# On the Approximate Controllability of Fractional Evolution Equations with Generalized Riemann-Liouville Fractional Derivative

Authors: N. I. Mahmudov; M. A. McKibben
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We discuss the approximate controllability of fractional evolution equations involving generalized Riemann-Liouville fractional derivative. The results are obtained with the help of the theory of fractional calculus, semigroup theory, and the Schauder fixed point theorem under the assumption that the corresponding linear system is approximately controllable.
Finally, an example is provided to illustrate the abstract theory.


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## Body

## 1. Introduction

Many social, physical, biological, and engineering problems can be described by fractional partial differential equations. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations. In the last two decades, fractional differential equations (see, e.g., [1–4] and the references therein) have attracted many scientists, and notable contributions have been made to both the theory and applications of fractional differential equations. Several researchers have studied the existence results of initial and boundary value problems involving fractional differential equations. The motivation for those works arises from both the development of the theory of fractional calculus itself and the applications of such constructions in various fields, including physics, chemistry, aerodynamics, and electrodynamics of complex medium. Recently, Zhou and Jiao [5] discussed the existence of mild solutions of fractional evolution and neutral evolution equations in an arbitrary Banach space in which the mild solution is defined using the probability density function and semigroup theory. Using the same method, Zhou et al. [6] gave a suitable definition of a mild solution for an evolution equation involving a Riemann-Liouville fractional derivative. Using sectorial operators, Shu and Wang [7] gave a definition of a mild solution for fractional differential equations with order 1<α<2 and established existence results. Agarwal et al. [8] studied the existence and dimension of the set of mild solutions of semilinear fractional differential equations inclusions. Hilfer [9] proposed a generalized Riemann-Liouville fractional derivative, for short, which includes Riemann-Liouville fractional derivative and Caputo fractional derivative. Very recently, Gu and Trujillo [10] investigated a class of evolution equations involving Hilfer fractional derivatives.Recently, the approximate controllability of fractional semilinear evolution systems in abstract spaces has been studied by many researchers. In [11], Sakthivel et al. studied the approximate controllability of semilinear fractional differential systems. Kumar and Sukavanam [12, 13] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order by using the contraction principle and the Schauder fixed point theorem. Balasubramaniam et al. [14] derived sufficient conditions for the approximate controllability of impulsive fractional integrodifferential systems with nonlocal conditions in Hilbert space. Using the analytic resolvent method and the continuity of a resolvent in the uniform operator topology, Fan [15] derived existence and approximate controllability results of a fractional control system. Liu and Bin [16] studied existence of mild solutions and approximate controllability results for impulsive fractional abstract Cauchy problems involving Riemann-Liouville fractional derivatives. More recently, Mahmudov [17] formulated and proved a new set of sufficient conditions for the approximate controllability of fractional neutral type evolution equations in Banach spaces by using Schauder’s fixed point theorem. However, the approximate controllability of fractional evolution equations with Hilfer fractional derivative has not yet been studied.Motivated by the aforementioned papers, we study the approximate controllability of a class of fractional evolution equations:(1)D0+ν,μxt=Axt+But+ft,xt,t∈0,b,I0+1-ν1-μx0=x0,where D0+ν,μ is the Hilfer fractional derivative, 0≤ν≤1, 0<μ<1, the state x· takes value in a Hilbert space X, and A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators St,t>0 in X. The control function u takes values in a Hilbert space U,u∈L20,b,U, and B:U→X is a linear bounded operator. The function f:0,b×X→X will be specified in later sections.The focus of this paper is the study of the approximate controllability of fractional semilinear differential equations in Hilbert spaces. We will explore approximate controllability using techniques from [18]. The method is inspired by viewing the problem of approximate controllability as the limit of optimal control problems and replacing it via the convergence of resolvent operators (the resolvent condition, (R)).


## 2. Preliminaries

Define(2)Cν,μ(0,b,X)=x∈C0,b,X:limt→0+t1-λ1-μxtlimt→0+t1-λ1-μxtexistsandisfinitewith the norm ·ν,μ defined by(3)xν,μ=sup⁡0≤t≤bt1-λ1-μxt.Obviously, Cν,μ(0,b,X) is a Banach space.Let us recall the following definitions from fractional calculus.Definition 1 (see [1]).
