# Approximate Controllability of Fractional Integrodifferential Evolution Equations

Authors: R. Ganesh; R. Sakthivel; N. I. Mahmudov; S. M. Anthoni
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper addresses the issue of approximate controllability for a class of control system which is represented by nonlinear fractional integrodifferential equations with nonlocal conditions. By using semigroup theory,p-mean continuity and fractional calculations, a set of sufficient conditions, are formulated and proved for the nonlinear fractional control systems. More precisely, the results are established under the assumption that the corresponding linear system is approximately controllable and functions satisfy non-Lipschitz conditions. The results generalize and improve some known results.


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## Body

## 1. Introduction

In recent years, use theory of fractional calculus and fractional differential equations has gained importance and popularity due to its applications in various fields of science and engineering. Various physical phenomena in science and engineering can be successfully modeled by using fractional calculus theory. Due to its tremendous scope and applications, several papers have been devoted to study the existence of mild solutions of fractional differential equations (see [1–4] and references therein). On the other hand, controllability is an important property of a control system which plays an important role in the analysis and design of control systems [5–8]. Most literatures in this direction so far have been concerned with controllability of nonlinear differential equations in infinite-dimensional spaces without fractional derivatives (see [9] and references therein). Using generalized open mapping theorem, a set of sufficient conditions for constrained local relative controllability near the origin are formulated and proved for the semilinear systems with delayed controls in [10, 11].Recently, only few papers deal with the controllability of fractional dynamical systems [12–14]. Klamka [15, 16] derived a set of sufficient conditions for the local controllability of finite-dimensional fractional discrete-time semilinear systems. However, the problem of controllability for fractional systems has not been fully investigated, and there is still room open for further research in this area [17]. Moreover, the approximate controllable systems are more prevalent, and very often, approximate controllability is completely adequate in applications (see [18–21] and references therein). Therefore, it is important, in fact necessary, to study the weaker concept of controllability, namely, approximate controllability for nonlinear fractional integrodifferential systems. Motivated by this fact, in this paper, we consider the approximate controllability of the fractional nonlinear integrodifferential evolution equations with nonlocal initial condition in the following form:
(1)CDqx(t)=-Ax(t)+Bu(t)+tnf(t,x(t),(Hx)(t)),t∈J=[0,b],n∈Z+,q∈(0,1),x(0)=g(x)+x0∈Xγ,
where the state x(·) takes the values in a Hilbert space X, DCq denotes Caputo derivative, -A:D(A)→X is the infinitesimal generator of an analytic semigroup {T(t),t≥0} on X; the control function u(·) is given in L2(J,U); U is a Hilbert space; B is a bounded linear operator from U into X; the operator H is defined by (Hx)(t)=∫0b‍h(t,s,x(s))ds; the nonlinear term f:J×Xγ×Xγ→X(orXγ) is a given function, where here J=[0,b]; and Xγ=D(Aγ)(0<γ<1) is a Hilbert space with the norm ∥x∥γ=∥Aγx∥ for x∈Xγ. The functions f, h, and g will be specified later. In fact, our results in this paper are motivated by the recent work of [20], and the fractional integrodifferential equations are studied in [4]. The main objective of this paper is to derive conditions for the approximate controllability of (1) with non-Lipschitz coefficients, and the associated linear system is approximately controllable.


## 2. Preliminaries

In this section, we provide definitions, lemmas, and notations necessary to establish our main results [4]. Let ∥f∥Lp(J,R+) denote the Lp(J,R+) norm of f whenever f∈Lp(J,R+) for some p with 1<p<∞. Let 𝒞γ denote the Banach space C(J,Xγ) endowed with sup norm given by ∥x∥∞≡supt∈J∥x∥γ, for x∈𝒞γ, γ∈(0,1).Let us recall the following known results.The fractional integral of orderβ with the lower limit 0 for a function f is defined as
(2)Iβf(t)=1Γ(β)∫0t‍f(s)(t-s)1-βds,t>0,β>0,
provided the right-hand side is pointwise defined on [0,∞), where Γ(·) is the gamma function.Riemann-Liouville derivative of orderβ with lower limit zero for a function f:[0,∞)→R can be written as
(3)LDβf(t)=1Γ(n-β)dndtn∫0t‍f(s)(t-s)β+1-nds,t>0,n-1<β<n.The Caputo derivative of orderβ for a function f:[0,∞)→R can be written as
(4)Dβf(t)=LDβ(f(t)-∑k=0n-1‍tkk!f(k)(0)),t>0,n-1<β<n.Remark 1 (see [4]).
