x2t=xt;x0,x1,u2,t∈0,b.
Thus, we may defineu3=u2−w2 in U and derive the following inequality:(49)xb−Hb;φ,x1−ℐhν,xνBu2−ℒfν,xνBu2−ℒBu3≤xb−Hb;φ,x1−ℐhν,xνBu1−ℒfν,xνBu1−ℒBu2+ℒBw2−ℐhν,xνBu2−hν,xνBu1−ℒfν,xνBu2−fν,xνBu1≤ε32+ε33.
By the same way, we obtain the sequenceun:n≥1U satisfying(50)xb−Hb;φ,x1−ℐhν,xνBun−ℒfν,xνBun−ℒBun+1≤132+133+⋯+13n+1ε,Bun+1−BunL2≤MQMuNfbb+τ1−qMSNh+MQNfbb+τBun−Bun−1L2,where xnt=xt;x0,x1,un,t∈0,bandn=1,2,…. Because hypothesis S(3) holds, we can acquire that the sequence Bun:n≥1 is a Cauchy sequence in L20,b,E, and in this way, we can gain a function item v∈L20,b,E satisfying(51)limn⟶∞Bun=v.
On account of the mappingℒ:L20,b,E⟶E is linear-continuous, for any given ε>0, we can find a real integer number Nε satisfying(52)LBuNε+1−LBuNε<ε3.
Consequently, by inequalities (50) and (52), we derive(53)xb−Hb;φ,x1−ℐhν,xνBuNε−ℒfν,xνBuNε−ℒBuNε≤xb−Hb;φ,x1−ℐhν,xνBuNε−ℒfν,xνBuNε−ℒBuNε+1+ℒBuNε+1−ℒBuNε≤132+133+⋯+13n+1ε+ε3≤ε,where xnεt=xt;x0,x1,unε,t∈0,b.
This means thatξ∈Kbf¯. Thus, Kb0¯⊂Kbf; therefore, the fractional neutral differential equations with damping (1) is approximately controllable on [0, b].Theorem 3.
Postulate the range of operatorB is denoted by RB and it is dense in L20,b;E. If conditions S(0)–S(3) are true, the fractional neutral differential equations with damping (1) is approximately controllable on [0, b].Proof.
BecauseRB is dense in L20,b;E, for every function j⋅∈L20,b;E and ι>0, there is Bu⋅∈RB, where u⋅∈U, satisfying(54)Bu⋅−j⋅L20,b,E<ιj⋅L20,b,E.
Now, we have(55)ℒp−ℒBu≤MQ∫0bjs−Busds≤MQb1/2js−BusL20,b,E≤MQb1/2ιj⋅L20,b,E<ε.
Thus, from (50), we have(56)Bu⋅L20,b,E=Bu⋅−j⋅+j⋅L20,b,E≤Bu⋅−j⋅L20,b,E+j⋅L20,b,E≤ιj⋅L20,b,E+j⋅L20,b,E=ι+1j⋅L20,b,E.
It indicates that condition S(3) holds if we selectι>0. Therefore, FNDED (1) is approximately controllable on [0, b] by using Theorem 2.



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