Fractional Nonlocal Integrodifferential Equations via Implicit Evolution System

Journal of Applied Mathematics Volume 2012, Article ID 931975,17pages doi:10.1155/2012/931975

Research Article

Exact Null Controllability for

Fractional Nonlocal Integrodifferential Equations via Implicit Evolution System

Amar Debbouche

and Dumitru Baleanu

1Department of Mathematics, Faculty of Science, Guelma University, Guelma, Algeria

2Department of Mathematics and Computer Science, Cankaya University, Ankara, Turkey

3Institute of Space Sciences, P.O. Box, MG-23, 76900 Magurele-Bucharest, Romania

Correspondence should be addressed to Dumitru Baleanu,[email protected] Received 1 May 2012; Revised 15 July 2012; Accepted 16 July 2012

Academic Editor: F. Marcell´an

Copyrightq2012 A. Debbouche and D. Baleanu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

In this paper, we study the fractional nonlocal integrodifferential system of the form

d^αut

dt^α At, B1utut ft, B2ut _t

0

gt, s, B3usds, 1.1

B4u0−u0

_a

0

hutdt, 1.2

where 0 < α ≤ 1, t ∈ 0, a. Let −Abe the infinitesimal generator of a C₀-semigroup in a Banach spaceX, and{Bit : i 1,2,3,4} is a family of linear closed operators defined on dense sets S_i ⊃ DA, i 1,2,3,4, respectively, in X into X. It is assumed that u₀ ∈ X,

f :I×X → X,g :Δ×X → X andh: CI : X → X are given abstract functions. Here, I 0, aandΔ {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 to1–13and 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 fractals14.

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, see15,16. In our previous work17, 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 Byszewski18,19; subsequently, many authors have been studied the same question, see for instance20–23.

Deng24indicated that, using the nonlocal conditionu0 hu u₀to 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 problemu0 u0. Let us observe also that, since Deng’s papers, the functionhis considered

hu p k1

c_kutk,

1.3

wherec_k, k1,2, . . . , pare given constants and 0≤t₁<· · ·< t_p≤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 fractional0< α <1and classicalfirst orderhomogeneous evolution equation?” which is based on the work17and Pazy25. A new form of nonlocal condition is also presented.

Our paper is organized as follows. Section2is 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 semigroups25in order to introduce our new concept that is called implicit evolution system. In Section4, 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 Section5, 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 orderαwith the lower limit zero for a functionf ∈ C0,∞is defined as

I^αft 1 Γα

_t

0

fs

t−s^1−αds, t >0, 0< α <1, 2.1 provided the right side is pointwise defined on0,∞, whereΓis the gamma function.

Riemann-Liouville derivative of orderαwith the lower limit zero for a functionf ∈ C0,∞can be written as

LD^αft 1

Γ1−α d dt

_t

0

fs

t−s^αds, t >0, 0< α <1. 2.2 The Caputo derivative of orderαfor a functionf∈C0,∞can be written as

CD^αft ^LD^α

ft−f0

, t >0, 0< α <1. 2.3 Remark 2.2. 1Iff ∈C¹0,∞, then

CD^αft 1

Γ1−α _t

0

fs

t−s^αdsI^1−αft, t >0, 0< α <1. 2.4 2The Caputo derivative of a constant is equal to zero.

3 If f is an abstract function with values in X, then integrals which appear in Definition2.1are taken in Bochner’s sense.

Definition 2.3. By a strong solution of the nonlocal Cauchy problem1.1,1.2, we mean a functionuwith values inXsuch that

iuis a continuous function int∈Iandut∈DA,

iid^αu/dt^αexists and continuous on0, a, 0< α <1, andusatisfies1.1on0, aand 1.2.

It is suitable to rewrite1.1,1.2in the form

ut u0 1

Γα _t

0

t−η_α−1

×

−A η, B1u

η u

η f

η, B2u η

_η

0

g

η, s, B3us ds

dη,

2.5

see also26,27.

