# Exact Null Controllability for Fractional Nonlocal Integrodifferential Equations via Implicit Evolution System

Authors: Amar Debbouche; Dumitru Baleanu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system, then we prove the exact null controllability result of a class of fractional evolution nonlocal integrodifferential control system in Banach space. As an application that illustrates the abstract results, two examples are provided.


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## Body

## 1. Introduction

In this paper, we study the fractional nonlocal integrodifferential system of the form(1.1)dαu(t)dtα+A(t,B1u(t))u(t)=f(t,B2u(t))+∫0tg(t,s,B3u(s))ds‍,(1.2)B4(u(0)-u0)=∫0ah(u(t))dt‍,
where 0<α≤1, t∈[0,a]. Let -A be the infinitesimal generator of a C0-semigroup in a Banach space X, and {Bi(t):i=1,2,3,4} is a family of linear closed operators defined on dense sets Si⊃D(A), i=1,2,3,4, respectively, in X into X. It is assumed that u0∈X, f:I×X→X, g:Δ×X→X and h:C(I:X)→X are given abstract functions. Here, I=[0,a] and Δ={(s,t):0≤s≤t≤a}.Basic researches in differential equations have showed that many phenomena in nature are modeled more accurately using fractional derivatives and integrals; for more detail, we can refer to [1–13] and the references therein. There are many applications where the fractional calculus can be used, for example, viscoelasticity, electrochemistry, diffusion processes, control theory, heat conduction, electricity, mechanics, chaos, and fractals [14].Controllability is a fundamental concept in mathematical control theory and plays an important role in both finite and infinite dimensional spaces, that is, systems represented by ordinary differential equations and partial differential equations, respectively. So it is natural to extend this concept to dynamical systems represented by fractional differential equations. Several fractional partial differential equations and integrodifferential equations can be expressed abstractly in some Banach spaces, in many cases, the accurate analysis, design and assessment of systems subjected to realistic environments must take into account the potential of random loads and randomness in the system properties. Randomness is intrinsic to the mathematical formulation of many phenomena such as fluctuations in the stock market or noise in communication networks. Fu studied the controllability results of some kinds of neutral functional differential systems, see [15, 16]. In our previous work [17], we established the controllability of fractional evolution nonlocal impulsive quasilinear delay integrodifferential systems.The existence results to evolution equations with nonlocal conditions in Banach space were studied first by Byszewski [18, 19]; subsequently, many authors have been studied the same question, see for instance [20–23].Deng [24] indicated that, using the nonlocal condition u(0)+h(u)=u0 to describe for instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give better result than using the usual local Cauchy problem u(0)=u0. Let us observe also that, since Deng’s papers, the function h is considered
(1.3)h(u)=∑k=1pcku(tk)‍,
where ck,k=1,2,…,p are given constants and 0≤t1<⋯<tp≤a.In this paper, we introduce a new concept in the theory of Semigroup named “implicit evolution system” to show the reader “what is the main difference between the solutions of fractional (0<α<1) and classical (first order) homogeneous evolution equation?” which is based on the work [17] and Pazy [25]. A new form of nonlocal condition is also presented.Our paper is organized as follows. Section2 is devoted to a review of some essential results which will be used in this work to obtain our main results. In Section 3, we use the theory of semigroups [25] in order to introduce our new concept that is called implicit evolution system. In Section 4, we establish the existence, uniqueness, and regularity of mild solutions of a class of fractional evolution nonlinear integrodifferential systems with nonlocal conditions in Banach space. In Section 5, we prove the exact null controllability of a class of fractional evolution nonlocal integrodifferential control systems; the last section deals to give examples that provide the abstract results.


