Approximate Controllability of Impulsive Neutral Functional Differential Equations with State dependent Delay via Fractional Operators

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Approximate controllability of impulsive neutral

functional differential equations with state-dependent

delay via fractional operators

N. Y. Nadaf

Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore-614 114, Tamil Nadu, India.

nadaf [email protected]

M. Mallika Arjunan

Department of Mathematics, C. B. M. College, Kovaipudur, Coimbatore- 641 042, Tamil Nadu, India.

[email protected]

ABSTRACT

In this article, the problem of approximate controllability for non-linear impulsive neutral differential systems with state-dependent delay is studied under the assumption that the corresponding lin-ear control system is approximately controllable. Using Schauder’s fixed point theorem and fractional powers of operators with semi-group theory, sufficient conditions are formulated and proved.

Keywords:

Approximate controllability, Impulsive neutral functional differ-ential equations, Semigroup theory, State-dependent delay, Fixed point..

1. INTRODUCTION

In this paper, we study the approximate controllability of the impul-sive neutral functional differential equation with state-dependent delay in the form:

d

dt[x(t) +g(t, xt)] =Ax(t) +Bu(t) +f(t, xρ(t,xt)),

t∈J= [0, b], (1)

x0=φ∈ B, (2)

∆x(tk) =Ik(xt_k), k= 1,2, ..., m, (3) whereA is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators on a Banach spaceX,B

is a bounded linear operator from a Banach spaceU intoX. The notationxs represents the function defined byxs : (−∞,0] → X, xs(θ) = x(s+θ),belongs to some abstract phase spaceB

described axiomatically andρ:J× B →(−∞, b]is a continuous function. Furtherf, g:J× B →X,andIk(·) :B →Xare appro-priate functions, the controlu(·) ∈ L2(J, U),a Banach space of admissible control functions. Here0< t1 < t2 <· · ·< tm < b are pre-fixed numbers and∆ξ(t)represents the jumpξat timet

which is defined by∆ξ(t) =ξ(t+₎₋_ξ(t−

).

The theory of impulsive differential equations has become an im-portant area of investigation in recent years. Relative to this

the-ory, we only refer the interested reader to [1] and the monographs [2, 3, 4].

The concept of controllability is an important property of a con-trol system which plays an important role in many concon-trol prob-lems such as stabilization of unstable systems by feedback con-trol. Therefore, in recent years controllability problems for vari-ous types of linear and nonlinear deterministic and stochastic dy-namic systems have been studied by many authors, see for in-stance [5, 6, 7, 8, 9, 10, 11]. In particular, approximate control-lable systems are more prevalent and very often approximate con-trollability is completely adequate in applications. There are many papers on the approximate controllability of the various types of nonlinear systems under different conditions, see for instance [12, 13, 14, 15, 17, 18] and references therein. Approximate con-trollability for semilinear deterministic and stochastic control sys-tems can be found in Mahmudov [19]. More recently, Sakthivel et al. [14] derived a set of sufficient conditions for the approxi-mate controllability of nonlinear deterministic and stochastic sys-tems with unbounded delay by using the Schauder’s fixed point theorem.

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systems with state-dependent delay by using the Schauder’s fixxed point theorem and fractional powers of operators with semigroup theory.

The remainder of this paper is organized as follows. In section 2, we give some notations, definitions and basic results about semi-group theory and approximate controllability. In section 3, first we give the existence of solutions for the problem(1)−(3)by us-ing the Schauder’s fixed point theorem, and then sufficient condi-tions for approximate controllability of impulsive neutral differen-tial systems with state-dependent delay are established.

2. PRELIMINARIES

In this section, we recall some notations, definitions and lemmas which are used in this paper.

A functionx : [σ, µ] → X is said to be a normalized piecewise continuous function on[σ, µ]ifxis piecewise continuous and left continuous on[σ, µ]. We denote by P C([σ, µ], X) the space of normalized piecewise continuous functions from[σ, µ]intoX. In particular, we introduce the spaceP Cformed by all normalized piecewise continuous functionsx : [0, b] → X such thatx(·)is continuous att 6= tk, x(t−_k) = x(tk)andx(t+_k)exists, fork = 1, ..., m.It is clear that P Cendowed with the norm ||x||P C = sups∈J||x(s)||is a Banach space.

LetA:D(A)→ X be the infinitesimal generator of an analytic semigroupT(t), t ≥ 0,of bounded linear operators on a Banach spaceX. Let0∈ρ(A), then it is possible to define the fractional power(−A)α_,_for₀ _{< α} _≤ ₁_{, as closed linear operator on its}

domainD(−A)α_{. Furthermore, the subspace}_D(₋_A)α_{is dense in} X, and the expression

||x||α=||(−A)αx||, x∈D(−A)α

defines a norm onD(−A)α_.

