# Exact Controllability for Hilfer Fractional Differential Inclusions Involving Nonlocal Initial Conditions

Authors: Jun Du; Wei Jiang; Denghao Pang; Azmat Ullah Khan Niazi
Journal: Complexity
 (2018)

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## Abstract

The exact controllability results for Hilfer fractional differential inclusions involving nonlocal initial conditions are presented and proved. By means of the multivalued analysis, measure of noncompactness method, fractional calculus combined with the generalized Mo¨nch fixed point theorem, we derive some sufficient conditions to ensure the controllability for the nonlocal Hilfer fractional differential system. The results are new and generalize the existing results. Finally, we talk about an example to interpret the applications of our abstract results.


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## Body

## 1. Introduction

Fractional calculus generalizes the standard integer calculus to arbitrary order. It provides a valuable tool for the description of memory and hereditary properties of diversified materials and processes. In the past twenty years, the subject of the fractional calculus is picking up considerable popularity and importance. We can refer to the monographs of Diethelm and Freed [1], Kilbas et al. [2], Miller and Ross [3], Podlubny [4], and Zhou [5]. Fractional differential equations and inclusions involving Caputo derivative or Riemann-Liouville derivative have obtained more and more results (see [6–15]). Recently, Hilfer [16] initiated an extended Riemann-Liouville fractional derivative, named Hilfer fractional derivative, which interpolates Caputo fractional derivative and Riemann-Liouville fractional derivative. This operator appeared in the theoretical simulation of dielectric relaxation in glass forming materials. Hilfer et al. [17] initially presented linear differential equations with the new Hilfer fractional derivative and applied operational calculus to solve such generalized fractional differential equations. Subsequently, Furati et al. [18] and Gu and Trujillo [19] generalized to consider nonlinear problems and proved the existence, nonexistence, and stability results for initial value problems of nonlinear fractional differential equations with Hilfer fractional derivative in a suitable weighted space of continuous functions.Control theory is an interdisciplinary branch of engineering and mathematics that deals with influence behavior of dynamical systems. Controllability is one of the fundamental concepts in mathematical control theory, it means that it is possible to steer a dynamical system from an arbitrary initial state to arbitrary final state using the set of admissible controls. Recently, the controllability conditions for various linear and nonlinear integer or fractional order systems have been considered in many papers by using different methods [20–33] and the references. There have also been some results [20–24, 32, 33] about the investigations of the exact controllability of systems represented by nonlinear evolution equations in infinite dimensional space. But when the semigroup or the control action operator B is compact, then the controllability operator is also compact and the applications of exact controllability results is just restricted to the finite dimensional space [20]. Therefore, we investigate the exact controllability of the fractional evolution systems only involving noncompact semigroups.The nonlocal initial problems have been initially proposed by Byszewski et al. [34, 35] to generalize the study of the canonical initial problem, comes from physical science. For instance, it used to determine the unknown physical parameters in some inverse heat condition problems. It has been found that the nonlocal initial condition is more exact to describe the nature phenomena than the classical initial condition, since more data is taken into account, therefore abating the negative influences induced by a possible inaccurate single estimation taken at the start time. For more discussion on this type of differential equations and inclusions, we can see papers [36–42] and references given therein.Boucherif and Precup [36] proved the existence for mild solutions to the following nonlocal initial problem for first-order evolution equations using Schaefer fixed point theorem:
(1)x′t+Axt=ft,xt,t∈J,x0+∑k=1makxtk=0,0<t1<t2<⋯<tm<b,where A:DA⊆X→X is the infinitesimal generator of a C0-semigroup Ttt≥0 on a Banach space X and f:J×X→X is a known function.Liang and Yang [33] concerned the controllability for the following fractional integrodifferential evolution equations involving nonlocal conditions using the Mo¨nch fixed point theorem:
(2)Dqxt+Axt=ft,xt,Gxt+But,t∈J,x0=∑k=1mckxtk,0<t1<t2<⋯<tm<b,where Dq is the Caputo derivative of order q∈0,1, −A:DA⊆X→X is the infinitesimal generator of C0-semigroup Ttt≥0 of uniformly bounded linear operator, the control function u is known in L2J,U; U is a Banach space, B is a linear bounded operator from U to X; f is a known function and Gxt=∫0tKt,sxsds is a Volterra integral operator.Du et al. [43] generalized the results of [33] and gave the controllability for a new class of fractional neutral integrodifferential evolution equations with infinite delay and nonlocal conditions using Mönch fixed point theorem. However, it should be emphasized that to the best of our knowledge, the exact controllability of Hilfer fractional differential system has not been investigated yet. Motivated by [19, 30, 33, 36, 43], in this paper, we concern the controllability of the following fractional differential inclusions involving a more general fractional derivative with nonlocal initial conditions:
(3)D0+p,qxt∈Axt+Ft,xt+But,t∈0,a,I0+1−q1−px0+=∑i=1naixti,0<t1<t2<⋯<tn<a,where D0+p,q is the Hilfer fractional derivative of order p (p obeys 1/2<p≤1) and type q (q obeys 0≤q≤1) which will be given in Section 2; E is separable and A is bounded, so S· is a uniformly continuous semigroup and St=eAt. The nonlinear term F:J×E→2E\∅ is multivalued function. Let J=0,a,J′=0,a,a>0 are two finite intervals of ℝ; ai∈ℝ,ai≠0i=1,2,…,n,n∈N. The control function u takes values in L2J,U, with U as a Banach space; B is a linear bounded operator from U to E.In this paper, by means of a concrete nonlocal function, we do not have to suppose the compactness and Lipschitz conditions on the nonlocal function but only assume thataii=1,2,…,n satisfy the hypothesis (H0) (see Section 3). Furthermore, the proofs of our main results are based on fractional calculus theory, the multivalued analysis, measure of noncompactness method, in addition to the O’Regan-Precup fixed point theorem, which is an extension of the Mönch fixed point theorem.