The fractional integral of orderα>0 with the lower limit a for a function f:a,∞→R is defined as(4)Ia+αf(t)=1Γ(α)∫at‍f(s)t-s1-αds,t>0,α>0,provided that the right-hand side is pointwise defined on [0,∞), where Γ is the gamma function.Definition 2 (see [9]).
The Hilfer derivative of order0≤ν≤1 and 0<μ<1 with lower limit a is defined as(5)Da+ν,μf(t)=I0+ν1-μddtIa+1-ν1-μft,for functions such that the expression on the right-hand side exists.Remark 3.
Whenν=0, 0<μ<1, the Hilfer fractional derivative coincides with the classical Riemann-Liouville fractional derivative:(6)Da+0,μf(t)=ddtIa+1-μft=Da+μLf(t).When ν=1, 0<μ<1, the Hilfer fractional derivative coincides with the classical Caputo fractional derivative:(7)Da+1,μf(t)=Ia+1-μddtft=Da+μCf(t).Forx∈X, we define two families of operators Sν,μ(t):t≥0 and Pμt:t≥0 by(8)Sν,μ(t)=I0+ν1-μPμ(t),Pμt=tμ-1Tμ(t),Tμ(t)=∫0∞‍μθΨμ(θ)S(tμθ)dθ,where(9)Ψμ(θ)=∑n=1∞‍-θn-1n-1!Γ(1-nμ)sin(nπα),θ∈(0,∞),is a function of Wright-type defined on (0,∞) which satisfies(10)Ψα(θ)≥0,∫0∞‍Ψα(θ)dθ=1,∫0∞‍θζΨμ(θ)dθ=Γ1+ζΓ1+μζ,ζ∈-1,∞.Lemma 4 (see [10]).
The operatorsSν,μ and Pμ have the following properties.(i)
For any fixedt>0, Sν,μ(t) and Pμ(t) are linear and bounded operators, and (11)Pμ(t)x≤Mtμ-1Γμx,Sν,μ(t)x≤Mtν-11-μΓν1-μ+μx.(ii)
Pμ(t):t>0 is compact, if S(t):t>0 is compact.In this paper we adopt the following definition of mild solution of the initial-value problem (1); see [10].Definition 5.
A solutionx(·;u)∈C(0,b,X) is said to be a mild solution of (1) if for any u∈L2(0,b,U) the integral equation(12)x(t)=Sν,μ(t)x0+∫0t‍Pμt-sBus+fs,xsdsis satisfied, for all 0≤t≤b.Letx(b;u) be the state value of (12) at the terminal time b corresponding to the control u. We introduce the set R(b)={x(b;u):u∈L2(0,b,U)}, which is called the reachable set of system (12) at terminal time T, and denote its closure in X by R(b)¯.Definition 6.
The system (1) is said to be approximately controllable on 0,b if R(b)¯=X; that is, given an arbitrary ɛ>0 it is possible to steer from the point x0 to within a distance ɛ from all points in the state space X at time b.Remark 7.
(i) Whenν=0, the fractional equation (12) simplifies to the classical Riemann-Liouville fractional equation which has been studied by Zhou et al. in [6]. In this case(13)S0,μ(t)=Pμ(t)=tμ-1Tμ(t),0<t≤b.(ii) When ν=1, the fractional equation (12) simplifies to the classical Caputo fractional equation which has been studied by Zhou and Jiao in [5]. In this case(14)S1,μ(t)=Sμ(t),0≤t≤b,where Sμ(t) is defined in [5].