(i) Iff(t)∈Cn[0,∞), then
(5)Dβf(t)=1Γ(n-β)∫0t‍fn(s)(t-s)β+1-nds=In-βfn(s),t>0,0≤n-1<β<n.
(ii) The Caputo derivative of a constant is equal to zero.
(iii) Iff is an abstract function with values in X, then integrals which appear in the above results are taken in Bochner’s sense.
For additional details concerning the fractional derivative, we refer the reader to [3].
To define the mild solution for the control system (1), we associate problem (1) to the following integral equation [4]:
(6)x(t)=𝒯(t)[x0+g(x)]+∫0t‍(t-s)q-1𝒮(t-s)Bu(s)ds+∫0t‍(t-s)q-1sn𝒮(t-s)f(s,x(s),(Hx)(s))ds,t∈J,
where 𝒯(t)=∫0∞‍ξq(θ)T(tqθ)dθ, 𝒮(t)=q∫0∞‍θξq(θ)T(tqθ)dθ, ξq(θ)=(1/q)θ-1-1/qw¯q(θ-1/q)≥0, w¯q(θ)=(1/π)∑n=1∞‍(-1)n-1θ-qn-1(Γ(nq+1)/n!)sin(nπq), θ∈(0,∞),  and ξq is a probability density function defined on (0,∞); that is ξq(θ)≥0, θ∈(0,∞), and ∫0∞‍ξq(θ)dθ=1.Definition 2.
A functionx(·;x0,u)∈C(J,Xγ) is said to be a mild solution of (1) if for any u(·)∈L2(J,U) the integral equation (6) is satisfied.Definition 3.
The system (1) is said to be approximately controllable on the interval J if ℜ(b,x0)¯=X, where
(7)ℜ(b,x0)={xb(x0;u)(0):u(·)∈L2(J,U)}
is called the reachable set of system (1) at terminal time b, and its closure in X is denoted by ℜ(b,x0)¯; let xb(x0;u) be the state value of (1) at terminal time b corresponding to the control u and the initial value x0∈X.Consider the following linear fractional differential control system(8)Dtqx(t)=Ax(t)+Bu(t),t∈J=[0,b],0<q<1,x(0)=x0.
The approximate controllability for the system (8) is a natural generalization of approximate controllability for the linear first order control system (see [18]). It is convenient at this point to introduce the controllability operator associated with the linear system
(9)Γ0b=∫0b‍(b-s)q-1𝒮(b-τ)BB*𝒮*(b-τ)dτ,
where B* denotes the adjoint of B and 𝒮*(t) is the adjoint of 𝒮(t). It is straightforward that the operator Γ0b is a linear bounded operator. Let R(α,Γ0b)=(αI+Γ0b)-1 for α>0.Lemma 4.
The linear fractional control system (8) is approximately controllable on J if and only if αR(α,Γ0b)→0 as α→0+ in the strong operator topology.The proof of this lemma is a straightforward adaptation of the proof of Theorem 2  of [18].Lemma 5 (see [4]).
For eachψ∈Lp(J,X) with 1≤p<+∞, one has limh→0∫0b‍∥ψ(t+h)-ψ(t+h)∥pdt=0, where ψ(s)=0 for s does not belong to J.Lemma 6 (see [4]).
