# Controllability of Impulsive Fractional Functional Integro-Differential Equations in Banach Spaces

Authors: C. Ravichandran; J. J. Trujillo
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

This paper is concerned with the controllability problem for a class of mixed type impulsive fractional integro-differential equations in Banach spaces. Sufficient conditions for the controllability result are established by using suitable fixed point theorem combined with the fractional calculus theory and solution operator under some weak conditions. The example is given in illustrate the theory. The results of this article are generalization and improved of the recent results on this issue.


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## Body

## 1. Introduction

In the past few decades, the fractional calculus, that is, calculus of integrals and derivatives of any arbitrary real or complex order has gained considerable popularity and importance, based on the wide applications in engineering and sciences such as fluid flow, rheology, dynamical processes in self-similar and porous structures, diffusive transport akin to diffusion, and electrical networks. For more details about fractional calculus theory and fractional differential equations with applications see the monographs of Baleanu et al. [1, 2], Kilbas et al. [3], Lakshmikantham et al. [4], Miller and Ross [5], Podlubny [6], and the papers of [7–14].Differential equations with impulsive conditions constitute an important field of research due to their numerous applications in ecology, medicine biology, electrical engineering, and other areas of science and technology. There has been a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments, see for instance the monographs by Lakshmikantham et al. [15], Bainov and Simeonov [16], Samoilenko and Perestyuk [17] and the papers of [18–24].On the other hand, the controllability for fractional dynamical system has become an interesting research area to this field and one of the fundamental concepts in modern mathematical control theory. Very recently, the authors shu et al. [25] studied the existence of solutions for impulsive fractional differential equations, assuming the operator A to be a sectorial, and the results are obtained by using Banach contraction theorem and Leray-schauder’s alternative fixed point theorem. Shu et al. [26] established the existence and uniqueness of solutions for class of fractional partial semilinear functional differential equations with finite delay, here assuming A is the infinitesimal generator of an analytic semigroup and by using Banach fixed point theorem.Tomar and Dabas [27] extended the results of [25] into a controllability of impulsive fractional semilinear evolution equations with nonlocal conditions with A as the α-resolvent family and the results are obtained by using Banach contraction principle. Many researchers [28–39] investigated the existence and controllability problem combined with fractional derivative with (or without) impulsive conditions. From above the collection of the literature survey, up till now, there is no work reported on this topic, and inspired by the above mentioned works [25–27, 40] we will establish the controllability of impulsive fractional mixed type functional integro-differential equations with finite delay of the form
(1)Dαx(t)=Ax(t)+f(t,xt,∫0t‍h(t,s,xs)ds,∫0T‍k(t,s,xs)ds)+Bu(t),t∈J≔[0,T],t≠tk,k=1,2,…,m,(2)Δx|t=tk=Ik(x(tk-)),k=1,2,…,m,(3)x(t)=ϕ(t),t∈[-r,0],
where Dα is Caputo fractional derivative of order 0<α<1, A:D(A)⊂X→X is the bounded linear operator of an α-resolvent family {Sα(t):t≥0} defined on a Banach space X, B:U→X is a bounded linear operator, 0=t0<t1<t2<⋯<tm<tm+1=T, h:J×J×D→X, and f:J×D×X×X→X are given functions, where D={ϕ:[-r,0]→X such that ϕ is continuous everywhere except for a finite number of points s at which ϕ(s-) and ϕ(s+) exists and ϕ(s-)=ϕ(s)}, ϕ∈D(0<r<∞), 0=<t0<t1<⋯<tk<⋯<tm<tm+1=T,  Δx|t=tk=x(tk+)-x(tk-), x(tk+)=limh→0+x(tk+h) and x(tk-)=limh→0-x(tk+h) represent the right and left limits of x(t) at t=tk, respectively.For any continuous function,x is defined on the interval [-r,T]-{t1,t2,…,tm} and any t∈J. We denote by xt be the element of D defined by
(4)xt(θ)=x(t+θ),θ∈[-r,0].
Here, xt(·) represents the history of the time t-r, upto the present time t. For ϕ∈D, then ∥ϕ∥D=sup{|ϕ(θ)|:θ∈[-r,0]}.We consider the problems (1)–(3) to study the controllability results using the solution operator and fixed-point theorems. The rest of this paper is organized as follows. In Section 2, we present some necessary definitions and preliminary results that will be used to prove our main results. The proof of our main results is given in Section 3. Finally, an example is included in Section 4.