Furthermore, we have the following properties appeared in [23]. LEMMA 1. ([23] ). The following properties hold.

(i) If0 < β < α ≤ 1, thenXα ⊂ Xβ and the embedding is

compact whenever the resolvent operator ofAis compact. (ii) For every0< α≤1there existsCα>0such that

k(−A)αR(t)|| ≤Cα

tα, 0< t≤b.

In this work we will employ an axiomatic definition of the phase spaceB, which is similar to those introduced in [23]. Specifically,

Bwill be a linear space of functions mapping(−∞,0]intoX en-dowed with a seminorm|| · ||Band we will assume thatBsatisfies the following axioms:

(A1) Ifx: (−∞, σ+b]→ X, b >0,is such thatxσ ∈ Band x|[σ,σ+b]∈P C([σ, σ+b], X),then for everyt∈[σ, σ+b) the following conditions hold:

(i) xtis inB,

(ii) ||x(t)|| ≤H˜||xt||B,

(iii) ||xt||B ≤ K(t˜ − σ)sup{||x(s)|| : σ ≤ s ≤ t}+

˜

M(t−σ)||xσ||B, whereH >˜ 0is a constant;K,˜ M˜ :

[0,∞)→[1,∞),K˜is continuous,M˜ is locally bounded, andH,˜ K,˜ M˜ are independent ofx(·).

(A2) The spaceBis complete.

For more details about phase space axioms and examples, we refer the reader to [24].

Letxb(φ;u)be the state value of (1)-(3) at terminal time b cor-responding to the controluand the initial valueφ. Introduce the set

<(b, φ) ={xb(φ;u)(0) :u()∈L2(J, U)},

which is called the reachable set of system (1)-(3) at terminal time

b, its closure inXis denoted by(<(b, φ)).

DEFINITION 1. A function x : (−∞, b] → X is called a mild solution of the abstract Cauchy problem (1)-(3) if x0 =

φ; xρ(s,xs) ∈ B for everys ∈ J; the functiont → AT(t−

s)g(s, xs)is integrable on[0, t), for everyt∈[0, b],and the inte-gral equation

x(t) =T(t)[φ(0) +g(0, φ)]−g(t, xt)−

Zt

0

AT(t−s)g(s, xs)ds

+ Z t

0

T(t−s)[Bu(s) +f(s, xρ(s,xs))]ds

+ X

0<tk<t

T(t−tk)Ik(xt_k), t∈J,

is satisfied.

DEFINITION 2. A system (1)-(3) is said to be approximate con-trollable on the interval [0, b] if (<(b, φ)) is dense in X, i.e.,

(<(b, φ)) =X.

It is convenient at this point to define operators

Γb

0=

Zb

0

T(b−s)BB∗T∗(b−s)ds,

R(α,Γb

0) = (αI+ Γb0) −1_,

whereB∗_{denotes the adjoint of}_B_and_T∗_{is the adjoint of}_T_{. It is} straight forward that the operatorΓb

0is a linear bounded operator. Assume the following:

(S1)αR(α,Γb0)→0asα→0

+_{in the strong operator topology.} It is known from [25] that the assumption(S1)holds if and only if the linear system

x0(t) =Ax(t) +Bu(t), t∈J= [0, b], (4)

x0=φ∈ B, (5)

is approximate controllable onJ.

LEMMA 2. ([26],Lemma 2.1] ). Letx : (−∞, b] → X be a function such thatx0=φandx|J∈P C.Then

||xs||B≤(Ma+J0φ)||φ||B+Kasup{||x(θ)||:θ∈[0, max{0, s}]},

s∈ R(ρ−)∪J.

where Ma = sups∈JM(s), Ka = sups∈JK(s), J φ

0 =

sups∈R(ρ−₎Jφ(s).

3. APPROXIMATE CONTROLLABILITY

To establish our results, we list the following assumptions on sys-tem (1)-(3).

(H1) ρ:J× B →(−∞, b]is continuous.

(H2) LetR(ρ−) ={ρ(s, ψ) : (s, ψ)∈J× B, ρ(s, ψ)≤0}.The functiont→φtis well defined fromR(ρ−)intoBand there exists a continuous and bounded functionJφ_{: (ρ}−

)→(0,∞)

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(H3) The functionf:J× B →Xsatisfies the following condi-tions:

(i) Letx : (−∞, b]→ X be such thatx0 = φandx|J ∈ P C. The functiont→f(t, xρ(t,xt))is measurable onJ and the functiont→f(s, xt)is continuous onR(ρ−)∪J

for everys∈J.