The operators𝒯 and 𝒮 have the following properties.(i)
For fixedt≥0, 𝒯(t) and 𝒮(t) are linear and bounded operators. For any x∈X,
(10)∥𝒯(t)x∥≤M∥x∥,∥𝒮(t)x∥≤MqΓ(q+1)∥x∥.(ii)
{𝒯(t),t≥0} and {𝒮(t),≥0} are strongly continuous.(iii)
For everyt>0, 𝒯(t) and 𝒮(t) are also compact operators.(iv)
For anyx∈X, β∈(0,1), and γ∈(0,1) one has
(11)A𝒮(t)x=A1-β𝒮(t)Aβx,t∈J,∥Aγ𝒮(t)∥≤MγqΓ(2-γ)Γ(1+q(1-γ))t-γq,0<t≤b.(v)
For fixedt≥0 and any x∈Xγ, one has
(12)∥𝒯(t)x∥γ≤M∥x∥γ,∥𝒮(t)x∥γ≤MqΓ(q+1)∥x∥γ.(vi)
𝒯γ(t) and 𝒮γ(t), t>0 are uniformly continuous; that is, for each fixed t>0 and ϵ>0, there exists h>0 such that
(13)∥𝒯γ(t+ϵ)-𝒯γ(t)∥γ<ϵ,fort+ϵ≥0,|ϵ|<h,∥𝒮γ(t+ϵ)-𝒮γ(t)∥γ<ϵ,fort+ϵ≥0,|ϵ|<h,
where𝒯γ(t)=∫0∞‍ξq(θ)Tγ(tqθ)dθ, 𝒮γ(t)=q∫0∞‍θξq(θ)Tγ(tqθ)dθ.


## 3. Main Result

In this section, we present our main result on approximate controllability of control system (1). We prove that under certain conditions, approximate controllability of the linear system (8) implies the approximate controllability of nonlinear fractional system (1). In order to establish the result, we require the following assumptions.(H1)
The functionf:J×Xγ×Xγ→Xγ is Carathéodory, and there exists a positive function ρ∈Lp(J,R+) for some p with 1<p<∞ such that ∥f(t,x,y)∥γ≤ρ(t) for all x,y∈Xγ and t∈J.(H2)
The functiong:𝒞γ→Xγ is completely continuous, and there exist β1,β2>0 such that
(14)∥g(x)∥γ≤β1∥x∥∞+β2.(H3)
The functionh:Dh×Xγ→Xγ is continuous, and there exist L1,L2>0 such that ∥h(t,s,x)-h(t,s,y)∥γ≤L1∥x-y∥γ+L2 for each (t,s)∈Dh={(t,s)∈R2,0≤s,t≤b} and x,y∈Xγ.(H4)
The linear fractional control system (8) is approximately controllable.(H5)
The functionf:J×Xγ×Xγ→Xγ is continuous and uniformly bounded, and there exists N>0 such that ∥f(t,x,y)∥≤N for all (t,x,y)∈J×Xγ×Xγ.(H6)
The semigroupT(t) is compact.In order to prove the required result, forα>0, we define the operator Fα on C(J,Xγ) as
(15)(Fαx)(t)=z(t),
where
(16)z(t)=𝒯(t)[x0+g(x)]+∫0t‍(t-s)q-1𝒮(t-s)Bv(s)ds+∫0t‍(t-s)q-1sn𝒮(t-s)f(s,x(s),(Hx)(s))ds,t∈J,(17)v(t)=B*𝒮*(b-t)R(α,Γ0b)p(x(·)),p(x(·))=xb-𝒯(b)[x0+g(x)]-∫0b‍(b-s)q-1sn𝒮(b-s)×f(s,x(s),(Hx)(s))ds.Theorem 7.
Under the conditions(H1), (H2), (H3), and (H6), the control system (1) admits a mild solution on [0,b], and here MB=∥B∥ and M1=∥ρ∥Lp2(J,R+).Proof.
The main aim in this section is to find conditions for solvability of systems (16) and (17) for α>0. In the Banach space C(J,Xγ), consider a set
(18)Br0={x(·)∈C(J,Xγ):∥x∥γ≤r},
where r is the positive constant. Now, it will be shown that, using Schauder’s fixed point theorem, for all α>0, the operator Fα:C(J,Xγ)→C(J,Xγ) has a fixed point.
First, we prove that for an arbitraryα>0 and there is a positive constant r0=r0(α) such that Fα:Br0→Br0.