## 3. Main Results

To investigate the approximate controllability of system (12), we impose the following conditions:(H1)
St,  t>0, is compact;(H2)
the functionf:0,b×X→X satisfies the following:
(a)
ft,·:X→X is continuous for each t∈0,b,(b)
for eachx∈X, f·,x:0,b→X is strongly measurable;(H3)
there is a constantμ1∈0,μ and n∈L1/μ10,b,R+ such that, for every x∈X and almost all t∈0,b, we have(15)ft,x≤nt.Consider the following linear fractional differential system:(16)D0+ν,μxt=Axt+But,t∈0,b,I0+1-ν1-μx0=x0.The approximate controllability for the linear fractional system (16) is a natural generalization of approximate controllability of linear first order control system. It is convenient at this point to introduce the following controllability and resolvent operators associated with (16):(17)L0b=∫0b‍Pμ(b-s)Bu(s)ds,Γ0b=∫0b‍Pμ(b-s)BB∗Pμ∗(b-s)ds,Rɛ,Γ0b=ɛɛI+Γ0b-1,respectively, where B∗ denotes the adjoint of B and Pμ∗(t) is the adjoint of Pμ(t). It is straightforward to show that the operator L0b is a linear bounded operator for 1/2<μ≤1.We also impose the following resolvent condition:(R)
for everyh∈X, ɛɛI+Γ0b-1h converges to zero as ɛ→0+ in strong topology.Remark 8.
The assumption (R) is equivalent to the approximate controllability of the linear system (16); see [19, 20].In order to formulate the controllability problem in the form in which the fixed point theorem is readily applicable, it is assumed that the corresponding linear system is approximately controllable. It will be shown that system (1) is approximately controllable provided that we can show for all ɛ>0 there exists a continuous function x∈C0,b,X such that(18)uɛt,x=B∗Pμ∗b-tɛI+Γ0b-1px,xt=Sν,μtx0+∫0t‍Pμt-sBus+fs,xsds,where(19)px=h-Sν,μbx0-∫0b‍Pμb-sfs,xsds.Based on this observation, our goal is to find conditions for the solvability of (18). Note also that it will be shown that the control in (18) drives the system (1) from x0 to(20)h-ɛɛI+Γ0b-1pxprovided that the system (18) has a solution.For allɛ>0, consider the operator Φɛ:Cν,μ(0,b,X)→Cν,μ(0,b,X) defined as follows:(21)Φɛxt:=Sν,μtx0+∫0t‍Pμt-sBuɛs,x+fs,xsds.Letx∈Cν,μ(0,b,X). Observe that (22)limt→0+t1-ν1-μSν,μtx0=limt→0+t1-ν1-μΓν1-μ∫0t‍t-sν1-μ-1sμ-1Pμsx0ds=limt→0+1Γν1-μ∫01‍1-sν1-μ-1sμ-1Pμtsx0ds=x0Γν1-μ+μ.Define t1-ν1-μΦɛxt as follows:(23)t1-ν1-μΦɛxt:=t1-ν1-μSν,μtx0+t1-ν1-μ∫0t‍Pμt-s×Buɛs,x+fs,xsds,0<t≤b,x0Γν1-μ+μ,t=0.It will be shown that for allɛ>0 the operator Φɛ:Cν,μ(0,b,X)→Cν,μ(0,b,X) has a fixed point. To prove this we will employ the Schauder fixed point theorem.Lemma 9.
Let0≤ν≤1 and 1/2<μ≤1. If assumptions (H1)–(H3) hold, then for any ɛ>0 the control function uɛt,x has the following properties:(i)
uɛt,x≤(MBMb-tμ-1/ɛΓμ)(h+(Mbν-11-μ/Γν1-μ+μ)x0+(M/Γμ)((1-μ1)1-μ1bμ-μ1/(μ-μ1)1-μ1)n1/μ1),(ii)
for anyt∈0,b we have limn→∞uɛ(t,xn)-uɛ(t,x)=0, where MB=B, n1/μ1 is L1/μ1 norm of n.Proof.
(i) By the definition ofuɛt,x we have(24)uɛt,x≤B∗Pμ∗b-tɛI+Γ0T-1px≤MBMb-tμ-1ΓμɛI+Γ0T-1px≤MBMb-tμ-1ɛΓμpx≤MBMb-tμ-1ɛΓμ·h+Sν,μbx0∫0b‍Pμb-sfs,xsdshhhhhh+∫0b‍Pμb-sfs,xsds.Using the Hölder inequality and (H3) yields(25)uɛt,x≤MBMb-tμ-1ɛΓμ·h+Mbν-11-μΓν1-μ+μx0hhhhhh+MΓμ∫0b‍b-sμ-1nsds≤MBMb-tμ-1ɛΓμ×h+Mbν-11-μΓν1-μ+μx0∫0t‍hhhhhhh+MΓμ∫0t‍t-s(μ-1)/(1-μ1)ds1-μ1hhhhhh·∫0t‍n1/μ1sdsμ1≤MBMb-tμ-1ɛΓμ·h+Mbν-11-μΓν1-μ+μx0MΓμ1-μ11-μ1bμ-μ1μ-μ11-μ1hhhhhh+MΓμ1-μ11-μ1bμ-μ1μ-μ11-μ1n1/μ1.