(ii) For eacht∈J, the functionf(t,) :B →Xis continu-ous.

(iii) For eachq >0there exists a functionλq ∈L1_{(J, R}+₎ such thatsup||ξ||≤q||f(t, ξ)|| ≤λq(t)for a.e.t∈J, and

lim inf q∞0

Z b

0

λq(t)

q dt= Λ<∞

(H4) There exist constants0< β <1, L1, L2, Lgsuch thatgis

Xβ-valued,(−A)β_g_{is continuous, and}

(i) ||(−A)β_{g(t, x)}_{|| ≤}_L

1||x||B+L2, t∈J, x∈ B, (ii) ||(−A)β_{g(t, x}

1)−(−A)βg(s, x2)|| ≤Lg[|t−s|+||x1−

x2||B], t, s ∈ J, xi ∈ B, i = 1,2with||(−A)−βk =

M0,||(−A)1−βk=N0.

(H5) The mapsIkare continuous and there exists a positive con-stantdksuch that||Ik(ξ)|| ≤dk.

(H6) The semigroupT(t), t >0is compact.

(H7) The functionf :J× B →X is continuous and uniformly bounded and there exists anN >0such that||f(t, ξ)|| ≤N

for all(t, ξ)∈J× B.

Now for convenience, we list the following notations

M= max{||T(t)||: 0≤t≤b}, M1=||B||,

K=||xb||+MH¯||φ||B+M M0(L1||φ||B+L2) +M

m X

k=1

dk,

K∗=MH¯||φ||B+M M0(L1||φ||B+L2) +M2M12

b αK

+M

m X

k=1

dk.

It will be shown that the system (1)-(3) is approximately control-lable, if for allα >0there exists a continuous functionx(·)∈Z

such that

u(t) =B∗T∗(b−t)R(α,Γb

0)p(x(·)),

x(t) =T(t)[φ(0) +g(0, φ)]−g(t, xt)

−

Z t

0

AT(t−s)g(s, xs)ds+ Z t

0

T(t−s)[Bu(s)

+f(s, xρ(s,xs))]ds+

X

0<tk<t

T(t−tk)Ik(xtk), t∈J,

where

p(x(·)) =xb−T(b)[φ(0) +g(0, φ)]−g(b, xb)

−

Z b

0

AT(b−s)g(s, xs)ds

−

Z b

0

T(b−s)f(s, xρ(s,xs))ds−

m X

k=1

T(b−tk)Ik(xt_k).

THEOREM 1. Assume that conditions(H1)−(H7)are satis-fied. Further, suppose that for allα >0

(1 +M2_M2 1

b

α)[(M0+ C1−βbβ

β )L1+ ΛM]Ka<1,

then the system (1)-(3) has a solution onJ.

PROOF. LetZ = {x ∈ P C : x(0) = φ(0)}be the space endowed with the uniform convergence topology. On the spaceZ, consider the setQ = {x∈ Z;||x|| ≤ r}, whereris a positive constant. Forα >0, define the operatorψ:Z →Zbyψx(t) = z(t),where

v(t) =B∗T∗(b−t)R(α,Γb0)p(x(·)),

z(t) =T(t)[φ(0) +g(0, φ)]−g(t,xt)¯

−

Z t

0

AT(t−s)g(s,xs)ds¯

+ Z t

0

T(t−s)[Bv(s) +f(s,xρ¯ (s,¯xs))]ds

+ X

0<tk<t

T(t−tk)Ik(¯xt_k), t∈J,

where

p(x(·)) =xb−T(b)[φ(0) +g(0, φ)] +g(b,xb)¯

+ Z b

0

AT(b−s)g(s,xs)ds¯

−

Z b

0

T(b−s)f(s,xρ¯(s,x¯s))ds−

m X

k=1

T(b−tk)Ik(¯xt_k).

andx¯: (−∞, b]→ X is such thatx¯0 =φandx¯ =xon[0, b]. It will be shown that for allα >0the operatorψ:Z →Zhas a fixed point.