## 2. Preliminary Results

Definition 2.1.
The fractional integral of orderα with the lower limit zero for a function f∈C([0,∞)) is defined as
(2.1)Iαf(t)=1Γ(α)∫0t‍f(s)(t-s)1-αds,t>0,0<α<1,
provided the right side is pointwise defined on [0,∞), where Γ is the gamma function.
Riemann-Liouville derivative of orderα with the lower limit zero for a function f∈C([0,∞)) can be written as
(2.2)LDαf(t)=1Γ(1-α)ddt∫0t‍f(s)(t-s)αds,t>0,0<α<1.
The Caputo derivative of order α for a function f∈C([0,∞)) can be written as
(2.3)CDαf(t)=LDα(f(t)-f(0)),t>0,0<α<1.Remark 2.2.
(1) Iff∈C1([0,∞)), then
(2.4)CDαf(t)=1Γ(1-α)∫0t‍f′(s)(t-s)αds=I1-αf′(t),t>0,0<α<1.
(2) The Caputo derivative of a constant is equal to zero.
(3) Iff is an abstract function with values in X, then integrals which appear in Definition 2.1 are taken in Bochner’s sense.Definition 2.3.
By a strong solution of the nonlocal Cauchy problem (1.1), (1.2), we mean a function u with values in X such that(i)
uis a continuous function in t∈I and u(t)∈D(A),(ii)
dαu/dtα exists and continuous on (0,a], 0<α<1, and u satisfies (1.1) on (0,a] and (1.2).It is suitable to rewrite (1.1), (1.2) in the form
(2.5)u(t)=u(0)+1Γ(α)∫0t‍(t-η)α-1×[-A(η,B1u(η))u(η)+f(η,B2u(η))+∫0η‍g(η,s,B3u(s))ds]dη,
see also [26, 27].
LetX and Y be two Banach spaces such that Y is densely and continuously embedded in X. We denote by Z every Banach space Z(I,X) endowed with the usual norm, which is given by ∥u∥Z=supt∈I∥u(t)∥, for u∈Z. The space of all bounded linear operators from X to Y is denoted by B(X,Y). We recall some definitions and known facts from Pazy [25].Definition 2.4.
LetS be a linear operator in X, and let Y be a subspace of X. The operator S~ defined by D(S~)={x∈D(S)∩Y:Sx∈Y} and S~x=Sx for x∈D(S~) is called the part of S in Y.Definition 2.5.
LetΩ be a subset of X and for every t∈I and B1u∈Ω, and let -A(t,B1u) be the infinitesimal generator of a C0-semigroup St,B1u(s), s≥0, on X. The family of operators {A(t,B1u),(t,B1u)∈I×Ω} is stable if there are constants M≥1 and ω such that
(2.6)ρ(A(t,B1u))⊃(ω,∞)∥∏j=1k‍R(λ:A(tj,B1u))∥≤M(λ-ω)-k
for λ>ω every finite sequences 0≤t1≤t2≤⋯≤tk≤a, 1≤j≤k.
The stability of{A(t,B1u),(t,B1u)∈I×Ω} implies that
(2.7)∥∏j=1k‍Stj,B1u(sj)∥≤Mexp(ω∑j=1k‍sj),sj≥0,
and any finite sequences 0≤t1≤t2≤⋯≤tk≤a, 1≤j≤k, k=1,2,….Definition 2.6.
LetSt,B1u(s), s≥0 be the C0-semigroup generated by A(t,B1u),(t,B1u)∈I×Ω. A subspace Y of X is called A(t,B1u)-admissible if Y is invariant subspace of St,B1u(s), and the restriction of St,B1u(s) to Y is a C0-semigroup in Y.
LetΩ⊂X be a subset of X such that for every (t,B1u)∈I×Ω, A(t,B1u) is the infinitesimal generator of a C0-semigroup St,B1u(s), s≥0 on X. We make the following assumptions.
(H1) The family {A(t,B1u),(t,B1u)∈I×Ω} is stable.
(H2) Y is A(t,B1u)-admissible for (t,B1u)∈I×Ω, and the family {A~(t,B1u),(t,B1u)∈I×Ω} of parts A~(t,B1u) of A(t,B1u) in Y is stable in Y.
(H3) For (t,B1u)∈I×Ω, D(A(t,B1u))⊃Y, A(t,B1u) is a bounded linear operator from Y to X and t→A(t,B1u) is continuous in the B(Y,X) norm ∥·∥.
(H4) There is a constant L>0 such that  (2.8)∥A(t,B1u)-A(t,B1v)∥Y→X≤L∥u-v∥X
holds for every B1u,B1v∈Ω, and t∈I.In the next section, we will introduce a new concept in the theory of semigroups.