(ii) Assume thatlimn→∞xn-xν,μ=0. Then we have(26)limn→∞xns=xs,0<s≤b.From (H2), it follows that(27)limn→∞fs,xns=fs,xsa.e.in0,b.Using (H3), we get(28)b-sμ-1fs,xns-fs,xs≤2b-sμ-1ns,hhhiiia.e.in0,b.Since s→2b-sμ-1ns is integrable on 0,b, by the Lebesgue dominated convergence theorem, we have(29)∫0b‍b-sμ-1fs,xns-fs,xsds⟶0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhiasn⟶∞.Further, it follows that (30)pxn-px≤MΓμ∫0b‍b-sμ-1fs,xns-fs,xsds⟶0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhasn⟶∞.Thus(31)uɛt,xn-uɛt,x=B∗Pμ∗b-tɛI+Γ0b-1pxn-px≤MBMɛΓμpxn-px⟶0,asn⟶∞.Lemma 10.
Let0≤ν≤1 and 1/2<μ≤1. Under assumptions (H1)–(H3), for any ɛ>0 there exists a positive number r:=rɛ such that ΦɛBr⊂Br, where(32)Br:=x∈Cν,μ(0,b,X):xν,μ≤r.Proof.
Letɛ>0 be fixed and x∈Br. Since xt is continuous, it follows from (H2) that ft,xt is a measurable function on 0,b. Using the Hölder inequality and (H3) yields(33)t1-ν1-μΦɛxt≤t1-ν1-μSν,μtx0+t1-ν1-μ∫0t‍Pμt-sfs,xsds+t1-ν1-μ∫0t‍Pμt-sBuɛs,xds=:I1+I2+I3.We estimate each of Ii, i=1,2,3, separately. By the assumption (H3), we have(34)I1≤t1-ν1-μSν,μtx0≤MΓν1-μ+μx0,I2≤t1-ν1-μ∫0t‍Pμt-sfs,xsds≤Mt1-ν1-μΓμ∫0t‍t-sμ-1fs,xsds≤Mt1-ν1-μΓμ∫0t‍t-sμ-1nsds≤Mt1-ν1-μΓμ∫0t‍t-s(μ-1)/(1-μ1)ds1-μ1·∫0t‍n1/μ1sdsμ1≤Mb1-ν1-μΓμ1-μ11-μ1bμ-μ1μ-μ11-μ1n1/μ1.Combining the estimates (33)-(34) yields(35)I1+I2<MΓν1-μ+μx0+Mb1-ν1-μΓμ1-μ11-μ1bμ-μ1μ-μ11-μ1n1/μ1:=Δ.Next, observe that(36)I3≤t1-ν1-μ∫0t‍Pμt-sBuɛs,xds=t1-ν1-μ∫0t‍ɛI+Γ0T-1Pμt-sBB∗Pμ∗b-shhhhhhhhhhhh·ɛI+Γ0T-1pxds≤t1-ν1-μ∫0t‍Pμt-sBB∗Pμ∗b-sds·ɛI+Γ0T-1px≤MB2M2t1-ν1-μΓ2μ·∫0t‍t-sμ-1b-sμ-1dsɛI+Γ0T-1px=MB2M2t1-ν1-μΓ2μb2μ-1μɛI+Γ0T-1px≤1ɛMB2M2t1-ν1-μΓ2μb2μ-1μpx≤1ɛMB2M2Γ2μb2μ-1μb1-ν1-μh+Δ.Thus,(37)t1-ν1-μΦɛxt≤Δ+1ɛMB2M2Γ2μb2μ-1μb1-ν1-μh+Δ.The last two inequalities imply that for large enough r>0 the following inequality holds:(38)t1-ν1-μΦɛzrt≤r.Therefore, Φɛ maps Br into itself.Lemma 11.