Step 1. For an arbitraryα > 0,there exists anr > 0such that

ψ(Q) ⊂ Q.If this is not true, then there exists anα > 0such that for everyr > 0, there existxr _∈ _Q_and_tr _∈ _J _{such that} r <||ψxr_(tr₎_||_._{For such}_{α >}_0,_{we find that}

r <||ψxr_(tr₎_||

≤ ||T(tr)||[||φ(0)||+||g(0, φ)||] +||g(tr, xr t)||

+ Z tr

0

||AT(tr−s)g(s, xr s)||ds

+ Z tr

0

||T(tr₋_s)_||||_B_||||_v(s)_||_ds

+ Z tr

0

||T(tr₋_s)_||||_{f(s, x}r ρ(s,xr

s)||ds

+ X

0<tk<t

||T(tr₋_t

k)||||Ik(xtk)||

≤MH¯||φ||B+M||(−A)−β||(L1||φ||B+L2)

+||(−A)−β||(L1||xrt)||B+L2)

+ Z tr

0

||(−A)1−β_T_(tr₋_s)_||||₍₋_A)β_{g(s, x}r s)||ds

+ Z tr

0

||T(tr−η)||||B||||B∗||||T∗(b−η)||||R(α,Γb0)||

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+ Z b

0

||(−A)1−βT(b−s)||||(−A)βg(s, xr s)||ds

+ Z b

0

||T(b−s)||||f(s, xr ρ(s,xr

s))||ds

+ m X

k=1

||T(b−tk)||||Ik(¯xt_k)||](η)dη

+ Z tr

0

||T(tr₋_s)_||||_f(s,_x_¯ ρ(s,xr

s))||ds

+ X

0<tk<t

||T(tr−tk)||||Ik(¯xt_k)||

≤MH¯||φ||B+M M0(L1||φ||B+L2) +M0(L1||xrt||B+L2)

+C1−βb β

β (L1||x r

s||B+L2) +M2M12

b

α[||xb||+M ¯ H||φ||B

+M M0(L1||φ||B+L2) +M0(L1||xrb||B+L2)

+C1−βb β

β (L1||x¯ r

s||B+L2) +M

Zb

0

||xr

ρ(s,(xr₎_s₎||Bds

+M

m X

k=1

dk] +M Z tr

0

||xr

ρ(s,(xr₎_s₎||Bds+M

m X

k=1

dk

≤MH¯||φ||B+M M0(L1||φ||B+L2)

+M0(L1((Ma+J0φ)||φ||B+Kar) +L2)

+C1−βb β

β (L1((Ma+J φ

0)||φ||B+Kar) +L2)

+M2_M2 1

b

α[||xb||+MH¯||φ||B+M M0(L1||φ||B+L2) +M0(L1((Ma+J0φ)||φ||B+Kar) +L2)

+C1−βb β

β (L1((Ma+J φ

0)||φ||B+Kar) +L2)

+M

Z b

0

||xr

ρ(s,(xr₎_s₎||Bds+M

m X

k=1

dk]

+M

Z tr

0

||xr

ρ(s,(xr)s)||Bds+M

m X

k=1

dk

For anyx∈Q, it follows from Lemma 2 that

kxr

ρ(s,(xr₎_s₎|| ≤(Ma+J₀φ)||φ||B+Kar=r∗, wherer∗is a positive constant. Hence we obtain

≤MH¯||φ||B+M M0(L1||φ||B+L2) +M0(L1r∗+L2)

+C1−βb β

β (L1r

∗

+L2) +M2M12

b

α[||xb||+M ¯ H||φ||B

+M M0(L1||φ||B+L2) +M0(L1r∗+L2)

+C1−βb β

β (L1r

∗

+L2) +M

Z b

0

λr∗(s)ds+M m X

k=1

dk]

+M

Z tr

0

λr∗(s)ds+M m X

k=1

dk

≤MH¯||φ||B+M M0(L1||φ||B+L2) +M0(L1r∗+L2)

+C1−βb β

β (L1r

∗

+L2) +M2M12

b

α[K+M0(L1r

∗

+L2)

+C1−βb β

β (L1r

∗

+L2) +M

Zb

0

λr∗(s)ds+M m X

k=1

dk]

+M

Ztr

0

λr∗(s)ds+M m X

k=1

dk

where K = ||xb|| +MH¯||φ||B + M M0(L1||φ||B +L2) +

MPm

k=1dk

r≤(M0+

C1−βbβ β )(L1r

∗

+L2)

+M2_M2 1

b α(M0+

C1−βbβ

β )(L1r

∗

+L2)

+M

Zb

0

λr∗(s)ds+M2M₁2b αM

Z b

0

λr∗(s)ds+K∗,

whereK∗=MH¯||φ||B+M M0(L1||φ||B+L2)

+M2M12

b αK+M

m X

k=1

dk

r≤(1 +M2_M2 1

b α)(M0+

C1−βbβ β )(L1r

∗

+L2)

+ (1 +M2_M2 1

b α)M

Zb

0

λr∗(s)ds+K∗

≤(1 +M2_M2 1

b

α)[(M0+ C1−βbβ

β )(L1r

∗

+L2)

+M

Zb

0

λr∗(s)ds] +K∗.