## 2. Preliminaries and Notations

LetCJ′,E and CJ,E denote the space of E-valued continuous functions from J′ to E and from J to E, respectively; Let r=p+q−pq.DefineY=x∈CJ′,E:limt→0+t1−rxt exists and is finite}, involving the norm ·Y defined by xY=supt∈J′t1−rxt. Then, Y is a Banach space. We also note that
(1)
Whenr=1, then Y=CJ,E and ·Y=·;
(2)
Letxt=tr−1yt for t∈J′, x∈Y if and only if y∈CJ,E, and xY=y.Forγ>0, define BγYJ′=x∈Y:xY≤γ. Thus BγY is a bounded closed and convex subset of Y.LetBγJ=y∈CJ,E:y≤γ, Then Bγ is a closed ball of the space CJ,E with the radius γ and center at 0. And Bγ is also a bounded closed and convex subset of CJ,E.Next, we list some definitions and properties in fractional calculus, multivalued analysis, semigroup theory, and measure of noncompactness.The following definitions concerning fractional calculus can be found in the books [2–4, 16].Definition 1.
The fractional integral for functionf from lower limit 0 and order α can be expressed by
(4)I0+αft=1Γα∫0tfst−s1−αds,α>0,t>0,where Γ is the gamma function, and right side of upper equality is point-wise defined on 0,+∞.Definition 2.
The Riemann-Liouville derivative of orderα with the lower limit 0 for function f:0,+∞→ℝ can be expressed by
(5)R−LD0+αft=1Γn−αdndtn∫0tfst−sα+1−nds,t>0,0≤n−1<α<n.Definition 3.
The Caputo derivative of orderα for function f:0,+∞→ℝ can be denoted by
(6)CD0+αft=R−LDαft−∑k=0n−1tkk!fk0,t>0,0≤n−1<α<n.Definition 4.
The left Hilfer derivative of order0<p≤1 and type 0≤q≤1 of function f is defined by
(7)D0+p,qft=I0+q1−pDI0+1−q1−pft,where D≔d/dt.Remark 1.
(i)
The operatorD0+p,q can be written as
(8)D0+p,qft=I0+q1−pDI0+1−rft=I0+q1−pDrft,r=p+q−pq.
(ii)
Whenq=0 and 0<p≤1, the Hilfer fractional derivative coincides with the Riemann-Liouville derivative:
(9)D0+p,0ft=ddtI0+1−pft=R−LD0+pft.
(iii)
Whenq=1 and 0<p≤1, the Hilfer fractional derivative coincides with the Caputo derivative:
(10)D0+p,1ft=I0+1−pddtft=CD0+pft.
LetPE be the set of all nonempty subsets of E. We will use the following notations:
(11)PclE≔Y∈PEYisclosed,PbE≔Y∈PEYisbounded,PcvE≔Y∈PEYisconvex,PcpE≔Y∈PEYiscompact.Remark 2.
(i)
A measurable functionu:J→E is Bochner integrable if and only if u is Lebesgue integrable.
(ii)
A multivalued mapF:E→2E is said to be convex valued (closed valued) if Fu is convex (closed) for all u∈E is said to be bounded on bounded sets if FB=∪u∈B is bounded in E for all B∈PbE.
(iii)
A multivalued mapF is said to be upper semicontinuous (u.s.c.) on E if for each u0∈E, the set Fu0 is a nonempty closed subset of E, and if for each open subset Ω of E containing Fu0, there exists an open neighborhood ∇ of u0 such that F∇⊆Ω.