LetXandYbe two Banach spaces such thatY is densely and continuously embedded inX. We denote byZevery Banach spaceZI, Xendowed with the usual norm, which is given byuZsup_t∈Iut, foru∈Z. The space of all bounded linear operators fromXto Y is denoted byBX, Y. We recall some definitions and known facts from Pazy25.

Definition 2.4. LetSbe a linear operator inX, and letY be a subspace ofX. The operatorS defined byDS {x∈DS∩Y :Sx∈Y}andSx Sxforx∈DS is called the part ofS inY.

Definition 2.5. LetΩbe a subset ofXand for everyt∈IandB₁u∈Ω, and let−At, B1ube the infinitesimal generator of aC₀-semigroupS_t,B1us,s ≥0, onX. The family of operators {At, B1u,t, B1u∈I×Ω}is stable if there are constantsM≥1 andωsuch that

ρAt, B1u⊃ω,∞

k

j1

R λ:A

t_j, B₁u

≤Mλ−ω^−k 2.6

forλ > ωevery finite sequences 0≤t₁≤t₂≤ · · · ≤t_k≤a, 1≤j≤k.

The stability of{At, B1u,t, B1u∈I×Ω}implies that

k

j1

S_tj,B1u s_j

≤Mexp

⎛

⎝ω k j1

s_j

⎞

⎠, s_j≥0, 2.7

and any finite sequences 0≤t₁≤t₂≤ · · · ≤t_k≤a, 1≤j≤k,k1,2, . . ..

Definition 2.6. LetS_t,B1us,s≥0 be theC₀-semigroup generated byAt, B1u,t, B1u∈I×Ω.

A subspaceY ofXis calledAt, B1u-admissible ifY is invariant subspace ofSt,B1us, and the restriction ofS_t,B1ustoY is aC₀-semigroup inY.

Let Ω ⊂ X be a subset of X such that for every t, B1u ∈ I ×Ω, At, B1u is the infinitesimal generator of a C0-semigroup St,B1us, s ≥ 0 on X. We make the following assumptions.

H1The family{At, B1u,t, B1u∈I×Ω}is stable.

H2YisAt, B1u-admissible fort, B1u∈I×Ω, and the family{At, B 1u,t, B1u∈ I×Ω}of partsAt, B 1uofAt, B1uinY is stable inY.

H3Fort, B1u∈I×Ω,DAt, B1u⊃Y,At, B1uis a bounded linear operator from Y toXandt → At, B1uis continuous in theBY, Xnorm · .

H4There is a constantL >0 such that

At, B1u−At, B1v_Y→X≤Lu−v_X 2.8 holds for everyB₁u, B₁v∈Ω, andt∈I.

In the next section, we will introduce a new concept in the theory of semigroups.

3. Implicit Evolution System

LetΩbe a subset ofX and{At, B1u,t, B1u ∈I×Ω}a family of operators satisfying the conditionsH1–H4. Ifu∈CI:Xhas values inΩ, then there is a unique evolution system U_αt, s;B₁u, 0< α≤1, 0≤s≤t≤a, inXsatisfying

iUαt, s;B1u ≤Me^ωt−sfor 0≤s≤t≤a, whereMandωare stability constants, ii ∂^α/∂t^αUαt, s;B1uyAs, B1usUαt, s;B1uyfory∈Yand 0≤s≤t≤a, iii ∂^α/∂s^αUαt, s;B₁uy−Uαt, s;B₁uAs, B1usyfory∈Y and 0≤s≤t≤a.

Remark 3.1. 1IfB1is the identity andα 1, thenUt, s;uis the explicit evolution system given in Pazy25and in Zaidman28.

2 Since, in our case, Ut, s;u is dependent ofα and B₁, so we call it an implicit evolution system generated by−At, B1u.

3For nonautonomous differential equations in a Banach space, the implicit evolution system is similar to our conceptα, u-resolvent family.

4We can deduce that1.1-1.2is well posed if and only if−At, B1uis the generator of the implicit evolution systemUt, s;u.

Further, we assume the following.