We note thatK∗is independent ofrandr∗→ ∞asr→ ∞.Now

lim inf r→∞

Z b

0

λr∗(s)

r ds= lim infr→∞

Zb

0

(λr∗(s) r∗ .

r∗

r)ds= ΛKa.

Hence we have forα >0, 1≤(1 +M2M12

b

α)[(M0+ C1−βbβ

β )L1+ ΛM]Ka,

which is contradiction to our assumption. Thusα >0, there exists anr >0such thatψmapsQinto itself.

Step 2. For each α > 0, the operator ψ maps Qinto a rela-tively compact subset ofQ.First, we prove that the setV(t) = {ψx(t);x∈Q}is relatively compact inXfor everyt∈J.The caset= 0is obvious. So, lettbe a fixed real number, and letτbe a given real number satisfying0< τ < t≤b,we define

(ψτx)(t) =T(t)[φ(0) +g(0, φ)]−g(t,xt)¯

−

Z t−τ

0

AT(t−s)g(s,x¯s)ds

+ Z t−τ

0

T(t−s)Bv(s)ds

+ Z t−τ

0

T(t−s)f(s,xρ¯(s,x¯s))ds

+ X

0<tk<t

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The setVτ(t) ={(ψτ_{x)(t) :}_x(_·₎_∈_Q_}_{is relatively compact set}

inX. That is, a finite set{yi,1≤i≤n}inXexists such that

Vτ(t)⊂ n [

i=1

¯

W(yi, /2),

whereW¯(yi, /2)is an open ball inXwith center atyiand radius

/2.

On the other hand,

||(ψx)(t)−(ψτ_x)(t)_||

≤ ||

Z t

t−τ

AT(t−s)g(s,¯xs)ds+ Z t

t−τ

T(t−s)[Bv(s)

+f(s,x¯ρ(s,¯xs))]dsk

≤

Z t

t−τ

||AT(t−s)g(s,xs)¯ ||ds+ Z t

t−τ

||T(t−η)||

(×)||B||kB∗kkT∗(b−η)kkR(α,Γb0)k[||xb||+||T(b)||[||φ(0)

+g(0, φ)||] +||g(b,xb)¯ ||+ Z b

0

||AT(b−s)g(s,xs)¯ ||ds

+ Z b

0

||T(b−s)||||f(s,xρ¯(s,x¯s))||ds

+ m X

k=1

||T(b−tk)||||Ik(¯xtk)||]ηdη

+ Z t

t−τ

||T(t−s)||||f(s,xρ¯ (s,¯xs))||ds

≤

Z t

t−τ C1−β

(t−s)1−β(L1||¯xs||B+L2)ds

+M2_M2 1

1 α

Z t

t−τ

[||xb||+MH¯||φ||B

+M M0(L1||φ||B+L2)

+M0(L1||xb¯ ||B+L2) +

C1−βbβ

β (L1||xs¯||B+L2)

+M

Z b

0

||xr

(s,x¯s)||Bds+M

m X

k=1

dk]ds

+M

Z t

t−τ ||xr

ρ(s,x¯s)||Bds

≤ C1−βτ β

β (L1r

∗

+L2) +M2M12

1 α

Z t

t−τ [||xb||

+MH¯||φ||B

+M M0(L1||φ||B+L2) +M0(L1r∗+L2)

+C1−βb β

β (L1r

∗

+L2)

+M

Z b

0

λr∗(s)ds+M m X

k=1

dk]ds+M Zt

t−τ

λr∗(s)ds

≤ C1−βτ β

β (L1r

∗

+L2) +M2M12

1 α

Z t

t−τ [K

+M0(L1r∗+L2)

+C1−βb β

β (L1r

∗₊_L 2) +M

Z b

0

λr∗(s)ds]ds

+M

Z t

t−τ

λr∗(s)ds

≤ C1−βτ β

β (L1r

∗

+L2) +M2M12

1

α[K+M0(L1r

∗

+L2) +

C1−βbβ β (L1r

∗

+L2)

+M

Z b

0

λr∗(s)ds]τ+M

Z t

t−τ

λr∗(s)ds

≤

2.

Consequently,

V(t)⊂ n [

i=1

¯ W(yi, ).

Hence for eacht∈[0, b], V(t)is relatively compact inX.