## 3. Implicit Evolution System

LetΩ be a subset of X and {A(t,B1u),(t,B1u)∈I×Ω} a family of operators satisfying the conditions (H1)–(H4). If u∈C(I:X) has values in Ω, then there is a unique evolution system Uα(t,s;B1u), 0<α≤1, 0≤s≤t≤a, in X satisfying(i)
∥Uα(t,s;B1u)∥≤Meω(t-s) for 0≤s≤t≤a, where M and ω are stability constants,(ii)
(∂α/∂tα)Uα(t,s;B1u)y=A(s,B1u(s))Uα(t,s;B1u)y for y∈Y and 0≤s≤t≤a,(iii)
(∂α/∂sα)Uα(t,s;B1u)y=-Uα(t,s;B1u)A(s,B1u(s))y for y∈Y and 0≤s≤t≤a.Remark 3.1.
(1) IfB1 is the identity and α=1, then U(t,s;u) is the explicit evolution system given in Pazy [25] and in Zaidman [28].
(2) Since, in our case,U(t,s;u) is dependent of α and B1, so we call it an implicit evolution system generated by -A(t,B1u).
(3) For nonautonomous differential equations in a Banach space, the implicit evolution system is similar to our concept(α,u)-resolvent family.
(4) We can deduce that (1.1)-(1.2) is well posed if and only if -A(t,B1u) is the generator of the implicit evolution system U(t,s;u).
Further, we assume the following.
(H5) For every u∈C(I:X) satisfying u(t)∈Ω for 0≤t≤a, we have
(3.1)U(t,s;u)Y⊂Y,0≤s≤t≤a
and U(t,s;u) is strongly continuous in Y for 0≤s≤t≤a.
(H6) Y is reflexive.
(H7) For every (t,B2u)∈I×Ω, f(t,B2u)∈Y.
(H8) The operator [B4(t)+λαI]-1 exists in B(X) for any λ with Reλ≤0 and
(3.2)∥[B4(t)+λαI]-1∥≤Cα|λ|+1,t∈I,
where Cα is a positive constant independent of both t and λ.
(H9) h:C(I:Ω)→Y is Lipschitz continuous in X and bounded in Y, that is, there exist constants k1>0 and k2>0 such that
(3.3)∥h(u)∥Y≤k1,∥h(u)-h(v)∥Y≤k2maxt∈I∥u-v∥X.
For the conditions (H9) and (H10), let Z be taken as both X and Y.
(H10) g:Δ×Z→Z is continuous, and there exist constants k3>0 and k4>0 such that
(3.4)∫0t‍∥g(t,s,B3u)-g(t,s,B3v)∥Zds≤k3∥u-v∥Z,u,v∈X,k4=max{∫0t∥g(t,s,0)∥Z‍ds:(t,s)∈Δ}.
(H11) f:I×Z→Z is continuous, and there exist constants k5>0 and k6>0 such that
(3.5)∥f(t,B2u)-f(t,B2v)∥Z≤k5∥u-v∥Z,u,v∈X,k6=maxt∈I∥f(t,0)∥Z.
Let us take M0=max∥U(t,s;u)∥Ω(Z), 0≤s≤t≤a, u∈Ω.
(H12) There exist positive constants r>0 and 0<λ<1 such that
(3.6)M0{∥u0∥+aCαk1+a[r(k3+k5)+k4+k6]}≤r,λ=Ka∥u0∥Y+a2Cαk1K+aM0Cαk2+a{K[(k3+k5)r+k4+k6]a+M0(k3+k5)}.
By a mild solution of (1.1), (1.2), we mean a function u∈C(I:X) with values in Ω and u0∈X satisfying the integral equation
(3.7)u(t)=U(t,0;u)u0+U(t,0;u)B4-1∫0a‍h(u(t))dt+∫0t‍U(t,s;u)[f(s,B2u(s))+∫0s‍g(s,η,B3u(η))dη]ds.
(H13) Further, there exists a constant K>0 such that for every u,v∈C(I:X) with values in Ω and every ω∈Y we have
(3.8)∥U(t,s;u)ω-U(t,s;v)ω∥≤K∥ω∥Y∫st‍∥u(τ)-v(τ)∥dτ.