## 2. Preliminaries

In this section, we mention some definitions and properties required for establishing our results. LetX be a complex Banach space with its norm denoted as ∥·∥X, and L(X) represents the Banach space of all bounded linear operators from X into X, and the corresponding norm is denoted by ∥·∥L(X). Let C(J,X) denote the space of all continuous functions from J into X with supremum norm denoted by ∥·∥C(J,X). In addition, Br(x,X) represents the closed ball in X with the center at x and the radius r.A two-parameter function of the Mittag-Leffler type is defined by the series expansion(5)Eα,β(z)=∑k=0∞‍zkΓ(αk+β)=12πi∫Ha‍μα-βeμμα-zdμ,dddα,β>0,z∈C,
where Ha is a Hankel path, that is, a contour which starts and ends at -∞ and encircles the disc |μ|≤|z|1/α contour clockwise. For short, Eα(z)=Eα,1(z). It is an entire function which provides a simple generalization of the exponent function: E1(z)=ez and the cosine function: E2(-z2)=cos(z) and plays an important role in the theory of fractional differential equations. The most interesting properties of the Mittag-Leffler functions are associated with their Laplace integral
(6)L(tβ-1Eα,β(-wαtα))(λ)1=λα-βλα+wα,Reλ>w1/α,w>0,
see [3, 6, 41] for more details.Definition 1 (see [40]).
Caputo derivative of orderα for a function f:[0,∞)→R is defined as
(7)dαdtαf(t)=1Γ(n-α)∫0t‍(t-s)n-α-1f(n)(s)ds,
for n-1<α<n, n∈N. If 0<α≤1, then
(8)dαdtαf(t)=1Γ(1-α)∫0t‍(t-s)-αf(1)(s)ds.
The Laplace transform of the Caputo derivative of order α>0 is given as
(9)L{Dtαf(t):λ}=λαf^(λ)-∑k=0n-1‍λα-k-1f(k)(0);n-1<α≤n.Definition 2 (see [42]).
LetA be a closed and linear operator with domain D(A) defined on a Banach space X and α>0. Let ρ(A) be the resolvent set of A. We call A the generator of an α-resolvent family if there exists w≥0 and a strongly continuous function Sα:R+→L(X) such that {λα:Reλ>w}⊂ρ(A) and
(10)(λαI-A)-1x=∫0∞‍e-λtSα(t)xdt,Reλ>w,x∈X.
In this case, Sα(t) is called the α-resolvent family generated by A.Definition 3 (see [43]).
LetA be a closed and linear operator with domain D(A) defined on a Banach space X and α>0. Let ρ(A) be the resolvent set of A. We call A the generator of an α-resolvent family if there exists w≥0 and a strongly continuous function Tα:R+→L(X) such that {λα:Reλ>w}⊂ρ(A) and
(11)λα-1(λαI-A)-1x=∫0∞‍e-λtTα(t)xdt,Reλ>w,x∈X.
In this case, Tα(t) is called the solution operator generated by A.The concept of the solution operator is closely related to the concept of a resolvent family [44, Chapter 1]. For more details on α-resolvent family and solution operators, we refer to [44, 45] and the references therein.