Step 3.ψmapsQinto equicontinuous family. We now show that the setV ={(ψx)(·)|x(·) ∈Q}is equicontinuous on[0, b].For

0< t1< t2≤b,we have

||z(t1)−z(t2)||

≤ ||T(t1)−T(t2)||||φ(0) +g(0, φ)||

+||g(t1,xt¯1)−g(t2,xt¯2)||

+ Z t1

0

||T(t2−s)−T(t1−s)||||Ag(s,xs)¯ ||ds

+ Z t2

t1

||AT(t2−s)||||g(s,xs)¯ ||ds

+ Z t1

0

||T(t2−s)−T(t1−s)||||B||||v(s)||ds

+ Z t2

t1

||T(t2−s)||||B||||v(s)||ds

+ Z t1

0

||T(t2−s)−T(t1−s)||||f(s,xρ¯ (s,¯xs))||ds

+ Z t2

t1

||AT(t2−s)||||f(s,xρ¯ (s,x¯s))||ds

+ X

0<tk<t1

||T(t2−tk)−T(t1−tk)||||Ik(¯xtk)||

+ X

t1<tk<t2

||T(t2−tk)||||Ik(¯xtk)||

≤ ||T(t1)−T(t2)||||φ(0) +g(0, φ)||

+Lg[(|t1−t2|+||xt¯1−¯xt2||B)]

+ Z t1

0

||T(t2−s)−T(t1−s)||N0(L1||xs¯ ||B+L2)ds

+ Z t2

t1

M N0(L1||xs¯ ||B+L2)ds

+M M2 1

1 α

Z t1

0

||T(t2−η)−T(t1−η)||[||xb||

(6)

+M0(L1||xb¯ ||B+L2) +

C1−βbβ

β (L1||xs¯ ||B+L2)

+M

Z b

0

||xρ¯ (s,x¯s)||Bds+M

m X

k=1

dk]dη

+M2_M2 1

1 α

Z t2

t1

[||xb||+MH¯||φ||B

+M M0(L1||φ||B+L2) +M0(L1||xs¯ ||B+L2)

+C1−βb β

β (L1||xs¯ ||B+L2) +M Z b

0

||xρ¯ (s,¯xs)||Bds

+M

m X

k=1

dk]dη

+ Z t1

0

||T(t2−s)−T(t1−s)||||¯xρ(s,x¯s)||Bds

+M

Z t2

t1

||¯xρ(s,x¯s)||Bds+M

X

t1<tk<t2 dk

+ X

0<tk<t1

||T(t2−tk)−T(t1−tk)||dk

≤ ||T(t1)−T(t2)||||φ(0) +g(0, φ)||

+Lg[(|t1−t2|+||xt1¯ −xt2¯ ||B)]

+ Z t1

0

||T(t2−s)−T(t1−s)||N0(L1r ∗

+L2)ds

+ Z t2

t1

M N0(L1r∗+L2)ds

+M M12

1 α

Zt1

0

||T(t2−η)−T(t1−η)||[||xb||

+MH¯||φ||B+M M0(L1||φ||B+L2)

+M0(L1r∗+L2) +

C1−βbβ β (L1r

∗

+L2)

+M

Z b

0

λr∗(s)ds+M m X

k=1

dk]dη

+M2_M2 1

1 α

Z t2

t1

[||xb||+MH¯||φ||B+M M0(L1||φ||B+L2)

+M0(L1r∗+L2) +

C1−βbβ β (L1r

∗

+L2)

+M

Z b

0

λr∗(s)ds+M m X

k=1

dk]dη

+ Z t1

0

||T(t2−s)−T(t1−s)||λr∗(s)ds

+M

Z t2

t1

λr∗(s)ds+

X

0<tk<t1

||T(t2−tk)−T(t1−tk)||dk

+M X

t1<tk<t2 dk.

The right hand side does not depend on any particular choices of

x(·)∈Qand tends to zero ast1−t2→0, since the compactness ofT(t)fort > 0implies the continuity in the uniform operator

topology. This proves thatV is right equicontinuous att∈(0, b).

The other cases’ right equicontinuity at zero and left equicontinuity att ∈ (0, b]are similar. ThusψmapsQinto an equicontinuous family of functions.

Step 4. The mapψis continuous onQ. Let{xn_}

n∈Nbe a sequence inQandx∈Qsuch thatxn→xin P C.From(H2)−(H6),we have

(i) Ik, k= 1,2, ..., nis continuous. (ii) g(t,x¯n

t)→g(t,¯xt)for eacht∈Jand since ||g(t,x¯n

t)−g(t,xt)¯ ||<2M0Lgr∗. (iii) Ag(s,¯xn

t)→Ag(s,¯xt)for eachs∈Jand since Ag(s,x¯n

t)−Ag(s,xt)¯ ||<2N0Lgr∗. (iv) f(s,x¯n

ρ(s,¯xs))→f(s,x¯ρ(s,¯xs))for eachs∈Jand since

||f(s,¯xnρ(s,x¯s))−f(s,xρ¯ (s,¯xs))||<2λr∗(s).