H5For everyu∈CI :Xsatisfyingut∈Ωfor 0≤t≤a, we have

Ut, s;uY ⊂Y, 0≤s≤t≤a 3.1

andUt, s;uis strongly continuous inY for 0≤s≤t≤a.

H6Y is reflexive.

H7For everyt, B2u∈I×Ω,ft, B2u∈Y.

H8The operatorB4t λ^αI⁻¹exists inBXfor anyλwith Reλ≤0 and B4t λ^αI⁻¹ ≤ Cα

|λ|1, t∈I, 3.2

whereCαis a positive constant independent of bothtandλ.

H9h:CI :Ω → Y is Lipschitz continuous inX and bounded inY, that is, there exist constantsk₁>0 andk₂>0 such that

hu_Y ≤k1, hu−hv_Y ≤k₂max

t∈I u−v_X. 3.3

For the conditionsH9andH10, letZbe taken as bothXandY.

H10g :Δ×Z → Zis continuous, and there exist constantsk₃ >0 andk₄ >0 such that

_t

0

gt, s, B3u−gt, s, B3v

Zds≤k₃u−v_Z, u, v∈X, k4max

t 0

gt, s,0

Zds:t, s∈Δ

.

3.4

H11f :I×Z → Zis continuous, and there exist constantsk5 > 0 andk6 > 0 such that

ft, B2u−ft, B2v

Z≤k5u−v_Z, u, v∈X, k₆max

t∈I ft,0

Z. 3.5

Let us takeM₀maxUt, s;u_ΩZ, 0≤s≤t≤a,u∈Ω.

H12There exist positive constantsr >0 and 0< λ <1 such that

M₀{u0aC_αk₁ark3k₅ k₄k₆} ≤r,

λKau0_Y a²Cαk1KaM0Cαk2a{Kk3k5rk4k6aM0k3k5}. 3.6

By a mild solution of1.1,1.2, we mean a functionu ∈ CI : Xwith values inΩand u₀∈Xsatisfying the integral equation

ut Ut,0;uu0Ut,0;uB₄⁻¹ _a

0

hutdt

_t

0

Ut, s;u

fs, B2us _s

0

g

s, η, B₃u η

dη

ds.

3.7

H13Further, there exists a constantK > 0 such that for everyu, v ∈ CI : Xwith values inΩand everyω∈Ywe have

Ut, s;uω−Ut, s;vω ≤Kω_{Y t}

s

uτ−vτdτ. 3.8

4. Existence Results

Theorem 4.1. Letu0 ∈Y andΩ {u∈ X : uY ≤ r},r >0. If−At, B1uis the generator of an implicit evolution systemUt, s;uand the assumptions (H₅)∼(H13) are satisfied, then1.1,1.2 has a unique mild solution onI.

Proof. LetSbe a nonempty closed subset ofCI:Xdefined by

S{u:u∈CI:X,u_Y ≤r}, t∈I. 4.1

Consider a mappingPonSdefined by

P ut Ut,0;uu0Ut,0;uB₄⁻¹ _a

0

hutdt

_t

0

Ut, s;u

fs, B2us _s

0

g

s, η, B₃u η

dη

ds.

4.2

Foru∈S, we have

P ut_Y ≤ Ut,0;uu0

Ut,0;uB⁻¹_{4 a}

0

hutdt

_t

0

Ut, s;u fs, B2us−fs,0 fs,0

_s

0

g

s, η, B₃u η

−g s, η,0

dη

_s

0

g s, η,0

dη

ds.

≤M0u0aM0Cαk1

_t

0

M₀{k5usk₆k₃usk₄}ds

≤M0u0aM0Cαk1aM0{k5usk6k3usk4}

≤M0{u0aCαk1ark3k5 k4k6}

≤r.

4.3 Thus,P mapsSinto itself. Now, we will show thatPis a strict contraction onSwhich will ensure the existence of a unique continuous function satisfying3.7onI.