## 3. Controllability Results

In this section, we present and prove the controllability for the system (1)–(3). In order to prove the controllability results, we need the following results which are taken from [25, 41]. If α∈(0,1) and A∈Aα(θ0,w0), then for any x∈X and t>0, we have
(12)∥Sα(t)∥≤Mewt,∥Tα(t)∥≤Cewt(1+tα-1),1t>0,w>w0.
Let M~S≔sup0≤t≤T∥Sα(t)∥L(X), M~T≔sup0≤t≤TCeωt(1+t1-α), where L(X) is the Banach space of bounded linear operators from X into X equipped with its natural topology. So, we have
(13)∥Sα(t)∥L(X)≤M~S,∥Tα(t)∥L(X)≤tα-1M~T.
Let us consider the set functions PC([-r,T],X)={x:[-r,T]→X:x∈C((tk,tk+1],X), k=0,1,2,…,m, and there exist x(tk-) and x(tk+), k=1,2,…,m with x(tk-)=x(tk)}. Endowed with the norm
(14)∥x∥PC=supt∈[-r,T]∥x(t)∥X,
the space (PC([-r,T],X),∥·∥PC) is a Banach space.Lemma 4 (see [25, 27, 40]).
Iff satisfies the uniform Hölder condition with the exponent β∈(0,1) and A is a sectorial operator, then the unique solution of the Cauchy problem
(15)Dαx(t)=Ax(t)+f(t,xt,∫0t‍h(t,s,xs)ds,∫0T‍k(t,s,xs)ds)+Bu(t),t>t0,t0∈R,0<α<1x(t)=ϕ(t),t∈[-r,0]
is given by
(16)x(t)=Sα(t-t0)(x(t0+))+∫t0t‍Tα(t-s)f(s,xs,∫0t‍h(s,τ,xτ)dτ,+∫t0t‍Tα(t-s)ddf∫0T‍k(s,τ,xτ)dτ)ds+∫0t‍Tα(t-s)Bu(s)ds,
where
(17)Sα(t)=Eα,1(Atα)=12πi∫Br^‍eλtλα-1λα-Adλ,Tα(t)=tα-1Eα,α(Atα)=12πi∫Br^‍eλt1λα-Adλ,Br^ denotes the Bronwich path, Sα(t) is called the α-resolvent family, and Tα(t) is the solution operator generated by A.Now, we define the mild solution of a system (1)–(3).Definition 5.
A functionx(·)∈PC is called a mild solution of the system (1)–(3) if x(t)=ϕ(t) on [-r,0]; Δx|t=tk=Ik(x(tk-)),k=1,2,…,m and satisfies the following integral equation:
(18)x(t)={Sα(t)ϕ(0)+∫0t‍Tα(t-s)×f(∫0T‍k(s,τ,xτ)dτs,xs,∫0s‍h(s,τ,xτ)dτ,×f∫0T‍k(s,τ,xτ)dτ)ds+∫0t‍Tα(t-s)Bu(s)ds,t∈(0,t1];Sα(t-t1)(x(t1-)+I1(x(t1-)))+∫t1t‍Tα(t-s)×f(∫0T‍k(s,τ,xτ)dτs,xs,∫0s‍h(s,τ,xτ)dτ,×f∫0T‍k(s,τ,xτ)dτ)ds+∫t1t‍Tα(t-s)Bu(s)ds,t∈(t1,t2];⋮Sα(t-tm)(x(tm-)+Im(x(tm-)))+∫tmt‍Tα(t-s)×f(∫0T‍k(s,τ,xτ)dτs,xs,∫0s‍h(s,τ,xτ)dτ,×f∫0T‍k(s,τ,xτ)dτ)ds+∫tmt‍Tα(t-s)Bu(s)ds,t∈(tm,T].
From Lemma 4 we can verify that definition.Note that, mild solution depends on control functionsu(·). The solution of (1)–(3) under a control u(·) denoted by x(·;u) is called the trajectory (state) function of (1) under u(·). The set of all possible terminal states, denoted by