## 4. Existence Results

Theorem 4.1.
Letu0∈Y and Ω={u∈X:∥u∥Y≤r}, r>0. If -A(t,B1u) is the generator of an implicit evolution system U(t,s;u) and the assumptions (H5)~(H13) are satisfied, then (1.1), (1.2) has a unique mild solution on I.Proof.
LetS be a nonempty closed subset of C(I:X) defined by
(4.1)S={u:u∈C(I:X),∥u∥Y≤r},t∈I.
Consider a mapping P on S defined by
(4.2)(Pu)(t)=U(t,0;u)u0+U(t,0;u)B4-1∫0a‍h(u(t))dt+∫0t‍U(t,s;u)[f(s,B2u(s))+∫0s‍g(s,η,B3u(η))dη]ds.
For u∈S, we have
(4.3)∥Pu(t)∥Y≤∥U(t,0;u)u0∥+∥U(t,0;u)B4-1∫0ah(u(t))dt‍∥+∫0t‍∥U(t,s;u)∥{∫∥f(s,B2u(s))-f(s,0)∥+∥f(s,0)∥∫0t‍∥U(t,s;u;u)∥+∥∫0s‍[g(s,η,B3u(η))-g(s,η,0)]dη∥+∥∫0sg(s,η,0)dη‍∥}ds.≤M0∥u0∥+aM0Cαk1+∫0t‍M0{k5∥u(s)∥+k6+k3∥u(s)∥+k4}ds≤M0∥u0∥+aM0Cαk1+aM0{k5∥u(s)∥+k6+k3∥u(s)∥+k4}≤M0{∥u0∥+aCαk1+a[r(k3+k5)+k4+k6]}≤r.
Thus, P maps S into itself. Now, we will show that P is a strict contraction on S which will ensure the existence of a unique continuous function satisfying (3.7) on I.
Ifu,v∈S, then
(4.4)∥Pu(t)-Pv(t)∥≤∥U(t,0;u)u0-U(t,0;v)u0∥+∥U(t,0;u)B4-1∫0a‍h(u(t))dt-U(t,0;v)B4-1∫0a‍h(v(t))dt∥+∫0t∥U(t,s;u)[f(s,B2u(s))+∫0s‍g(s,η,B3u(η))dη]‍Pv(t)-U(t,s;v)[f(s,B2v(s))+∫0s‍g(s,η,B3v(η))dη]∥ds≤∥U(t,0;u)u0-U(t,0;v)u0∥+∥U(t,0;u)B4-1∫0a‍h(u(t))dt-U(t,0;v)B4-1∫0a‍h(u(t))dt∥+∥U(t,0;v)B4-1∫0a‍h(u(t))dt-U(t,0;v)B4-1∫0a‍h(v(t))dt∥+∫0t‍{∥U(t,s;u)[f(s,B2u(s))+∫0s‍g(s,η,B3u(η))dη]U(t,s)-U(t,s;v)[f(s,B2u(s))+∫0s‍g(s,η,B3u(η))dη]∥U(t,s;u)+∥U(t,s;v)[f(s,B2u(s))+∫0s‍g(s,η,B3u(η))dη]U(t,s;u)(s)-U(t,s;v)[f(s,B2v(s))+∫0s‍g(s,η,B3v(η))dη]∥}ds≤Ka∥u0∥Ymaxτ∈I∥u(τ)-v(τ)∥+a2Cαk1Kmaxτ∈I∥u(τ)-v(τ)∥+aM0Cαk2maxτ∈I∥u(τ)-v(τ)∥+∫0t‍{K∥f(s,B2u(s))+∫0s‍g(s,η,B3u(η))dη∥Yamaxτ∈I∥u(τ)-v(τ)∥PPv(t)+M0[∫∥f(s,B2u(s))-f(s,B2v(s))∥Pvv(t)+M0+∫0s∥g(s,η,B3u(η))-g(s,η,B3v(η))∥‍dη]}ds≤(Ka∥u0∥Y+a2Cαk1K+aM0Cαk2)maxτ∈I∥u(τ)-v(τ)∥+a{K[(k3+k5)r+k4+k6]a+M0(k3+k5)}maxτ∈I∥u(τ)-v(τ)∥.