(iv)
A multivalued mapF is said to be completely continuous if FB is relatively compact for every B∈PbE. If the multivalued map F is completely continuous with nonempty compact values, then F is u.s.c. if and only if F has a closed graph, that is, un→u, yn→y, yn∈Fu0 imply y∈Fu. We say that F has a fixed point if there is u∈E, such that u∈Fu.
(v)
A multivalued mapF:J→PclE is said to be measurable if for each u∈E, the function y:J→R defined by yt=du,Ft=infu−z,z∈Ft is measurable.Lemma 1 (see [44]).
LetE be a Banach space. The multivalued map satisfies the following: for each t∈J, Ft,·:E→Pb,cl,cvE is u.s.c.; for each x∈E, the function F·,x:E→Pb,cl,cvE is strongly measurable and the set SF,x=f∈L1J,E:ft∈Ft,xt,fora.e.t∈J is nonempty. Let Γ be a linear continuous mapping from L1J,E to CJ,E, then the operator
(12)Γ∘SF:CJ,E→Pb,cl,cvCJ,E,x↦Γ∘SFx=ΓSF,x,is a closed graph operator in CJ,E×CJ,E.
Consider(13)Kpt=p∫0∞θξpθStpθdθ,t≥0,where ξpθ=1/pθ−1−1/2ωpθ−1/2, and
(14)ωpθ=1π∑n=1∞−1n−1θ−pn−1Γnp+1n!sinnπp,θ∈0,∞,and ξp is a probability density function defined on 0,∞, that is ∫0∞ξpθdθ=1.Remark 3 (see [2]).
As we all know from [2] that Kp can be denoted by the Mittag-Leffler functions:
(15)Kpt=∑i=0∞AitpiΓpi+p.
The following essential propositions can be found in the papers [19, 38].Lemma 2.
IfSt≤N,t∈J, then for each x∈E,
(i)
Kptx≤N/Γpx;
(ii)
tp−1Kptx≤Ntp−1/Γpx and I0+q1−ptp−1Kptx≤Ntr−1/Γrx.Lemma 3
(Example 2.1.3 [45]).
For eachG⊂CJ,E and t∈J, define Gt=gt:g∈G. If G is equicontinuous and bounded, then βGt is continuous on J and
(16)βG=maxt∈JβGt.
Here,β is the Hausdorff noncompact measure on E defined on every bounded subset U of Banach space E by
(17)βU=infε>0,Uhasafiniteε−netinE.Lemma 4 (see Lemma 5 [46]).
LetG⊆L1J,E be a countable subset with gt≤φt, for almost everywhere t∈J and any g∈G, where φ∈L1J,ℝ+. Then
(18)β∫Jgtdt:g∈G≤∫JβGtdt.
To end this section, we reintroduce the O’Regan-Precup fixed point theorem.Lemma 5 (see Theorem 3.2 [47]).
LetD be a subset of Banach space E which is closed and convex. Ω is a relatively open subset of D, and T:Ω¯→PcvD. Suppose graph (T) is closed, T maps compact sets into relatively compact sets, and that for some x0∈Ω, the following two conditions are satisfied:
(19)Z⊆D,Z⊂convx0∪TZZ¯=G¯withG⊆Zcountable⇒Z¯compact,x∉1−λx0+λTx,forallx∈Ω¯\Ω,λ∈0,1.
ThenT has a fixed point.


## 3. Controllability Results

We first consider linear Hilfer fractional differential equations of the form(20)D0+p,qxt=Axt+ht,t∈J′,I0+1−rx0+=∑i=1naixti,where h∈CJ,E.Assume that there exists the bounded operatorK:E→E given by
(21)K≔I−∑i=1naiΦp,rA,ti−1,where Φp,rA,t=I0+q1−ptp−1Kpt.By means of [48], we can present the sufficient conditions for the existence and boundedness of the operator K.Lemma 6.
If the hypothesis (H0)∑i=1nai<Γr/Nt11−r holds, the operator K defined in (21) exists and is bounded.Proof 1.
From the hypothesisH0, we have
(22)∑i=1naiΦp,rA,ti≤NΓr∑i=1nai·tir−1<Nt1r−1Γr∑i=1nai<1.
By operator spectrum theorem, the operatorK≔I−∑i=1naiΦp,rA,ti−1 exists and is bounded. Furthermore, by Neumann expression, we get
(23)K≤∑n=0∞∑i=1naiΦp,rA,tin=11−∑i=1naiΦp,rA,ti≤11−Nt1r−1/Γr∑i=1nai.
Using Lemma6 and [19], we give the following definition of mild solution for the Hilfer fractional system (20) involving nonlocal initial conditions.Definition 5.