(19)[KT(f)≔{x(T;u)∈X:u∈L2([0,T];U)}],
is called the reachable set of system (1) at terminal time T.Definition 6.
The system (1)–(3) is said to be controllable on J if KT(f)=X.Now we list the following hypothesis:(H1)
f:J×D×X×X→X is continuous and there exist functions L∈L1(J,R+) such that
(20)∥f(t,xt,u1,v1)-f(t,yt,u2,v2)∥X≤L[∥x-y∥+∥u1-u2∥+∥v1-v2∥],forx,y∈PC,ui,vi∈X,i=1,2.(H2)
h:J×J×D→X is continuous and there exists a constant M1>0 such that for all (t,s)∈J×J(21)∥∫0t‍[h(t,s,xs)-h(t,s,ys)]ds∥X≤M1∥x-y∥PC.(H3)
k:J×J×D→X is continuous and there exists a constant M2>0 such that for all (t,s)∈J×J(22)∥∫0T‍[k(t,s,xs)-k(t,s,ys)]ds∥X≤M2∥x-y∥PC.(H4)
The linear operatorsWk:L2([tk-1,tk]:U)→X, defined by
(23)Wku=∫tk-1tk‍Tα(t-s)Bu(s)ds,
has an invertible operator Wk-1 taking values in L2([tk-1,tk]:U)/KerWk and there exists a positive constant Mk such that ∥BWk-1∥≤Ωk and Ω=max{Ωk} (For the construction of the operator W and its inverse, see [46]).(H5)
The functionIk:X→X is continuous and there exists ρk>0 such that
(24)∥Ik(x)-Ik(y)∥X≤ρk∥x-y∥,x,y∈X,k=1,2,…,m.Theorem 7.
If the hypotheses(H1)–(H5) are satisfied, then the impulsive fractional integro-differential system (1)–(3) is controllable on J provided
(25)[M~S(1+ρm)+M~TαTαΩm×(M~S(1+ρm)+M~TL(1+M1+M2)Tαα)+M~TαTαL(1+M1+M2)]<1.Proof.
LetZ∈PC(J,X) be any arbitrary function, now to transfer the system (1) from initial state to Z(T) consider the control
(26)u(t)={W1-1[-∫0t1‍Tα(t-s)f(s,xs,∫0s‍h(s,τ,xτ)dτ,∫0T‍k(s,τ,xτ)dτ)dsZ(t1)-Sα(t1)ϕ(0)-∫0t1‍Tα(t-s)W1-1×f(s,xs,∫0s‍h(s,τ,xτ)dτ,W1-1×f∫0T‍k(s,τ,xτ)dτ)ds](t),111111111111111111t∈(0,t1];W2-1[-∫t2t1‍Tα(t2-s)f(s,xs,∫0s‍h(s,τ,xτ)dτ,∫0T‍k(s,τ,xτ)dτ)dsZ(t2)-Sα(t2-t1)W1-1×[x(t1-1)+I1x(t1-1)]W1-1-∫t2t1‍Tα(t2-s)W1-1×f(∫0T‍k(s,τ,xτ)dτs,xs,∫0s‍h(s,τ,xτ)dτ,W1-1×f∫0T‍k(s,τ,xτ)dτ)ds](t),11111111111111111t∈(t1,t2];⋮Wm+1-1[+∫tmt‍Tα(T-s)f(s,xs,∫0s‍h(s,τ,xτ)dτ,∫0T‍k(s,τ,xτ)dτ)dsZ(T)-Sα(T-tm)W1-1×[x(tm-)+Imx(tm-)]W1-1+∫tmt‍Tα(T-s)W1-1×f(s,xs,∫0s‍h(s,τ,xτ)dτ,W1-1×f∫0T‍k(s,τ,xτ)dτ)ds](t),11111111111111111t∈(tm,T].
We define the operator N:PC([-r,T],X)→PC([-r,T],X) by