Thus,
(4.5)∥Pu(t)-Pv(t)∥≤λmaxτ∈I∥u(τ)-v(τ)∥,
which means that P is a strict contraction map from S into S, and therefore by the Banach contraction principle there exists a unique fixed point u∈S such that Pu=u. Hence, u is a unique mild solution of (1.1), (1.2) on I.Theorem 4.2.
Assume the following.(i)
Conditions (H1)~(H13) hold.(ii)
The functionsB2(t)ω and B3(t)ω are uniformly Hölder continuous in t∈I for every element ω in S2∩S3.(iii)
There are numbersL1, L2>0 and p,q∈(0,1] such that
(4.6)∥f(t1,B2u)-f(t2,B2v)∥≤L1(|t1-t2|p+∥B2u-B2v∥),∥g(s1,η,B3u)-g(s2,η,B3v)∥≤L2|s1-s2|q
for all t1,t2∈I and all (s1,η),(s2,η)∈Δ.Then, the problem (1.1), (1.2) has a unique strong solution on I.Proof.
Applying Theorem4.1, the problem (1.1), (1.2) has a mild solution u∈S. Now, we will show that u is a unique strong solution of the considered problem on I.
According to (ii),∥B2u-B2v∥ is uniformly Hölder continuous in t∈I for every element u in S2∩S3, also (iii) implies that t→f(t,B2u) and t→∫0t‍g(t,s,B3u)ds are uniformly Hölder continuous on I ([20, 26]).
Set(4.7)V(t)=f(t,B2u)+∫0t‍g(t,s,B3u)ds.
Clearly V(t) is uniformly Hölder continuous in t∈I.
Consider the following nonlocal Cauchy problem:(4.8)dαv(t)dtα+A(t,B1u(t))u(t)=V(t),B4(u(0)-u0)=∫0a‍h(u(t))dt.
From Pazy, (4.8) has a unique solution v on I given by
(4.9)v(t)=U(t,0;u)u0+U(t,0;u)B4-1∫0a‍h(u(t))dt+∫0t‍U(t,s;u)V(s)ds.
Noting that [21], each term on the right hand side of (4.9) belongs to D(A), thus v(t)∈D(A), using the uniqueness of V(t), we have that u(t)=v(t). Hence, u is the unique strong solution of (1.1), (1.2) on I.In next section, some results are obtained from Sakthivel et al. [29, 30].