Let0≤ν≤1 and 1/2<μ≤1. If assumptions (H1)–(H3) hold, then the set Φɛx:x∈Br is an equicontinuous family of functions on 0,b.Proof.
For0<t<t+h≤b, we have(39)t+h1-ν1-μΦɛzt+h-t1-ν1-μΦɛzt≤t+h1-ν1-μSν,μt+hx0-t1-ν1-μSν,μtx0+∫tt+h‍t+h-s1-ν1-μt+h-sμ-1hhhhhhhhhh∫tt+h‍·Tμt+h-sBuɛs,z+fs,zsds+∫0tt+h-s1-ν1-μt+h-sμ-1hhhhhhhhhh-t-s1-ν1-μt-sμ-1hhhhhhhh∫0tt+h-s1-ν1-μt+h-sμ-1×Tμt+h-sBuɛs,z+fs,zsds+∫0t‍t-s1-ν1-μt-sμ-1hhhhhhhh·Tμt+h-s-Tμt-shhhhhhhh∫0t‍·Buɛs,z+fs,zsds≤I4+I5+I6+I7.
For0<t<t+h≤b, we have(40)I4≤t+h-s1-ν1-μSν,μt+hh-t-s1-ν1-μSν,μtx0.
By Lemma4, we know that t1-ν1-μSν,μt is uniformly continuous on 0,b, which enables us to deduce that limh→0+I4=0.
By condition (H3), we deduce thatlimh→0+I5=0.
Noting that(41)t+h-s1-ν1-μt+h-sμ-1t-s1-ν1-μt-sμ-1t+h-s1-ν1-μt+h-sμ-1-t-s1-ν1-μt-sμ-1ms≤t-s1-ν1-μt-sμ-1ms,and ∫0t‍t-s1-ν1-μt-sμ-1msds exists, it follows from the Lebesgue dominated convergence theorem that(42)∫0t‍t+h-s1-ν1-μt+h-sμ-1t-s1-ν1-μt-sμ-1hhht+h-s1-ν1-μt+h-sμ-1-t-s1-ν1-μt-sμ-1ms⟶0,as h→0+. It follows that limh→0+I6=0.
Forɛ>0 sufficiently small, we have(43)I7≤∫0t-ɛ‍+∫t-ɛt‍t-s1-ν1-μt-sμ-1·Tμt+h-s-Tμt-s·Buɛs,z+fs,zsds.
Since the compactness ofTμt  t>0 implies the continuity of Tμt  t>0 in the uniform operator topology, it can be easily seen that limh→0+I7=0.
The caset=0 and 0<h≤b follows from (23).
Thus, the setΦɛx:x∈Br is an equicontinuous family of functions in Cν,μ(0,b,X).Lemma 12.
Let0≤ν≤1 and 1/2<μ≤1. Let assumptions (H1)–(H3) hold. For any t∈0,b the set Vt:=Φɛxt:x∈Br is relatively compact in X.Proof.