A functionx∈CJ′,X is called a mild solution for the Hilfer fractional system (20) if it satisfies the following equation:
(24)xt=Φp,rA,t∑i=1naiKh¯ti+h¯t,t∈J′,where h¯t=∫0tΦp,pA,t−shsds.Remark 4.
By virtue of [19], a mild solution for Hilfer fractional evolution (20) with the initial condition is
(25)xt=Φp,rA,tI0+1−rx0++∫0tΦp,pA,t−shsds.
Specially,(26)xti=Φp,rA,tiI0+1−rx0++∫0tiΦp,pA,ti−shsds.
Using (20) and (26), we get
(27)I−∑i=1naiΦp,rA,ti·I0+1−rx0+=∑i=1nai∫0tiΦp,pA,ti−shsds.
SinceI−∑i=1naiΦp,rA,ti exists a bounded inverse operator which is denoted by K, so
(28)I0+1−rx0+=∑i=1naiK∫0tiΦp,pA,ti−shsds.
And hence,(29)xt=Φp,rA,t∑i=1naiKh¯ti+h¯t,t∈J′,it is indeed (24).Using Definition5, we give a new definition of the mild solution for the Hilfer fractional nonlocal differential inclusions (3) as follows:Definition 6.
A functionx∈CJ′,E is called a mild solution of the Hilfer fractional nonlocal differential inclusions (3) if for any u∈L2J,U, the following integral equation is satisfied:
(30)xt=Φp,rA,t∑i=1naiKf¯ti+f¯t,t∈J′,where f¯t=∫0tΦp,pA,t−sBus+fsds, Φp,pA,t=tp−1Kpt and f∈SF,x.To present and prove the main results of this paper, we enumerate the following hypotheses:(H1)
E is a separable Banach space, A is bounded, hence S·=eA· is a uniformly continuous semigroup.
(H2)
The multivalued mapF:J×E→Pb,cl,cvE satisfies the following:
(2a)
For everyt∈J, Ft,·:E→Pb,cl,cv is u.s.c., for each x∈E, the function F·,x:J→Pb,cl,cv is strongly measurable. The set SF,x=f∈L1J,E:ft∈Ft,xt,foralmosteverywheret∈Jis nonempty;
(2b)
There exists a functionξ1∈L1/p1J,ℝ+, p1∈0,p and a continuous nondecreasing function ψ:0,∞→0,∞, such that for any t,x∈J×E, we have Ft,xt=supft:ft∈Ft,xt≤ξ1tψxY,limm→∞infψm/m=Λ<∞;
(2c)
There exists a constantp2∈0,p and a function ξ2∈L1/p2J′,ℝ+ s.t. βFt,Z≤ξ2tβt1−rZ for any countable subset Z⊂E.
(H3)
(3a) Linear operatorV:L2J,U→E defined by
(31)Vu=a1−rΦp,rA,a∑i=1naiK∫0tiΦp,pA,ti−sBusds+∫0aΦp,pA,a−sBusdsis reversible, the inverse operator denoted by V−1 and takes values in L2J,UkerV, and there exist two constants Nu>0, Nv>0 such that
(32)B≤Nu,V−1≤Nv;
(3b) There exists a constantp3∈0,p and ξ3∈L1/p3J′,ℝ+ such that
(33)βV−1Zt≤ξ3tβZ,t∈J,for any countable subset Z⊂E.
(H4)
limη→∞supηar−1Npx1+N1NrNp+1ψη−1>1, where Nr=ΓrN/ΓpΓr−Nt1r−1∑i=1nai, Np=NuNvNra1−q1−p/pa1−q1−p/p and di=biap−pi, bi=1−pi/p−pi1−pi, Ni=diξisL1/pi, i=1,2,3.For anyx∈BγYJ′, define an operator T as follows:
(34)Txt=Φp,rA,t∑i=1naiK∫0tiΦp,pA,ti−sfs+Busds+∫0tΦp,pA,t−sfs+Busds,t∈J′,where f∈SF,x.It is evident to see thatlimt→0t1−rTxt=1/Γr∑i=1naixti.For anyy∈BγJ, let xt=tr−1yt for t∈J′, then x∈BγYJ′. Define T as follows
(35)Tyt=t1−rTxt,t∈J′,1Γr∑i=1naixti,t=0.Clearly,x is a mild solution of (3) in Y if and only if y=Ty has a solution y∈CJ,E.