Ifu, v∈S, then P ut−P vt

≤ Ut,0;uu0−Ut,0;vu0

Ut,0;uB₄⁻¹ _a

0

hutdt−Ut,0;vB₄⁻¹ _a

0

hvtdt

_t

0

Ut, s;u

fs, B2us _s

0

g

s, η, B3u η

dη

−Ut, s;v

fs, B2vs _s

0

g

s, η, B₃v η

dη ds

≤ Ut,0;uu0−Ut,0;vu0

Ut,0;uB₄⁻¹ _a

0

hutdt−Ut,0;vB₄⁻¹ _a

0

hutdt

Ut,0;vB⁻¹_{4 a}

0

hutdt−Ut,0;vB₄⁻¹ _a

0

hvtdt

_t

0

Ut, s;u

fs, B2us _s

0

g

s, η, B₃u η

dη

−Ut, s;v

fs, B2us _s

0

g

s, η, B3u η

dη

Ut, s;v

fs, B2us _s

0

g

s, η, B3u η

dη

−Ut, s;v

fs, B2vs _s

0

g

s, η, B₃v η

dη

ds

≤Kau0_Ymax

τ∈I uτ−vτa²Cαk1Kmax

τ∈I uτ−vτ aM₀C_αk₂max

τ∈I uτ−vτ

_t

0

K

fs, B2us _s

0

g

s, η, B3u η

dη Y

amax

τ∈I uτ−vτ

M0 fs, B2us−fs, B2vs

_s

0

g

s, η, B₃u η

−g

s, η, B₃v η dη

ds

≤

Kau0_Ya²C_αk₁KaM₀C_αk₂

maxτ∈I uτ−vτ a{Kk3k₅rk₄k₆aM₀k3k₅}max

τ∈I uτ−vτ.

4.4

Thus,

P ut−P vt ≤λmax

τ∈I uτ−vτ, 4.5

which means thatP is a strict contraction map fromSintoS, and therefore by the Banach contraction principle there exists a unique fixed pointu∈Ssuch thatP u u. Hence,uis a unique mild solution of1.1,1.2onI.

Theorem 4.2. Assume the following.

iConditions (H1)∼(H13) hold.

iiThe functions B₂tω and B₃tω are uniformly H¨older continuous in t ∈ I for every elementωinS₂∩S₃.

iiiThere are numbersL₁,L₂>0 andp, q∈0,1such that

ft1, B2u−ft2, B2v ≤L1

|t1−t2|^pB2u−B2v , g

s1, η, B3u

−g

s2, η, B3v ≤L2|s1−s2|^q 4.6 for allt₁, t₂∈Iand alls1, η,s2, η∈Δ.

Then, the problem1.1,1.2has a unique strong solution onI.

Proof. Applying Theorem4.1, the problem1.1,1.2has a mild solutionu∈S. Now, we will show thatuis a unique strong solution of the considered problem onI.

According toii,B2u−B2vis uniformly H ¨older continuous int∈Ifor every element uinS2∩S3, alsoiiiimplies thatt → ft, B2uandt → _t

0gt, s, B3udsare uniformly H ¨older continuous onI20,26.

Set

Vt ft, B2u _t

0

gt, s, B3uds. 4.7

ClearlyVtis uniformly H ¨older continuous int∈I.

Consider the following nonlocal Cauchy problem:

d^αvt

dt^α At, B1utut Vt, B₄u0−u₀

_a

0

hutdt.

4.8

From Pazy,4.8has a unique solutionvonIgiven by

vt Ut,0;uu0Ut,0;uB⁻¹_{4 a}

0

hutdt

_t

0

Ut, s;uVsds. 4.9

Noting that21, each term on the right hand side of4.9belongs toDA, thusvt∈DA, using the uniqueness ofVt, we have thatut vt. Hence,uis the unique strong solution of1.1,1.2onI.

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

d^αut

dt^α At, B1utut Φμ

t Ψ

t, ft, B2ut, _t

0

gt, s, B3usds

,

B₄u0−u_{0 a}

0

hutdt,

5.1

where the unknownu·takes values in the Banach spaceX, the control functionμbelongs to the spacesL²I, H, a Banach space of admissible control functions withH, a Banach space.