Let0<t≤b be fixed and let λ be a real number satisfying 0<λ<t. For δ>0 define the operator Φɛλ,δ on Br by(44)Φɛλ,δxt:=1Γ(ν1-μ)Sλμδ·∫λt‍sμ-1t-s1-ν1-μhhh·∫δ∞‍μθΨμθSsμθ-λμδdθdsx0+μSλμδ∫0t-λ‍∫δ∞‍θt-sμ-1Ψμθhhhhhhhhhhhhhhh·St-sμθ-λμδdθhhhhhhhhhhh·Buɛs,z+fs,zsds.Since St is a compact operator, the set Φɛλ,δxt:x∈Br is relatively compact in X. Moreover, for each x∈Br, we have(45)Φɛxt-Φɛλ,δxt≤1Γ(ν1-μ)·∫0t‍sμ-1t-s1-ν1-μ∫0δ‍μθΨμθSsμθdθdsx0+1Γ(ν1-μ)·∫0λ‍sμ-1t-s1-ν1-μ∫δ∞‍μθΨμ(θ)S(sμθ)dθdsx0+μ∫0t‍∫0δ‍θt-sμ-1ΨμθSt-sμθhhhhhhhhhhh∫0t‍∫0δ‍·Buɛs,x+fs,xsdθds+μ∫t-λt‍∫δ∞‍θt-sμ-1ΨμθSt-sμθhhhhhhhhhhhhhh∫t-λt‍∫δ∞‍·Buɛs,x+fs,xsdθds=:I8+I9+I10+I11.A similar argument as before yields(46)I8≤μMΓ(ν1-μ)∫0t‍sμ-1t-s1-ν1-μds∫0δ‍θΨμθdθx0≤μMΓν1-μ1t1-ν1-μ·∫01‍1-sν1-μ-1sμ-1ds∫0δ‍θΨμθdθx0≤μMΓν1-μ1t1-ν1-μ·Bν1-μ,μ∫0δ‍θΨμθdθx0,I9≤μMΓ(ν1-μ)·∫0λ‍sμ-1t-s1-ν1-μds∫δ∞‍θΨμθdθx0≤μMbν1-μ-1Γ(ν1-μ)Γ1+μλμμ∫δ∞‍θΨμθdθx0,where we have used the equality(47)∫0∞‍θμβΨμθdθ=Γ1+βΓ1+μβ.From (46), it follows that(48)I8⟶0,I9⟶0asλ⟶0+,δ⟶0+.Similarly,(49)I10⟶0,I11⟶0asλ⟶0+,δ⟶0+.Consequently, for each x∈Br,(50)Φɛxt-Φɛλ,δxt⟶0asλ⟶0+,δ⟶0+.Therefore, there exist relatively compact sets arbitrarily close to the set Φɛxt:x∈Br. Hence, the set Φɛxt:x∈Br is relatively compact in X.Lemma 13.
Let0≤ν≤1 and 1/2<μ≤1. If assumptions (H1)–(H3) hold, then the operator Φɛ:Cν,μ(0,b,X)→Cν,μ(0,b,X) is continuous on Br.Proof.
Observe that, for allt∈0,b, xn,x∈Br, we have (51)t1-ν1-μΦɛxnt-t1-ν1-μΦɛxt≤Mt1-ν1-μΓμ·∫0tt-sμ-1‍hhhhhhh·fs,xns-fs,xsMbuɛs,xn-uɛs,xhhhhhhhhhfs,xns-fs,xsfs,xns-fs,xs+Mbuɛs,xn-uɛs,xds.The rest of the proof is similar to the proof of Lemma 9.Theorem 14.
If assumptions (H1)–(H3) hold and1/2<μ≤1, then there exists a solution to (18).Proof.
According to infinite-dimensional version of the Ascoli-Arzela theorem if (i) fort∈0,b, the set Vt:=Φɛxt:x∈Br is relatively compact in X; (ii) family Φɛx:x∈Br is uniformly bounded and equicontinuous, and then Φɛx:x∈Br is relatively compact family in Cν,μ(0,b,X). Properties (i) and (ii) follow from Lemmas 10–12. By Lemma 13, for any ɛ>0, the operator Φɛ is continuous. Thus from the Schauder fixed point theorem Φɛ has a fixed point. Therefore, the fractional control system (18) has a solution on 0,b. The proof is complete.We are now in a position to state and prove the main result of the paper.Theorem 15.
Let0≤ν≤1 and 1/2<μ≤1. Suppose that conditions (H1)–(H3) (R) are satisfied. Then system (1) is approximately controllable on 0,b.Proof.