## 5. Exactly Null Controllability Results

Consider fractional nonlocal evolution integrodifferential control system of the form(5.1)dαu(t)dtα+A(t,B1u(t))u(t)=(Φμ)(t)+Ψ(t,f(t,B2u(t)),∫0t‍g(t,s,B3u(s))ds),B4(u(0)-u0)=∫0a‍h(u(t))dt,
where the unknown u(·) takes values in the Banach space X, the control function μ belongs to the spaces L2(I,H), a Banach space of admissible control functions with H, a Banach space. Further, Φ is a bounded linear operator from H into X, the function Ψ:I×X×X→X is given, and the others terms are defined as above.For allu0∈X and admissible control μ∈L2(I,H), the problem (5.1) admits a mild solution given by
(5.2)uμ(t)=U(t,0;u)u0+U(t,0;u)B4-1∫0a‍h(u(t))dt+∫0t‍U(t,s;u)[Φμ(s)+Ψ(s,f(s,B2u(s)),∫0s‍g(s,τ,B3u(τ))dτ)]ds.Definition 5.1.
We will say that system (5.1) is exactly null controllable on the interval I if for a11 u0∈X, there exists a control μ∈L2(I,H), such that the mild solution u(t) of (5.1) corresponding to μ verifies u(0)-B4-1∫0a‍h(u(t))dt=u0 and u(a)=0.
In order to prove the controllability result, in addition, we consider the following conditions.
(H14)  Ψ:I×X×X→X is continuous, and there exist constants N1 and N2 such that for all xi,yi∈X, i=1,2, we have
(5.3)∥Ψ(t,x1,y1)-Ψ(t,x2,y2)∥≤N1[∥x1-x2∥+∥y1-y2∥],N2=maxt∈I∥Ψ(t,0,0)∥.
(H15) Let
(5.4)ρ+aM0M1M2(ρ+σ)+σ≤r,
where σ=aM0{N1[(k3+k5)r+k4+k6]+N2} and ρ=M0∥u0∥+aM0Cαk1, and let
(5.5)λ={Ka∥u0∥+a2Cαk1K+aM0Cαk2}+2a{M0M1M2(ρ+ap)}+a{Kqa+p},
where p=M0N1(k3+k5), q=N1[(k3+k5)r+k4+k6]+N2, and 0≤λ<1.
(H16) The bounded linear operator W:L2(I,H)→X defined by
(5.6)Wμ=∫0a‍U(a,s;u)Φμds
has an induced inverse operator W~-1 which takes values in L2(I,H)/ker W and there exist positive constants M1, M2, such that ∥Φ∥≤M1 and ∥W~-1∥≤M2.Theorem 5.2.
If hypotheses (H1)~(H16) are satisfied, then the control nonlocal fractional integrodifferential system (5.1) is exactly null controllable on I.Proof.
LetSr={u:u∈C(I:X),u(0)-B4-1∫0a‍h(u(t))dt=u0, ∥u∥≤r, t∈I}.
We define an operatorQ:Sr→Sr by
(5.7)(Quμ)(t)=U(t,0;u)u0+U(t,0;u)B4-1∫0a‍h(u(t))dt+∫0t‍U(t,η;u)ΦW~-1[-U(a,0;u)u0-U(a,0;u)B4-1∫0a‍h(u(t))dt(t)+∫0t‍U(η;u)ΦW~-1-∫0a‍U(a,s;u)Ψ(∫s,f(s,B2u(s)),(t)+∫0t‍U(η;u)ΦW~-1-∫0a‍U(a,s;u)Ψ∫0s‍g(s,τ,B3u(τ))dτ)ds](η)dη+∫0t‍U(t,s;u)Ψ(s,f(s,B2u(s)),∫0s‍g(s,τ,B3u(τ))dτ)ds.
Using the hypothesis (H14), for an arbitrary function u(·), we define the control
(5.8)μ(t)=W~-1[-U(a,0;u)u0-U(a,0;u)B4-1∫0a‍h(u(t))dtW~-1f-∫0a‍U(a,s;u)Ψ(s,f(s,B2u(s)),∫0s‍g(s,τ,B3u(τ))dτ)ds](t).