Further,Φis a bounded linear operator fromHintoX, the functionΨ :I×X×X → X is given, and the others terms are defined as above.

For allu0 ∈X and admissible controlμ ∈ L²I, H, the problem5.1admits a mild solution given by

u_μt Ut,0;uu0Ut,0;uB⁻¹_{4 a}

0

hutdt

_t

0

Ut, s;u

Φμs Ψ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

ds.

5.2

Definition 5.1. We will say that system 5.1 is exactly null controllable on the intervalI if for a11u₀ ∈ X, there exists a controlμ ∈ L²I, H, such that the mild solutionutof5.1 corresponding toμverifiesu0−B⁻¹_{4 a}

0hutdtu0andua 0.

In order to prove the controllability result, in addition, we consider the following conditions.

H14 Ψ:I×X×X → Xis continuous, and there exist constantsN1andN2such that for allx_i, y_i∈X,i1,2, we have

Ψ

t, x₁, y₁

−Ψ

t, x₂, y₂ ≤N₁

x1−x₂ y₁−y₂ , N2max

t∈I Ψt,0,0. 5.3

H15Let

ρaM0M1M2

ρσ

σ≤r, 5.4

whereσaM0{N1k3k5rk4k6 N2}andρM0u0aM0Cαk1, and let

λ

Kau0a²Cαk1KaM0Cαk2

2a

M0M1M2

ρap a

Kqap

, 5.5

wherepM0N1k3k5,qN1k3k5rk4k6 N2, and 0≤λ <1.

H16The bounded linear operatorW :L²I, H → Xdefined by

Wμ

_a

0

Ua, s;uΦμds 5.6

has an induced inverse operatorW⁻¹ which takes values inL²I, H/kerW and there exist positive constantsM₁,M₂, such thatΦ ≤M₁andW⁻¹ ≤M₂.

Theorem 5.2. If hypotheses (H1)∼(H16) are satisfied, then the control nonlocal fractional integrodif- ferential system5.1is exactly null controllable onI.

Proof. LetS_r {u:u∈CI:X, u0−B⁻¹_{4 a}

0hutdtu₀,u ≤r,t∈I}.

We define an operatorQ:Sr → Srby Quμ

t Ut,0;uu0Ut,0;uB⁻¹_{4 a}

0

hutdt

_t

0

U t, η;u

ΦW⁻¹

−Ua,0;uu0−Ua,0;uB⁻¹_{4 a}

0

hutdt

− _a

0

Ua, s;uΨ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

ds η

dη

_t

0

Ut, s;uΨ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

ds.

5.7

Using the hypothesisH14, for an arbitrary functionu·, we define the control μt W⁻¹

−Ua,0;uu0−Ua,0;uB⁻¹_{4 a}

0

hutdt

− _a

0

Ua, s;uΨ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

ds

t.

5.8

Using this controller, we will show that the operatorQhas a fixed point. This fixed point is then a solution of5.2.

Clearly,Qμua 0, which means that the controlμsteers system5.1from the initial stateu₀to origin in timea, provided we can obtain a fixed point of the nonlinear operatorQ.

Now, we show thatQmapsS_r into itself.

We have

Qu_μ t

≤

Ut,0;uu0Ut,0;uB₄⁻¹ _a

0

hutdt

_t

0

U

t, η;u ΦW⁻¹

Ua,0;uu0Ua,0;uB⁻¹_{4 a}

0

hutdt

_a

0

Ua, s;u Ψ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

−Ψs,0,0

Ψs,0,0

ds

dη

_t

0

Ut, s;u

× Ψ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

−Ψs,0,0

Ψs,0,0

ds.

≤M₀u0aM₀C_αk₁

aM0M1M2M0u0aM0Cαk1aM0{N1k5rk6k3rk4 N2} aM₀{N1k5rk₆k₃rk₄ N₂}

≤r.