Letx∈Br0, and then, for t∈J and 1/p<q(⇒(pq-1))>0, using Holders inequality, Lemma 6, and conditions (H1) and (H2) in (17), we obtain
(19)∥v(t)∥γ≤∥B*𝒮*(b-t)R(α,Γ0b)∥×[∫0b‍∥xb∥γ-∥𝒯(b)[x0+g(x)]∥γ+∫0b‍(b-s)q-1sn×∥𝒮(b-s)f(s,x(s),(Hx)(s))∥γds∫0b‍]≤MBMqαΓ(q+1)[(p-1np2+p-1)(p-1)/p2∥xb∥γ+M(∥x0∥γ+β1∥x∥∞+β2)+M1MqΓ(q+1)(p-1pq-1)(p-1)/p×(p-1np2+p-1)(p-1)/p2bn+q-1/p2].
It follows from (16) that
(20)∥z(t)∥γ≤∥𝒯(t)[x0+g(x)]∥γ+∫0t‍(t-s)q-1∥𝒮(t-s)Bv(s)∥γds+∫0t‍(t-s)q-1sn∥𝒮(t-s)f(s,x(s),(Hx)(s))∥γds≤M(∥x0∥γ+β1∥x∥∞+β2)+MBMqΓ(q+1)(p-1qp-1)(p-1)/pb((q-1)p+p-1)/pb∥v(s)∥γ+M1MqΓ(q+1)(p-1pq-1)(p-1)/p(p-1np2+p-1)(p-1)/p2×bn+q-1/p2.
From the above two inequalities, we get that ∥(Fαx)(t)∥γ≤r0. This follows that Fα maps Br0 into itself.
For eachα>0, we prove the operator Fα maps Br0 into a relatively compact subset of Br0. First, we show that V(t)={(Fαx)(t):x∈Br0} is relatively compact in Xγ for every t∈J. The case t=0 is obvious. Let t∈(0,b] be fixed, and for each ε∈(0,t), arbitrary δ>0, and x∈Br0, we define the operator Fαε,δ by
(21)(Fαε,δx)(t)=𝒯[x0+g(x)]+q∫0t-ε‍∫δ∞‍θ(t-s)q-1ξq(θ)T((t-s)qθ)×Bv(s)dθds+q∫0t-ε‍∫δ∞‍θ(t-s)q-1snξq(θ)T((t-s)qθ)×f(s,x(s),(Hx)(s))dθds=T(εqδ)×∫δ∞‍ξq(θ)(T(tqθ)-T(εqδ))dθ[x0+g(x)]+(T(εqδ))q×∫0t-ε‍∫δ∞‍θ(t-s)q-1ξq(θ)×(T((t-s)qθ)-T(εqδ))Bv(s)dθds+T(εqδ)q∫0t-ε‍∫δ∞‍θ(t-s)q-1snξq(θ)×(T((t-s)qθ)-T(εqδ))×f(s,x(s),(Hx)(s))dθds=T(εqδ)y(t,ε).
Since T(εqδ) is compact in Xγ and y(t,ε) is bounded in Br0, the set Vε(t)={(Fαε,δx)(t):x∈Br0} is relatively compact in Xγ [4]. On the other hand, using (H1) and Holder’s inequality, we have(22)∥(Fαx)(t)-(Fαε,δx)(t)∥γ≤q∫0t‍∫0δ‍θ(t-s)q-1ξq(θ)×∥T((t-s)qθ)[Bv(s)+snf(s,x(s),(Hx)(s))]∥γdθds+q∫t-εt‍∫δ∞‍θ(t-s)q-1ξq(θ)×∥T((t-s)qθ)[Bv(s)+snf(s,x(s),(Hx)(s))]∥γdθds≤MBMq(p-1qp-1)(p-1)/pt1+q-1/p∥v(s)∥γ∫0δ‍θξq(θ)dθ+M1Mq(p-1pq-1)(p-1)/p(p-1np2+p-1)(p-1)/p2bn+q-1/p2×∫0δ‍θξq(θ)dθ+MBMqΓ(q+1)(p-1qp-1)(p-1)/pε1+q-1/p∥v(s)∥γ+M1MqΓ(q+1)(p-1pq-1)(p-1)/p(p-1np2+p-1)(p-1)/p2εn+q-1/p2.This implies that there are relatively compact sets Vε(t) arbitrarily close to the set V(t) for each t∈(0,b]. Thus, V(t) is relatively compact in Xγ for all t∈(0,b] since it is compact at t=0. Therefore, we have the relatively compactness in Xγ for all t∈[0,b].