From the Lebesgue dominated convergence theorem, we obtain

||ψzn−ψz|| ≤ ||g(t,¯xn

t)−g(t,xt)¯ ||

+ Z t

0

||AT(t−s)g(s,x¯ns)−AT(t−s)g(s,¯xs)||ds

+ Z t

0

||T(t−η)||||B||||B∗||||T∗(b−η)||||R(α,Γb

0)||

(×)h||g(b,x¯nb)−g(b,¯xb)||

+ Z b

0

||T(b−s)||||Ag(s,x¯ns)−Ag(s,¯xs)||ds

+ Z b

0

||T(b−s)||||f(s,¯xnρ(s,x¯s))−f(s,xρ¯ (s,¯xs))||ds

+ X

0<tk<t

||T(b−tk)||||Ik(¯xn

tk)−Ik(¯xtk)||

i (η)dη

+ Z t

0

||T(t−s)||||f(s,x¯nρ(s,¯xs))−f(s,¯xρ(s,x¯s))||ds

+ X

0<tk<t

||T(t−tk)||||Ik(¯xtnk)−Ik(¯xtk)||

≤ ||g(t,¯xn

t)−g(t,xt)¯ ||

+M

Z t

0

||Ag(s,x¯n

s)−Ag(s,xs)¯ ||ds

+M M2 1

1 α

Z b

0

[||g(b,¯xn

b)−g(b,xb)¯ ||

+M

Z t

0

||Ag(s,x¯ns)−Ag(s,xs)¯ ||ds

+M

Z b

0

||f(s,x¯nρ(s,¯xs))−f(s,xρ¯(s,x¯s))||ds

+M X

0<t_k<t

||Ik(¯xntk)−Ik(¯xtk)||]dη

+M

Z t

0

||f(s,¯xn

ρ(s,x¯s))−f(s,xρ¯ (s,¯xs))||ds

+M X

0<tk<t

||Ik(¯xn

(7)

→0asn→ ∞. Which shows the continuity ofψ.

Hence all the conditions of Schauder’s fixed point theorem[27]are satisfied, and consequently the operatorψhas a fixed point inQ. Thus the problem(1)−(3)has a solution onJ.

THEOREM 2. Assume that linear system(4)−(5)is approxi-mately controllable onJ. If the conditions(H4),(H6),(H7)and

(S1)are satisfied then the system(1)−(3)is approximately con-trollable.

PROOF. Letxα(·)be a fixed point ofψinQany fixed point of

ψis a mild solution of(1)−(3)under the control

uα(t) =B∗T∗(b−t)R(α,Γb

0)p(xα)

and satisfies the inequality

xα(b) =xb+αR(α,Γb0)p(xα). (6) By the conditions(H4)and(H7),

Zb

0

||f(s,x¯α ρ(s,x¯α

s)||

2_ds_≤_bN2_,

Z b

0

||(−A)β_g(s,_x_¯α s)||

2_ds_≤_b_N_¯2_.

Consequently, the sequence {f(s,¯xα

ρ(s,x¯αs))} and

{(−A)β_g(s,_x_¯α

s)}is bounded inL2(J, X). Then there is a subse-quence, still denoted by{f(s,x¯α

ρ(s,x¯α

s))}and {(−A)

β_g(s,_x_¯α s)}

that weakly converges to, say, f(s) and g(s) in L2(J, X) respectively. Define

p(x(·)) =xb−T(b)[φ(0) +g(0, φ)] +g(b,xb)¯

+ Z b

0

AT(b−s)g(s,x¯αs)ds

−

Z b

0

T(b−s)f(s,x¯α ρ(s,x¯α

s))ds−

m X

k=1

T(b−tk)Ik(¯xt_k), w=xb−T(b)[φ(0) +g(0, φ)] +g(b,xb)¯

+ Z b

0

AT(b−s)g(s)ds−

Z b

0

T(b−s)f(s)ds

− m X

k=1

T(b−tk)Ik(¯xt_k).

It follows by the compactness of the operators l(·) →

R·

0(−A)

−β_T₍_{· −}_s)l(s)ds _: _L

2(J, X) → C(J, X)and l(·) →

R·

0(−A)T(· −s)l(s)ds:L2(J, X)→C(J, X)we obtain that

||p(xα)−w||

=||

Z b

0

(−A)−βT(b−s)[(−A)βg(s,x¯αs)−g(s)]ds||

+||

Zb

0

T(b−s)[f(s,x¯αρ(s,x¯α

s))−f(s)]ds|| →0

as α→0+_.