5.9

Thus,QmapsS_rinto itself. Now, foru, v∈S_r, we have Qu_μt−Qv_μt

≤ Ut,0;uu0−Ut,0;vu0

Ut,0;uB₄⁻¹ _a

0

hutdt−Ut,0;vB₄⁻¹ _a

0

hvtdt

_t

0

U t, η;u

ΦW⁻¹

−Ua,0;uu0−Ua,0;uB⁻¹_{4 a}

0

hutdt

− _a

0

Ua, s;uΨ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

ds

−U t, η;v

ΦW⁻¹

−Ua,0;vu0−Ua,0;vB₄⁻¹ _a

0

hvtdt

− _a

0

Ua, s;vΨ

s, fs, B2vs, _s

0

gs, τ, B3vτdτ

ds

dη

_t

0

Ut, s;uΨ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

−Ut, s;vΨ

s, fs, B2vs, _s

0

gs, τ, B3vτdτ ds.

≤ Ut,0;uu0−Ut,0;vu0

Ut,0;uB₄⁻¹ _a

0

hutdt−Ut,0;vB₄⁻¹ _a

0

hutdt

Ut,0;vB⁻¹_{4 a}

0

hutdt−Ut,0;vB₄⁻¹ _a

0

hvtdt

_t

0

U t, η;u

ΦW⁻¹

−Ua,0;uu0−Ua,0;uB⁻¹_{4 a}

0

hutdt

− _a

0

Ua, s;uΨ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

ds

−U t, η;v

ΦW⁻¹

−Ua,0;vu0−Ua,0;vB₄⁻¹ _a

0

hvtdt

− _a

0

Ua, s;vΨ

s, fs, B2vs, _s

0

gs, τ, B3vτdτ

ds

dη

_t

0

Ut, s;uΨ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

−Ut, s;vΨ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

Ut, s;vΨ

s, fs, B2us, _s

0

gs, τ, B3uτdτ

−Ut, s;vΨ

s, fs, B2vs, _s

0

gs, τ, B3vτdτ

ds.

≤

Kau0a²Cαk1KaM0Cαk2

maxτ∈I uτ−vτ

2a{M0M₁M₂M0u0aM₀C_αk₁aM₀N₁k5k₃}max

τ∈I uτ−vτ

a{aKN1k5rk₆k₃rk₄ N₂ M₀N₁k5k₃}max

τ∈I uτ−vτ

≤λmax

τ∈I uτ−vτ.

5.10

Therefore, Qis a contraction mapping, and hence there exists a unique fixed pointu ∈ X, such thatQut ut. Any fixed point ofQis a mild solution of5.1onIwhich satisfies ua 0. Thus, system5.1is exactly null controllable onI.

6. Examples

To illustrate the abstract results, we give the following examples.

Example 6.1. Consider the nonlinear integropartial differential equation of fractional order

∂^αux, t

∂t^α

|^q|^≤2m a_q

x, t;b_qux, t

D^q_xux, t Fx, t, w1

_t

0

Gx, t, s, w2sds, 6.1

with nonlocal condition

ux,0 p k1

ckux, tk gx, 6.2

where 0< α≤1, 0≤t₁<· · ·< t_p≤a,x∈Rⁿ,D^q_xD^q_x¹₁· · ·D^q_xⁿ_n,D_xi ∂/∂x_i,q q1, . . . , q_nis ann-dimensional multi-index,|q|q₁· · ·q_n, andw_i,i1,2, is given by

w_ix, t

|^q|^≤2m−1

b_qix, tD^qxux, t

Ω

|^q|^≤2m−1

c_qix,tDy^qu y, t

dy. 6.3

LetL2Rⁿbe the set of all square integrable functions onRⁿ. We denote byC^mRⁿthe set of all continuous real-valued functions defined onRⁿwhich have continuous partial derivatives of order less than or equal tom. ByC^m₀Rⁿ, we denote the set of all functionsf ∈ C^mRⁿ with compact supports. LetH^mRⁿbe the completion ofC^m₀Rⁿwith respect to the norm

f ²

m

|^q|^≤m

Rⁿ

!!!D^q_xfx!!!²dx. 6.4

It is supposed that the following hold.