(36)ut;x=a1−rV−1x1−Φp,rA,a∑i=1naiK∫0tiΦp,pA,ti−sfsds−∫0aΦp,pA,a−sfsdst,t∈J,f∈SF,x.For brevity, let us take the following notations(37)Pt;x=But;x+ft,f∈SF,x;P~x=∑i=1naiK∫0tiΦp,pA,ti−sPs;xds.In view of Lemma2, we obtain the following lemma that will be useful in the proof of the main results.Lemma 7.
Under the hypothesis (H2) (2b), (H3) (3a), for eachy∈BγJ, set xt=tr−1yt, t∈J′, we have
(38)Pt;x≤NuNva1−rx1+N1Nrψy+ξ1tψy,P~x≤Np∑i=1naix1+N1NrNp+1∑i=1naiψy.Proof 2.
By Lemma2, for each y∈BγJ, set t=tr−1yt,t∈J′, it is easy to get
(39)But;x≤NuNva1−rx1+N2KψxY∑i=1naiΓrΓp∫0titi−sp−1ξ1sds+a1−rNψxYΓp∫0aa−sp−1ξ1sds≤NuNva1−rx1+N2ψy∑i=1naid1ξ1L1/p1ΓpΓr−Nt1r−1∑i=1nai+Nd1a1−rψyΓpξ1L1/p1≤NuNva1−rx1+ΓrNN1ΓpΓr−Nt1r−1∑i=1naiψy≜NuNva1−rx1+N1Nrψy.
Hence, we easily see that(40)Pt;x≤NuNva1−rx1+N1Nrψy+ξ1tψy;P~x≤NΓr∑i=1naiΓpΓr−Nt1r−1∑i=1nai∫0titi−sp−1Ps;xds≤NΓr∑i=1naiΓpΓr−Nt1r−1∑i=1nai∫0titi−sp−1NuNva1−rx1+N1Nrψy+ξ1sψyds≤ΓrNNuNva1−q1−p∑i=1naiΓp+1Γr−Nt1r−1∑i=1naix1+N1Nrψy+N1Nr∑i=1naiψy≜Np∑i=1naix1+N1NrNp+1∑i=1naiψy.
This completes the proof.
Next, we derive the controllability results for the Hilfer fractional nonlocal differential inclusions (3).Theorem 1.
Assume the hypotheses (H0)–(H4) hold, then the Hilfer fractional nonlocal differential inclusions (3) are exact controllable on J provided that
(41)Λ≜NN21+N3NrNua1−rNrΓr∑i=1n∣ai∣+1Γp<1.Proof 3.
According to (H2) (2a) and [22], for each x∈CJ′,E, the multivalued function t→Ft,xt has a measurable selection, and in view of (H2) (2b), this selection belongs to SF,x. Thus, we can define a multivalued function T′:CJ,E→2CJ,E as follows. For every y∈CJ,E, let xt=tr−1yt∈Y,t∈J′, and a function z∈T′y if and only if
(42)zt=Krt∑i=1naiKf¯ti+t1−rf¯t,t∈J′,1Γr∑i=1naixti,t=0.where Krt=t1−rΦp,rA,t, f¯t=∫0tt−sp−1Kpt−sBus+fsds and f∈SF,x.
Note that according to (H4), there isR∗ such that for all R>R∗,
(43)a1−rNpx1+N1NrNp+1ψR<R.
DenoteΩ=y∈CJ,E:y∞<R0 and D=y∈CJ,E:y∞≤R0, where R0≥γ and R0=R∗+1. We just prove that the multivalued operator mathcalT′:D→2CJ,E meets the conditions of Lemma 5. Obviously, since the values of F are convex, the values of T′ are also convex.Claim 1.
Each solution to the inclusion(44)y∈λT′y,y∈D,λ∈0,1satisfies y∉D−Ω.
Lety be a solution of (44). Then, by reminding the definition of T′, the hypothesis (H2) (2b) and recalling also Lemma 2, for each t∈J, we derive
(45)yt≤N2∑i=1naiΓpΓr−Nt1r−1∑i=1nai∫0titi−sp−1Ps;xds+Nt1−rΓp·∫0tt−sp−1Ps;xds≤N2NuNvap+1−r∑i=1naipΓpΓr−Nt1r−1∑i=1naix1+N1Nrψy+NNuNvap+2−2rpΓp·x1+N1Nrψy+Na1−rψyΓp∫0tt−sp−1ξ1sds+N2ψy∑i=1naiΓpΓr−Nt1r−1∑i=1nai∫0titi−sp−1ξ1sds≤N2NuNvap+1−r∑i=1naipΓpΓr−Nt1r−1∑i=1nai+NNuNvΓrap+2−2r−N2NuNvap+2−2rt1r−1∑i=1naipΓpΓr−Nt1r−1∑i=1nai·x1+N1Nrψy+Na1−rψyΓr−N2a1−rt1r−1ψy∑i=1naiΓpΓr−Nt1r−1∑i=1nai+N2ψy∑i=1naiΓpΓr−Nt1r−1∑i=1nai·d1ξ1L1/p1≤ΓrNNuNvap+2−2rΓp+1Γr−Nt1r−1∑i=1naix1+N1Nrψy+Γra1−rNd1ξ1L1/p1ψyΓpΓr−Nt1r−1∑i=1nai≤ΓrNNuNvap+2−2rx1Γp+1Γr−Nt1r−1∑i=1nai+a1−rN1NrψR0ΓrNNuNvap+1−rΓp+1Γr−Nt1r−1∑i=1nai+1≜a1−rNpx1+N1NrNp+1ψR0.