Using this controller, we will show that the operator Q has a fixed point. This fixed point is then a solution of (5.2).
Clearly,Qμu(a)=0, which means that the control μ steers system (5.1) from the initial state u0 to origin in time a, provided we can obtain a fixed point of the nonlinear operator Q.
Now, we show thatQ maps Sr into itself.
We have(5.9)∥(Quμ)(t)∥≤∥U(t,0;u)u0+U(t,0;u)B4-1∫0a‍h(u(t))dt∥+∫0t‍∥U(t,η;u)∥∥ΦW~-1∥[∥U(a,0;u)u0+U(a,0;u)B4-1∫0a‍h(u(t))dt∥+∫0t‍∥U(t,η;u)∥∥ΦW~-1∥+∫0a‍∥U(a,s;u)∥{∫∥∫Ψ(∫s,f(s,B2u(s)),∫0s‍g(s,τ,B3u(τ))dτ)B4-1+∫0t‍∥U∥∥ΦW~-1∥∫0s‍g(s,τ,B3u(τ))dτ)∫0s‍g(s,τ,B3u(τ))dτ)-Ψ(s,0,0)∥+∥Ψ(s,0,0)∥}ds]dη+∫0t‍∥U(t,s;u)∥(t,s)×{∥Ψ(s,f(s,B2u(s)),∫0s‍g(s,τ,B3u(τ))dτ)-Ψ(s,0,0)∥+∥Ψ(s,0,0)∥}ds.≤M0∥u0∥+aM0Cαk1+aM0M1M2[M0∥u0∥+aM0Cαk1+aM0{N1(k5r+k6+k3r+k4)+N2}]+aM0{N1(k5r+k6+k3r+k4)+N2}≤r.
Thus, Q maps Sr into itself. Now, for u,v∈Sr, we have
(5.10)∥Quμ(t)-Qvμ(t)∥≤∥U(t,0;u)u0-U(t,0;v)u0∥+∥U(t,0;u)B4-1∫0a‍h(u(t))dt-U(t,0;v)B4-1∫0a‍h(v(t))dt∥+∫0t‍{∥U(t,η;u)ΦW~-1[-U(a,0;u)u0-U(a,0;u)B4-1∫0a‍h(u(t))dtU(t,η;u)ΦΦΦW~-1ΦW~-1-∫0a‍U(a,s;u)Ψ(s,f(s,B2u(s)),∫0s‍g(s,τ,B3u(τ))dτ)ds]U(t,0;u)-U(t,η;v)ΦW~-1[-U(a,0;v)u0-U(a,0;v)B4-1∫0a‍h(v(t))dtU-U(t,η;v)ΦΦW~-1(t,0;u)-∫0a‍U(a,s;v)Ψ(∫s,f(s,B2v(s)),U-U(t,η;v)ΦΦW~-1(t,0;u)∫0s‍g(s,τ,B3v(τ))∫0s‍g(s,τ,B3v(τ))dτ)ds]∥}dη+∫0t∥U(t,s;u)Ψ(s,f(s,B2u(s)),∫0s‍g(s,τ,B3u(τ))dτ)‍(t,0;u)-U(t,s;v)Ψ(s,f(s,B2v(s)),∫0s‍g(s,τ,B3v(τ))dτ)∥ds.≤∥U(t,0;u)u0-U(t,0;v)u0∥+∥U(t,0;u)B4-1∫0a‍h(u(t))dt-U(t,0;v)B4-1∫0a‍h(u(t))dt∥+∥U(t,0;v)B4-1∫0a‍h(u(t))dt-U(t,0;v)B4-1∫0a‍h(v(t))dt∥+∫0t‍{∥U(t,η;u)ΦW~-1[-U(a,0;u)u0-U(a,0;u)B4-1∫0a‍h(u(t))dtU(t,η;u)ΦW~-1ΦΦΦW~-1-∫0a‍U(a,s;u)Ψ(s,f(s,B2u(s)),∫0s‍g(s,τ,B3u(τ))dτ)ds]U(t,0;v)-U(t,η;v)ΦW~-1[-U(a,0;v)u0-U(a,0;v)B4-1∫0a‍h(v(t))dtU(t,0;v)-U(t,η;v)ΦΦW~-1-∫0a‍U(a,s;v)Ψ(∫s,f(s,B2v(s)),U(t,0;v)-U(t,η;v)ΦΦW~-1∫0s‍g(s,τ,B3v(τ))∫0s‍g(s,τ,B3v(τ))dτ)ds]∥}dη+∫0t{∥U(t,s;u)Ψ(s,f(s,B2u(s)),∫0s‍g(s,τ,B3u(τ))dτ)‍(t,0;v)-U(t,s;v)Ψ(s,f(s,B2u(s)),∫0s‍g(s,τ,B3u(τ))dτ)∥(t,0;v)+∥U(t,s;v)Ψ(s,f(s,B2u(s)),∫0s‍g(s,τ,B3u(τ))dτ)(t,0;v;v)-U(t,s;v)Ψ(s,f(s,B2v(s)),∫0s‍g(s,τ,B3v(τ))dτ)∥}ds.≤(Ka∥u0∥+a2Cαk1K+aM0Cαk2)maxτ∈I∥u(τ)-v(τ)∥+2a{M0M1M2[M0∥u0∥+aM0Cαk1+aM0N1(k5+k3)]}maxτ∈I∥u(τ)-v(τ)∥+a{aK[N1(k5r+k6+k3r+k4)+N2]+M0N1(k5+k3)}maxτ∈I∥u(τ)-v(τ)∥≤λmaxτ∈I∥u(τ)-v(τ)∥.
Therefore, Q is a contraction mapping, and hence there exists a unique fixed point u∈X, such that Qu(t)=u(t). Any fixed point of Q is a mild solution of (5.1) on I which satisfies u(a)=0. Thus, system (5.1) is exactly null controllable on I.