Next we show thatV(t)={(Fαx)(t):x∈Br0} is an equicontinuous family of functions on [0,b]. By the compactness of the set, we can prove that the functions Fαx,x∈Br0 are equicontinuous at t=0. For any x∈Br0 and 0<t1<t2≤b, we have
(23)∥z(t2)-z(t1)∥γ≤∥(𝒯(t2)-𝒯(t1))[x0+g(x)]∥γ+∥∫0t1‍(t2s)q-1[𝒮(t2-s)-𝒮(t1-s)]Bv(s)ds∥γ+∥∫0t1‍[(t2-s)q-1-(t1-s)q-1]𝒮(t1-s)Bv(s)ds∥γ+∥∫t1t2‍(t2-s)q-1𝒮(t2-s)Bv(s)ds∥γ+∥∫0t1‍(t2-s)q-1sn[𝒮(t2-s)-𝒮(t1-s)]×f(s,x(s),(Hx)(s))ds∫0t1‍∥γ+∥∫0t1‍[(t2-s)q-1-(t1-s)q-1]sn𝒮(t2-s)×f(s,x(s),(Hx)(s))ds∫0t1‍∥γ+∥∫t1t2‍(t2-s)q-1sn𝒮(t2-s)f(s,x(s),(Hx)(s))ds∥γ≤I1+I2+I3+I4+I5+I6+I7.
Now, we have to prove that I1, I2, I3, I4, I5, I6, and I7 tend to 0 independently of x∈Br0 as t2-t1→0. By Lemma 6, one can easily show that limt1→t2I1=0. By using Lemmas 5 and 6 and the Lagrange mean value theorem and following the similar procedure as in the proof of Theorem 3.2  of [4], one can deduce that limt1→t2I5,I6,I7=0. In a similar way, we can obtain
(24)I2≤Γ(2)Γ(q+1)M1+γγ((p-1)2(q-1)p2+(p-1)2)(p-1)2/p2×b((q-1)p2+(p-1)2)/p2×(∫0b‍[(t2-s)-qγ-(t1-s)-qγ]p2/(p-1)ds)(p-1)/p2×bMB∥v(s)∥,I3≤MqΓ(q+1)(∫0b‍[(t2-s)q-1-(t1-s)q-1]p/(p-1)ds)(p-1)/p×bMB∥v(s)∥,I4≤MqΓ(q+1)(p-1pq-1)(p-1)/p(t2-t1)((q-1)p+p-1)/p×MB∥v(s)∥(t2-t1).
By the Lagrange mean value theorem and Lemmas 5 and 6, it can be easily seen that I2, I3, and I4 tend to 0 as t2-t1→0. Thus, the right-hand side of (23) tends to 0 as t2-t1→0 which means that {(Fαx):x∈Br0} is a family of equicontinuous functions. It can be easily seen that for all α>0, Fα is continuous on C(J,Xγ). Hence, by Arzela-Ascoli’s theorem, Fα is compact. By Schauder’s fixed point theorem, Fα has a fixed point x∈Br0. Thus, the control system (1) has at least one mild solution on [0,b].Theorem 8.
Assume that(H1)–(H6) hold. Then, system (1) is approximately controllable on [0,b].Proof.
Linear system (8) is approximately controllable, by Lemma 4, and it can be seen that approximate controllability of (8) is equivalent to convergence of the operator αR(α,Γ0b) to zero operator in the strong operator topology as α→0+. Let x^α(·) be a fixed point of Fα in Br. Any fixed point of Fα is a mild solution of (1) under the control u^α(t)=B*𝒮*(b-t)R(α,Γ0b)p(x^α) and satisfies x^α(b)=xb-αR(α,Γ0b)p(x^α). By the condition (H5), we have ∫0b‍∥f(s,x^α(s),(Hx^α)(s))∥2ds≤bN2, and consequently, the sequence {f(s,x^α(s),(Hx^α)(s))} is bounded in L2(J,Xγ). Then there is a subsequence denoted by {f(s,x^α(s),(Hx^α)(s))} that converges weakly to say f(s) in L2(J,Xγ). Now, the compactness of 𝒮(t) implies that 𝒮(b-s)f(s,x^α(s),(Hx^α)(s))→𝒮(b-s)f(s) in L2(J,Xγ), and we obtain
(25)∥p(x^α)-w∥=∥∫0b‍(b-s)q-1sn𝒮(b-s)×[f(s,x^α(s),(Hx^α)(s))-f(s)]ds∫0b‍∥≤sup0≤t≤b∥∫0t‍(t-s)q-1sn𝒮(t-s)×[f(s,x^α(s),(Hx^α)(s))-f(s)]ds∫0t‍∥⟶0asα⟶0+,
where
(26)w=xb-𝒯(b)[x0+g(x^α)]-∫0b‍(b-s)q-1𝒮(b-s)snf(s)ds.