iThe operatorA−"

|q|≤2maqx, t;bqux, tD^qxis uniformly elliptic onRⁿ. In other words, all the coefficientsa_q,|q| 2m, are continuous and bounded on Rⁿ, and there is a positive numbercsuch that

−1^m1

|^q|^2m e_q

x, t;b_q

ξ^q ≥c|ξ|^2m, 6.5

for allx∈Rⁿand allξ /0,ξ∈Rⁿ,ξ^qξ₁^q1· · ·ξ^q_nⁿ, and|ξ|²ξ₁²· · ·ξ²_n.

iiAll the coefficientsa_q,|q| 2m, satisfy a uniform H ¨older condition onRⁿ. Under these conditions, the operatorAwith domain of definition DA H^2mRⁿgenerates an evolution operator defined onL₂Rⁿ, and it is well known thatH^2mRⁿis dense in X L₂Rⁿand the initial functiongxis an element in Hilbert spaceH^2mRⁿ, see26, page 438. Applying Theorem4.1, this achieves the proof of the existence of mild solutions of the problem6.1,6.2. In addition,

iiiIf the coefficientsb_q,c_q,|q| ≤2m−1 satisfy a uniform H ¨older condition onRⁿand the operatorsFandGsatisfy.

There are numbersL1,L2≥0 andλ1,λ2∈0,1such that

|^q|^≤2m−1

Rⁿ

!!!F

x, t, D^q_xw₁

−F

x, s, D_x^qw^∗₁!!!²dx≤L₁

|t−s|^λ1!!w₁−w^∗₁!!²dx ,

|^q|^≤2m−1

Rⁿ

!!!G

x, t, η, D_x^qw₂

−G

x, s, η, D^q_xw₂!!!²dx≤L₂|t−s|^λ2,

6.6

for allt, s∈I,t, η,s, η ∈Δ, and allx∈Rⁿ. Applying Theorem4.2, we deduce that6.1, 6.2has 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

∂^αux, t

∂t^α ax, t,bux, t∂²ux, t

∂x² ζx, t Υ

t, μ₁t, ux, t, _t

0

μ₂t, s, ux, tds

,

ux,0 p k1

ckux, tk gx, x∈0, π, u0, t uπ, t 0, t∈I,

6.7 where 0< α≤1, 0≤t1 <· · ·< tp≤a, and the functionsax, t,·,bx, tare continuous.

Let us take

XL²0, π, S_r

y∈L²0, π: y ≤r

. 6.8

PutΦμxt ζx, t,x∈0, πwhereμt ζ·, tandζ:0, π×I → 0, πis continuous.

We defineAt,·:X → XbyAt,·wx ax, t,·wwith domainDA {w∈X: w, ware absolutely continuous,w ∈X,w0 wπ 0}. Assume that−At,·generates an evolution systemUt, s,·such that for every positive numbersn1andn2,Ut, s,· ≤n1

andUt, s,·As,· ≤n2.

Also, defineB₁t : DB1 S₁ → X byB1tzx bx, tz, for allz ∈ X and x∈0, π.

Assume that the linear operatorWthat is given by

Wμx _a

0

Ua, s;uζx, sds, x∈0, π, 6.9

has a bounded invertible operatorW⁻¹inL²I, H/kerW.

Let us assume that the nonlinear functionsΥ,μ₁, andμ₂satisfy the following Lipschitz conditions

Υ

t, z₁, y₁

−Υ

t, z₂, y₂ ≤c₁

z1−z₂ y₁−y₂ , μ₁t, x1−μ₁t, x2 ≤c₂x1−x₂,

_t

0

μ2t, s, v1−μ2t, s, v2 ds≤c3v1−v2,

6.10

wherec_j>0,j1,2,3,x_i, y_i, z_i, v_i∈X,i1,2, ands, t∈I.

All the conditions stated in Theorem 5.2 are satisfied. Hence, system6.7is exactly null controllable onI.

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