(27)Nx(t)={Sα(t)ϕ(t)+∫0t‍Tα(t-s)×f(∫0T‍k(s,τ,xτ)dτs,xs,∫0s‍h(s,τ,xτ)dτ,×f∫0T‍k(s,τ,xτ)dτ)ds+∫0t‍Tα(t-s)Bu(s)ds,t∈(0,t1];Sα(t-t1)(x(t1-)+I1(x(t1-)))11+∫t1t‍Tα(t-s)1×f(∫0T‍k(s,τ,xτ)dτs,xs,∫0s‍h(s,τ,xτ)dτ,×f∫0T‍k(s,τ,xτ)dτ)ds+∫t1t‍Tα(t-s)Bu(s)ds,t∈(t1,t2];⋮Sα(t-tm)(x(tm-)+Im(x(tm-)))+∫tmt‍Tα(t-s)×f(∫0T‍k(s,τ,xτ)dτs,xs,∫0s‍h(s,τ,xτ)dτ,×f∫0T‍k(s,τ,xτ)dτ)ds+∫tmt‍Tα(t-s)Bu(s)ds,t∈(tm,T].
Note that N is well defined on PC([-r,T],X).
For our convenience, let us take(28)C(s,x)=BW-1[∫0T‍k(s,τ,xτ)dτ)dsZ(t1)-Sα(t1)ϕ(0)BW-11-∫0t1‍Tα(t1-s)f(s,xs,∫0s‍h(s,τ,xτ)dτ,BW-1-∫0t1‍Tα(t1-s)f1∫0T‍k(s,τ,xτ)dτ)ds](s),
and for i=2,3,…,m+1.
(29)Di(s,x)=BWi-1[∫0T‍k(s,τ,xτ)dτZ(ti)-Sα(ti-ti-1)×[x(ti-1-1)+Ii-1(x(ti-1-1))]BWi-1-∫ti-1ti‍Tα(ti-s)f(s,xs,∫0s‍h(s,τ,xτ)dτ,BWi-1-∫ti-1ti‍Tα(ti-s)f1∫0T‍k(s,τ,xτ)dτ)ds](s).
From our assumptions, we have
(30)∥C(s,x)-C(s,y)∥≤Ω1M~TαTα[L(1+M1+M2)]∥x-y∥PC,∥Di(s,x)-Di(s,y)∥≤Ωi[TααM~S(1+ρi-1)+M~TL(1+M1+M2)Tαα]∥x-y∥PC.
Let us take t∈(0,t1] and x,y∈PC([-r,T],X). From (13) we have
(31)∥(Nx)(t)-(Ny)(t)∥X≤∫0t‍∥Tα(t-s)∥∥C(s,x)-C(s,y)∥ds+∫0t‍∥Tα(t-s)∥×[∥f(s,xs,∫0s‍h(s,τ,xτ)dτ,∫0T‍k(s,τ,xτ)dτ)×1-f(s,ys,∫0s‍h(s,τ,yτ)dτ,∫0T‍k(s,τ,yτ)dτ)∥]ds≤M~TαTαΩ1M~TαTαL[1+M1+M2]∥x-y∥PC+M~TαTαL[1+M1+M2]∥x-y∥PC≤[1+Ω1M~TαTα]M~TαTαL[1+M1+M2]∥x-y∥PC.
Fort∈(t1,t2], and by using (13)–(29) we have
(32)∥(Nx)(t)-(Ny)(t)∥X≤∥Sα(t-t1)∥[+∥I1(x(t1-))-I1(y(t1-))∥x(t1-)-y(t1-)∥+∥I1(x(t1-))-I1(y(t1-))∥]+∫0t‍∥Tα(t-s)∥∥D2(s,x)-D2(s,y)∥ds+∫0t‍∥Tα(t-s)∥×[∥f(s,xs,∫0s‍h(s,τ,xτ)dτ,∫0T‍k(s,τ,xτ)dτ)11-f(s,ys,∫0s‍h(s,τ,yτ)dτ,∫0T‍k(s,τ,yτ)dτ)∥]ds≤[M~S(1+ρ1)+M~TαTαΩ2×(M~S(1+ρ1)+M~TL(1+M1+M2)Tαα)+M~TαTαL(1+M1+M2)]∥x-y∥PC.
Similarly, for t∈(ti,ti+1](33)∥(Nx)(t)-(Ny)(t)∥X≤[M~S(1+ρi)+M~TαTαΩi×(M~S(1+ρi)+M~TL(1+M1+M2)Tαα)+M~TαTαL(1+M1+M2)]∥x-y∥PC,
and for t∈(tm,T](34)∥(Nx)(t)-(Ny)(t)∥X≤[M~S(1+ρm)+M~TαTαΩm×(M~S(1+ρm)+M~TL(1+M1+M2)Tαα)+M~TαTαL(1+M1+M2)]∥x-y∥PC.
Thus, for all t∈[0,T], we have
(35)∥(Nx)-(Ny)∥PC≤max1≤i≤m[M~S(1+ρm)+M~TαTαΩm11×(M~S(1+ρm)+M~TL(1+M1+M2)Tαα)11+M~TαTαL(1+M1+M2)]∥x-y∥PC.
Since max1≤i≤m[M~S(1+ρm)+(M~T/α)TαΩm(M~S(1+ρm)+M~TL(1+M1+M2)(Tα/α)) + (M~T/α)TαL(1+M1+M2)]<1, then N is a contraction, and so by Banach fixed point theorem there exists a unique fixed point x∈PC(J,X) such that (Nx)(t)=x(t). This fixed point is then a solution of the system (1)–(3), and clearly, x(T)=(Nx)(T)=Z(T), which implies that the system is controllable on J. This completes the proof.