## 6. Examples

To illustrate the abstract results, we give the following examples.Example 6.1.
Consider the nonlinear integropartial differential equation of fractional order(6.1)∂αu(x,t)∂tα+∑|q|≤2m‍aq(x,t;bqu(x,t))Dxqu(x,t)=F(x,t,w1)+∫0t‍G(x,t,s,w2(s))ds,
with nonlocal condition
(6.2)u(x,0)+∑k=1p‍cku(x,tk)=g(x),
where 0<α≤1, 0≤t1<⋯<tp≤a, x∈Rn, Dxq=Dx1q1⋯Dxnqn, Dxi=∂/∂xi, q=(q1,…,qn) is an n-dimensional multi-index, |q|=q1+⋯+qn, and wi, i=1,2, is given by
(6.3)wi(x,t)=∑|q|≤2m-1‍bqi(x,t)Dxqu(x,t)+∫Ω‍∑|q|≤2m-1‍cqi(x,t)Dyqu(y,t)dy.
Let L2(Rn) be the set of all square integrable functions on Rn. We denote by Cm(Rn) the set of all continuous real-valued functions defined on Rn which have continuous partial derivatives of order less than or equal to m. By C0m(Rn), we denote the set of all functions f∈Cm(Rn) with compact supports. Let Hm(Rn) be the completion of C0m(Rn) with respect to the norm
(6.4)∥f∥m2=∑|q|≤m‍∫Rn‍|Dxqf(x)|2dx.
It is supposed that the following hold.
(i) The operatorA=-∑|q|≤2m‍aq(x,t;bqu(x,t))Dxq is uniformly elliptic on Rn. In other words, all the coefficients aq, |q|=2m, are continuous and bounded on Rn, and there is a positive number c such that
(6.5)(-1)m+1∑|q|=2m‍eq(x,t;bq)ξq≥c|ξ|2m,
for all x∈Rn and all ξ≠0, ξ∈Rn, ξq=ξ1q1⋯ξnqn, and |ξ|2=ξ12+⋯+ξn2.
(ii) All the coefficientsaq, |q|=2m, satisfy a uniform Hölder condition on Rn. Under these conditions, the operator A with domain of definition D(A)=H2m(Rn) generates an evolution operator defined on L2(Rn), and it is well known that H2m(Rn) is dense in X=L2(Rn) and the initial function g(x) is an element in Hilbert space H2m(Rn), see [26, page 438]. Applying Theorem 4.1, this achieves the proof of the existence of mild solutions of the problem (6.1), (6.2). In addition,
(iii) If the coefficientsbq, cq, |q|≤2m-1 satisfy a uniform Hölder condition on Rn and the operators F and G satisfy.
There are numbersL1, L2≥0 and λ1, λ2∈(0,1) such that
(6.6)∑|q|≤2m-1‍∫Rn‍|F(x,t,Dxqw1)-F(x,s,Dxqw1*)|2dx≤L1(|t-s|λ1+|w1-w1*|2dx),∑|q|≤2m-1‍∫Rn‍|G(x,t,η,Dxqw2)-G(x,s,η,Dxqw2)|2dx≤L2|t-s|λ2,
for all t,s∈I, (t,η),(s,η)∈Δ, and all x∈Rn. Applying Theorem 4.2, we deduce that (6.1), (6.2) has a unique strong solution.The second example is concerned with the controllability result.Example 6.2.
Consider the fractional nonlocal evolution integropartial differential control system of the form(6.7)∂αu(x,t)∂tα+a(x,t,(bu)(x,t))∂2u(x,t)∂x2=ζ(x,t)+Υ(t,μ1(t,u(x,t)),∫0t‍μ2(t,s,u(x,t))ds),u(x,0)+∑k=1p‍cku(x,tk)=g(x),x∈[0,π],u(0,t)=u(π,t)=0,t∈I,
where 0<α≤1, 0≤t1<⋯<tp≤a, and the functions a(x,t,·), b(x,t) are continuous.
Let us take(6.8)X=L2[0,π],Sr={y∈L2[0,π]:∥y∥≤r}.
Put (Φμ)(x)(t)=ζ(x,t), x∈[0,π] where μ(t)=ζ(·,t) and ζ:[0,π]×I→[0,π] is continuous.
We defineA(t,·):X→X by (A(t,·)w)(x)=a(x,t,·)w′′ with domain D(A)={w∈X:w,w′ are absolutely continuous, w′′∈X, w(0)=w(π)=0}. Assume that -A(t,·) generates an evolution system U(t,s,·) such that for every positive numbers n1 and n2, ∥U(t,s,·)∥≤n1 and ∥U(t,s,·)A(s,·)∥≤n2.
Also, defineB1(t):D(B1)=S1→X by (B1(t)z)(x)=b(x,t)z, for all z∈X and x∈[0,π].
Assume that the linear operatorW that is given by
(6.9)Wμ(x)=∫0a‍U(a,s;u)ζ(x,s)ds,x∈[0,π],
has a bounded invertible operator W~-1 in L2(I,H)/kerW.
Let us assume that the nonlinear functionsΥ, μ1, and μ2 satisfy the following Lipschitz conditions
(6.10)∥Υ(t,z1,y1)-Υ(t,z2,y2)∥≤c1[∥z1-z2∥+∥y1-y2∥],∥μ1(t,x1)-μ1(t,x2)∥≤c2∥x1-x2∥,∫0t‍∥μ2(t,s,v1)-μ2(t,s,v2)∥ds≤c3∥v1-v2∥,
where cj>0, j=1,2,3, xi,yi,zi,vi∈X, i=1,2, and s,t∈I.
All the conditions stated in Theorem 5.2 are satisfied. Hence, system (6.7) is exactly null controllable on I.



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*Source: 931975-2012-09-02.xml*