Then, we obtain
(27)∥x^α(b)-xb∥≤∥αR(α,Γ0b)(w)∥+∥αR(α,Γ0b)∥∥p(x^α)-w∥≤∥αR(α,Γ0b)(w)∥+∥p(x^α)-w∥.
It follows from Lemma 4 and estimation (25) that ∥x^α(b)-xb∥→0asα→0+. This proves the approximate controllability of (1). The proof is completed.Example 9.
As an application of the obtained theory, we consider a control system which is represented by fractional partial differential equation. LetX=U=L2([0,1]), and let A:D(A)⊂X→X be the operator defined by Az=-z′′ with domain
(28)D(A)={z∈L2([0,1]):z,z′areabsolutelycontinuous,z′′∈L2([0,1]),z(0)=z(1)=0}.
Let D(A1/2)={z(·)∈X:∑n=1∞‍(z,en)en∈X}, and let X1/2=D(A1/2,∥·∥1/2), where ∥·∥1/2=∥A1/2∥2. For each z∈D(A1/2) and ∥A-1/2∥=1, the operator A1/2 is given by A1/2z=∑n=1∞‍n2(z,en)en, where en(y)=2sin(ny) and 0≤y≤1, n=1,2,… is the orthogonal set of eigenvectors of A. It is well known that -A generates a compact analytic semigroup of uniformly bounded linear operator.
Consider the fractional partial differential equation with control in the following form:(29)∂q∂tqx(t,y)=∂2∂y2x(t,y)+μ(t,y)+e-tet+e-tcos[x(t,y)+∫0b‍cos(ts)x(s,y)ds]+e-t,t∈[0,1],y∈[0,1],x(t,0)=x(t,1)=0,t>0,x(0,y)=∑i=0σ‍∫01‍k1(y,τ)x(ti,τ)dτ+∑i=0σ‍∫01‍k2(y,τ)∂∂τx(ti,τ)dτ,
where 0≤q≤1; μ:[0,1]×[0,1]→[0,1] is continuous; k1,k2∈L2([0,1]×[0,1]); σ∈N.
Letx(t)(y)=x(t,y), and define the bounded linear operator B:U→X by (Bu)(t)(y)=μ(t,y),0≤y≤1. Let C([0,1],X1/2) be a Banach space equipped with sup norm. Further, we define the operators f:[0,1]×X1/2→X1/2 and g:C([0,1],X1/2)→X1/2 by
(30)f(t,x(t),(Hx)(t))(y)=e-tet+e-tcos[x(t)+∫0b‍cos(ts)x(s)ds](y)+e-t,g(x)(y)=(∑i=0σ‍(𝒦x)(ti))(y),forx∈C(J,X1/2),
where (Hx)(t)(y)=∫0b‍cos(ts)x(s,y)ds,(𝒦ϕ)(τ)=∫01‍k1(y,τ)ϕ(τ)dτ+∫01‍k2(y,τ)(∂/∂τ)ϕ(τ)dτ for all ϕ∈X1/2. Moreover, the linear fractional control system corresponding to (29) is approximately controllable. Thus, with the above choices of A, B, f, and g, the system (29) can be written to the abstract form of (1). Therefore, all the conditions of Theorem 8 are satisfied. Hence, by Theorem 8, the fractional nonlinear integrodifferential system (29) is approximately controllable on [0,1].Note. The considered system (1) is of the more general form, and in particular, if functions f and h have various physical meanings, it is important to note that (1) has a great diversity. The result in this paper assumes that the linear system has a compact semigroup and consequently is not completely controllable. Moreover, the functions with Lipschitz condition are considerably strong when one discusses various applications in the real-world problems. Such an assumption is removed from this paper.



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