This inequality with (43) deduces y∞<R0, So y∉D−Ω.Claim 2.
Every function inT′y:y∈BγJ is equicontinuous.
For eachy∈BγJ, set xt=tr−1yt,t∈J′ such that z∈T′y. By (42), there is f∈SF,x, for τ1,τ2∈J, τ1<τ2, we get
(46)zτ2−zτ1≤Krτ2P~x−Krτ1P~x+τ21−r∫0τ2τ2−sp−1Kpτ2−sPs;xds−τ11−r∫0τ1τ1−sp−1Kpτ1−sPs;xds≤Krτ2−Krτ1P~x+∫τ1τ2τ21−rτ2−sp−1Kpτ2−sPs;xds+∫0τ1τ11−rτ1−sp−1Kpτ2−s−Kpτ1−sPs;xds+∫0τ1τ11−rτ1−sp−1−τ21−rτ2−sp−1Kpτ2−sPs;xds≤I1+I2+I3+I4,where
(47)I1=Krτ2−Krτ1P~x,I2=NΓp∫τ1τ2τ21−rτ2−sp−1Ps;xds,I3=NΓp∫0τ1τ11−rτ1−sp−1−τ21−rτ2−sp−1Ps;xds,I4=∫0τ1τ11−rτ1−sp−1Kpτ2−s−Kpτ1−sPs;xds.
From (H1), one can deduce thatI1→0 as τ2−τ1→0. From Lemma 7, we have
(48)I2≤NNuNva2−2rΓp+1x1+N1Nrψγτ2−τ1p+Nb1a1−rΓpτ2−τ1p−p1ξ1L1/p1ψγ,which implies that I2→0 as τ2−τ1→0.
(49)I3≤NNuNva1−rΓpx1+N1Nrψγ∫0τ1τ11−rτ1−sp−1−τ21−rτ2−sp−1ds+NΓpψγ∫0τ1τ11−rτ1−sp−1−τ21−rτ2−sp−1ξ1sds.
Noting that(50)τ11−rτ1−sp−1−τ21−rτ2−sp−1ξ1s≤τ11−rτ1−sp−1ξ1s,and ∫0τ1τ11−rτ1−sp−1ξ1sds,s∈0,τ1 exists, then by Lebesgue’s dominated convergence theorem, we obtain I3→0 as τ2−τ1→0.
Forε>0, we get
(51)I4≤∫0τ1−ετ11−rτ1−sp−1Kpτ2−s−Kpτ1−sPs;xds+∫τ1−ετ1τ11−rτ1−sp−1Kpτ2−s−Kpτ1−sPs;xds≤NuNva2−2rx1+N1Nrψγτ1p−εppsups∈0,τ1−εKpτ2−s−Kpτ1−s+τ11−r∫0τ1−ετ1−sp−1ξ1sdssups∈0,τ1−εKpτ2−s−Kpτ1−sψγ+2NNuNva2−2rεpΓp+1x1+N1Nrψγ+2Na1−rΓpb1εp−p1ψγξ1L1/p1.
The assumption (H1) guarantees thatI4→0 as τ2−τ1→0 and ε→0. We prove that Krt is uniformly continuous on J. Consequently, T′ is equicontinuous on BγJ.Claim 3.
The inference (13) holds with y0=0.
LetZ=convy0∪T′Z⊆D,Z¯=G¯ with G⊆Z countable. We assert that Z is relatively compact. In fact, since G is countable and G⊆Z=convy0∪T′Z, we can chase down a countable set H=zn:n≥1⊆T′Z with G⊆convy0∪H. Then, there exists yn∈Z with zn∈T′yn. This means that there is fn∈SF,xn such that for t∈J
(52)znt=Krt∑i=1naiKf¯nti+f¯nt,where
(53)f¯nt=∫0tt−sp−1Kpt−sBuns+fnsds.