Letɛ>0 and let xɛ be a fixed point of Φɛ in Brɛ. Then xɛ is a mild solution of (1) on 0,b under the control (52)uɛt,xɛ=B∗Sν,μ∗b-tɛI+Γ0T-1pxɛ,pxɛ=h-Sν,μbx0-∫0b‍Pμb-sfs,xɛsdsand satisfies the following equality:(53)xɛb=Sν,μbx0+∫0b‍b-sα-1Pμb-shhhhh·Buɛs,xɛ+fs,xɛsds=Sν,μbx0+-ɛI+ɛI+Γ0bɛI+Γ0b-1pxɛ+∫0b‍Pμb-sfs,xɛsds=h-ɛɛI+Γ0b-1pxɛ.It follows from (H3) that for all ɛ>0(54)∫0b‍fs,xɛs1/μ1ds≤∫0T‍n1/μ1sds.Consequently, the sequence f·,xɛ· is bounded. As such, there is a subsequence, still denoted by f·,xɛ·, that converges weakly to, say, f· in L1/μ10,b,X. Then(55)pxɛ-p=∫0b‍Pμb-sfs,xɛs-fsds≤sup⁡0≤t≤b∫0t‍Pμt-sfs,xɛs-fsds⟶0,as ɛ→0+, because of the compactness of an operator(56)f·⟶∫0·‍Pμ·-sfsds:L1/μ10,b,Xhhhhhhhhhhhhhhhhhhhh⟶C0,b,X,where(57)p=h-Sν,μx0-∫0b‍Pμb-sfsds.Then, by (53), the assumption (R), and ɛɛI+Γ0b-1≤1, it follows that(58)xɛb-h=ɛɛI+Γ0b-1pxɛ-p+ɛɛI+Γ0b-1p≤pxɛ-p+ɛɛI+Γ0b-1p⟶0as ɛ→0+. This gives the approximate controllability. The theorem is proved.Remark 16.
Theorem15 is a generalization of the existing results on the approximate controllability of fractional differential equations. When ν=0, the fractional control system (12) simplifies to the classical Riemann-Liouville fractional control equation which has been studied by Liu and Bin [16]. When ν=1, the fractional equation (12) simplifies to the classical Caputo fractional control system which has been studied by Sakthivel et al. [11].


## 4. Applications

The partial differential system arises in the mathematical modeling of heat transfer(59)D0+ν,3/4xt,θ=xθθt,θ+bθut+ft,xt,θ,xt,0=xt,π=0,t>0,I0+(1/4)1-νx0=x0,0<θ<π,0≤t≤b,where u∈L20,b,  X=L20,π,  h∈X,  0≤ν≤1,  μ=3/4, and f:R×R→R is continuous and uniformly bounded. Let B∈LR,X be defined as (60)Buθ=bθu,B∗v=∑n=1∞‍b,env,en,where 0≤θ≤π,  u∈R,  and bθ∈L20,π, and let A:X→X be operator defined by Az=z′′ with domain(61)DA=z∈X∣z,z′areabsolutelycontinuous,hhhhhhhz′′∈X,z0=zπ=0.Then(62)Az=∑n=1∞‍-n2z,enen,z∈DA,where enθ=2/πsinnθ, 0≤x≤π, n=1,2,…. It is known that A generates a compact semigroup St, t>0, in X and is given by(63)Stx=∑n=1∞‍e-n2tx,enen,x∈X.Moreover, for any x∈X we have (64)T3/4(t)=34∫0∞‍θΨ3/4(θ)S(t3/4θ)dθ,T3/4tx=34∑n=1∞‍∫0∞‍θΨ3/4θexp⁡-n2t3/4θdθx,enen.In order to show that the associated linear system is approximately controllable on0,b, we need to show that b-sα-1B∗Tμ(b-s)x=0⇒x=0. Indeed, observe that(65)b-sμ-1B∗Tμ(b-s)x=b-sμ-1·∑n=1∞‍b,en34∫0∞‍θΨ3/4θexp⁡-n2t3/4θdθx,en=b-sμ-134·∑n=1∞‍∫0∞‍θΨ3/4θexp⁡-n2t3/4θdθb,enx,en=0.So, x,en=0⇒x=0 provided that b,en=∫0π‍bθenθdθ≠0 for n=1,2,3,…. Therefore, the associated linear system is approximately controllable provided that ∫0π‍bθenθdθ≠0 for n=1,2,3,…. Because of the compactness of the semigroup St (and consequently T3/4) generated by A, the associated linear system of (59) is not exactly controllable but it is approximately controllable. Hence, according to Theorem 15, system (59) will be approximately controllable on 0,b.



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*Source: 263823-2015-05-18.xml*