## 4. Example

Consider the following fractional partial functional integro-differential equations of the form(36)Dtαz(t,ξ)=∂∂ξz(t,ξ)+m(ξ)u(t,ξ)+F(∫0b‍h1(t,w(x,ξ-r))ds)t,x(t,ξ),∫0t‍k1(t,w(x,ξ-r))ds,1∫0b‍h1(t,w(x,ξ-r))ds),1for(t,ξ)∈[0,T]×(0,π),t≠T2,z(t,0)=z(t,π)=0,t∈[0,T],z(0,ξ)=z0(ξ),0<ξ<π,Δz|t=T/2=I1(T-2),
where T>0,0<α<1. The the above example resembles the control system (1)–(3), if we take (i)
X=L2([0,π]) as the state space and z(t,·)={z(t,ξ):0≤ξ≤π} as the state.(ii)
Input trajectoryu(t,·)∈U as the control, where U is any Banach space.(iii)
A:D(A)⊂X→X is defined by D(A)=[y∈X:∂z/∂x,∂2z/∂x2∈X are absolutely continuous and y(0)=y(π)=0],and Au=∂2u/∂x2. Then Az=-∑n=1∞‍n2(y,yn)yn, y∈D(A), where zn(x)=2/πsin(nx), n∈N is the orthogonal set of eigen vectors of A. It is well known that A is the infinitesimal generator of an analytic semigroup {T(t)t≥0} in X and that is given by
(37)T(t)y=∑n=1∞‍e-n2t(y,yn)yn,
for all y∈X, and every t>0. From these expression it follows that, {T(t)t≥0} is a uniformly bounded compact semigroup, so that, R(λ,A)=(λI-A)-1 is a compact operator for λ∈ρ(A), that is, A∈𝒜α(θ0,ω0).(iv)
B:X→X by (Bu)(ξ)=m(ξ)u(ξ) for almost every ξ∈[0,π].(v)
I1:X→X is any function satisfying assumption (H3).Therefore, the the system (36) can be written to the abstract form (1)–(3), and all the conditions of Theorem 7 are satisfied. We can conclude that the system (36) is controllable on J.


## 5. Conclusions

In this article, abstract results concerning the controllability of impulsive fractional functional integro-differential equations involving Caputo fractional derivative in Banach spaces are obtained. By using fractional calculus theory and some standard fixed point theorem, we derived the controllability results. An example is provided to show the effectiveness of the proposed results.



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