FromZ⊆G¯⊆conv¯y0∪H, for t∈J, s∈0,t. By utilizing Lemmas 2, 4, (H2) (2c), (H3) (3b), and the properties of the noncompact measure, we can derive
(54)βBuns+fns≤Nuξ3sN2∑i=1naiΓpΓr−Nt1r−1∑i=1nai∫0titi−sp−1βfns:n≥1ds+Na1−rΓp∫0aa−sp−1βfns:n≥1ds+βfns:n≥1≤Nuξ3sN2∑i=1naiΓpΓr−Nt1r−1∑i=1nai∫0titi−sp−1ξ2sβyns:n≥1ds+Na1−rΓp∫0aa−sp−1ξ2sβyns:n≥1ds+ξ2sβyns:n≥1≤N2NuNra1−rξ3s+ξ2s·βZ,and
(55)βZt≤βG¯t≤βHt≤βKrt∑i=1naiK∫0titi−sp−1Kpti−sBuns+fnsds:n≥1+β∫0tt−sp−1Kpt−sBuns+fnsds:n≥1≤N2∑i=1naiΓpΓr−Nt1r−1∑i=1nai∫0titi−sp−1N2NuNra1−rξ3s+ξ2sds·βZ+NΓp∫0tt−sp−1N2NuNra1−rξ3s+ξ2sds·βZ≤NN21+N3NrNua1−rNrΓr∑i=1nai+1Γp·βZ≜Λ·βZ.
RemindingZ=convy0∪T′Z, by Claim 2, Z is equicontinuous. So we find from Lemma 3 that
(56)βZ=maxt∈JβZt≤ΛβZ.
SinceΛ<1, we obtain βZ=0. That is, Z¯ is compact.Claim 4.
T′ maps compact sets into relatively compact sets.
LetQ be a compact subset of Z. From Claim 2, T′Q is equicontinuous. Let t∈J, by the definition of T′, for any y∈Q and z∈T′y, there is fy∈SF,x such that
(57)zt=Krt∑i=1naiKf¯yti+f¯yt,t∈J,where f¯yt=∫0tt−sp−1Kpt−sBuys+fysds.
Therefore(58)βT′Qt≤βzt:z∈T′y,y∈Q≤βKrt∑i=1naiK∫0titi−sp−1Kpti−sBuys+fysds:y∈G+β∫0tt−sp−1Kpt−sBuys+fysds:y∈G≤NN21+N3NrNua1−rNrΓr∑i=1nai+1ΓpβQ=0.
Then, Lemma3 indicates βT′Q=maxt∈JβT′Qt=0, that is, the set T′Q is relatively compact.Claim 5.
The graphT′ is closed.
For anyyn∈BγJ, let xnt=tr−1ynt and x∗t=tr−1y∗t, t∈J′. Let yn→y∗n→∞, μn∈T′yn, μn→μ∗n→∞. We will show that μ∗∈T′y∗. Since μn∈T′yn, there is fn∈SF,xn such that for any t∈J,
(59)μnt=Krt∑i=1naiK∫0titi−sp−1Kpti−sfnsds+Krt∑i=1naiK∫0titi−sp−1·Kpti−sBV−1·a1−rx1−Kra∑i=1naiK∫0titi−τp−1Kpti−τfnτdτ−a1−r∫0aa−τp−1Kpa−τfnτdτds+t1−r∫0tt−sp−1Kpt−sfnsds+t1−r∫0tt−sp−1Kpt−sBV−1a1−rx1−Krs∑i=1naiK·∫0titi−τp−1·Kpti−τfnτdτ−a1−r∫0aa−τp−1Kpa−τfnτdτds.
So we just need to demonstrate the existence off∗∈SF,x∗ such that for any t∈J,
(60)μ∗t=Krt∑i=1naiK∫0titi−sp−1Kpti−sf∗sds+Krt∑i=1naiK∫0titi−sp−1·Kpti−sBV−1a1−rx1−Kra∑i=1naiK∫0titi−τp−1Kpti−τf∗τdτ−a1−r∫0aa−τp−1Kpa−τf∗τdτds+t1−r∫0tt−sp−1Kpt−s·f∗sds+t1−r∫0tt−sp−1Kpt−sBV−1a1−rx1−Krs∑i=1naiK·∫0titi−τp−1Kpti−τf∗τdτ−a1−r∫0aa−τp−1Kpa−τf∗τdτds.Take into account the linear continuous operator(61)Γ:L1/pJ,E→BγJ,where
(62)Γft=Krt∑i=1naiK∫0titi−sp−1Kpti−sfs−BV−1Krs∑i=1naiK·∫0titi−τp−1Kpti−τfτdτ+a1−r∫0aa−τp−1Kpa−τfτdτds+t1−r∫0tt−sp−1Kpt−sfs−BV−1·Krs∑i=1naiK∫0titi−τp−1·Kpti−τfτdτ+a1−r∫0aa−τp−1Kpa−τfτdτds.Clearly, we can get from Lemma1 that the operator Γ∘SF is a closed graph.Sinceμn→μ∗, n→∞, we can obtain that as n→∞,
(63)μnt−Krt∑i=1naiK∫0titi−sp−1Kpti−sBV−1a1−rx1ds−t1−r∫0tt−sp−1·Kpt−sBV−1a1−rx1ds−μ∗t−Krt∑i=1naiK∫0titi−sp−1Kpti−s·BV−1a1−rx1ds−t1−r∫0tt−sp−1Kpt−sBV−1a1−rx1ds→0.In addition, we get(64)μnt−Krt∑i=1naiK∫0titi−sp−1Kpti−sBV−1a1−rx1ds−t1−r∫0tt−sp−1Kpt−sBV−1a1−rx1ds∈ΓSF,xn.Sinceyn→y∗n→∞, we can obtain from Lemma 1 that
(65)μ∗t−Krt∑i=1naiK∫0titi−sp−1Kpti−sBV−1a1−rx1ds−t1−r∫0tt−sp−1Kpt−sBV−1a1−rx1ds=Krt∑i=1naiK∫0titi−sp−1Kpti−sf∗sds−Krt∑i=1naiK∫0titi−sp−1·Kpti−sBV−1Krs∑i=1naiK∫0titi−τp−1Kpti−τf∗τdτ+a1−r·∫0aa−τp−1Kpa−τf∗τdτds+t1−r∫0tt−sp−1Kpt−sf∗sds−t1−r∫0tt−sp−1Kpt−sBV−1Krs∑i=1naiK∫0titi−τp−1Kpti−τ·f∗τdτ+a1−r∫0aa−τp−1Kpa−τf∗τdτds.For somef∗∈SF,x∗, this infers that μ∗∈T′y∗. Hence, T′ has a closed graph. Thus, Claims 1–5 are completed. By use of Lemma 5, we know operator T′ has a fixed point in BγJ. Let xt=tr−1yt, then x is a mild solution of (3) and it satisfies xa=x1. Therefore, the Hilfer fractional nonlocal differential inclusions (3) are exact controllable on J.


## 4. Applications

Consider the following partial differential system(66)D0+3/4,2/3xt,s∈−∂∂sxt,s+e−2t1+etxt,s+ηt,s,t∈0,a,s∈0,1,xt,0=xt,1=0,t∈0,a,I0+1/31−pxt,st=0=∑i=1nΓ1112arctan12i2xti,s,s∈0,1,where n∈0,a is constants, η:J×0,1→0,1 is continuous.LetE=Ω≕C0,1 and A is defined by
(67)DA=w∈E:w′∈E,w0=w1=0,Aw=−w′,w∈DA.As we all know that−A generates an equicontinuous semigroup Stt≥0 in E and it satisfies
(68)Ttwυ=wt+υ,for w∈E. Thus, Stt≥0 is not compact in E and sup0≤t≤aTt≤1. Take
(69)xts=xt,s,D0+3/4,2/3xts=I0+1/6D0+11/12xt,s=1Γ1/6∫0tt−τ−5/6∂11/12∂τ11/12xτ,sdτ,ft,xts=e−2t1+etxt,s,uts=μt,s,ai=Γ1112arctan12i2,ti=i,i=1,2,…,n.Then, for anyy∈Bγ≔y∈CJ,E:y≤γ, where J=0,a, let xt=t−1/12yt, t∈J′≔0,a, then x∈BγYJ′, and we obtain
(70)ft,xts≤e−2t1+etxt,s≤e−2tγ1+et≤γ2.Thus, the hypothesisH2 holds for β=1/2 and ξ2t=1/2 for all t∈J′. By
(71)∑i=1nai≤Γ1112·∑i=1∞arctan12i2=Γ1112·π4<Γ1112,we verify that the hypothesis (H0) holds.Fors∈0,1, the operator V is defined as
(72)Vts=KraI−Γ1112∑i=1ni−1/12Kriarctan12i2−1Γ1112∑i=1narctan12i2·∫0ii−τ−1/4Kpi−τητ,sdτ+∫0aa−τ−1/4Kpa−τητ,sdτ,where Krtt≥0 and Kptt≥0 satisfy
(73)Krt=I0+1/6t−1/4Kpt,Kpt=∑i=0∞Ait3/4iΓ3/4i+3/4.IfV satisfies the hypothesis (H3), from Theorem 1, we get that the Hilfer fractional differential inclusion (66) involving nonlocal initial conditions is exact controllable on 0,a provided that (H4) and (41) are satisfied.



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*Source: 9472847-2018-08-19.xml*