Then from(6), we obtain

||xα(b)−xb||

=||αR(α,Γb0)p(xα)||

=||αR(α,Γb

0)(p(xα)−w+w)||

≤ ||αR(α,Γb

0)(w)||+||αR(α,Γ

b

0)||||(p(xα)−w)||

≤ ||αR(α,Γb0)(w)||+||(p(xα)−w)|| →0

asα→0+_{. This completes the approximate controllability of}₍₁₎₋

(3).

4. CONCLUSION

Sufficient conditions for approximate controllability results are es-tablished for a class of impulsive neutral functional differential equation with state-dependent delay. The proof of the main rem is based on the application of the Schauder’s fixed point theo-rem and fractional powers of operators with semigroup theory.

5. REFERENCES

[1] Y. V. Rogovchenko, Impulsive evolution systems: Main results and new trends, Dynam. Contin.,Discrete Impuls. Systems, 3 (1997), 57-88.

[2] M. Benchohra, J. Henderson and S. K. Ntouyas, Impul-sive Differential Equations and Inclusions, Series Con-temporary Mathematics and Its Applications, Vol. 2, Hindawi Publ. Corp.,2006.

[3] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Sci-entific Pub. Co., Singapore, 1989.

[4] A. Samoilenko and N. Peresyuk, Differential Equations with Impusive Effectes, World Scientific Pub. Co., Sin-gapore, 1995.

[5] M. Benchohra, A. Ouahab, Controllability results for functional semilinear differential inclusions in Frechet spaces, Nonlinear Anal. TMA, 61(2005), 405-423. [6] N. Abada, M. Benchohra and H. Hammouche, Existence

and controllability results for nondensely de-ned impul-sive semilinear functional differential inclusions, J. Dif-fer. Equ., 246 (2009), 3834-3863.

[7] J. Klamka, Constrained controllability of semilinear sys-tems with delays, Nonlinear Dynam., 56 (2009), 169-177.

[8] J. Klamka, Constrained controllability of semilinear sys-tems with delayed controls, Bull. Pol. Ac. Tech., 56 (2008), 333-337.

[9] L. Gorniewicz, S. K. Ntouyasand D. O’Regan, Control-lability of semilinear differential equations and inclu-sions via semigroup theory in Banach spaces, Rep. Math. Phys., 56 (2005), 437-470.

[10] L. Gorniewicz, S. K. Ntouyas and D. O’Regan, Exis-tence and controllability results for first and second order functional semilinear differential inclusions with nonlo-cal conditions, Numer. Funct. Anal. Optim., 28 (2007), 53-82.

(8)

[12] N. I. Mahmudov, Approximate controllability of evolu-tion systems with nonlocal Condievolu-tions, Nonlinear Anal. TMA, 68 (2008), 536-546.

[13] R. Sakthivel, Y. Ren and N. I. Mahmudov, Approxi-mate controllability of second order stochastic differen-tial equations with impulsive e-ects, Modern Phys. Lett. B 24 (2010), 1559-1572.

[14] R. Sakthivel, J. Juan, Nieto, N. I. Mahmudov, Ap-proximate controllability of nonlinear deterministic and stochastic systems with unbounded delay, Taiwan J. Math., 14 (2010), 1777-1797.

[15] R. Sakthivel, Approximate controllability of impulsive stochastic evolution Equations, Funkcialaj Ekvacioj, 52 (2009), 381-393.

[16] J. Klamka, Constrained approximate controllability, IEEE. T. Automat. Contr., 45 (2000), 1745-1749. [17] R. Sakthivel and E. R. Anandhi, Approximate

control-lability of impulsive differential equations with state-dependent delay, International Journal of Control, 82, 2 (2010), 387-393.

[18] B. Radhakrishnan and K. Balachandran, Controllability of impulsive neutral functional evolution integrodiffer-ential systems with infinite delay, Nonlinear Analysis, HS, 5 (2011), 655-670.

[19] N. I. Mahmudov, Approximate controllability of semi-linear deterministic and stochastic evolution equations in abstract spaces, SIAM J, Control Optim., 42 (2003), 1604-1622.

[20] Y. -K. Chang and W. S. Li, Solvability for impulsive neu-tral integrodifferential equations with state-dependent delay via fractional operators, J. Optim. Theory Appl., 144 (2010), 445-459.

[21] M. Mallika Arjunan and V. Kavitha, Existence results for impulsive neutral functional differential equations with state-dependent delay, Electronic Journal of Qualitative Theory of Differential equations, 26 (2009), 1-13. [22] A. Pazy, Semigroups of Linear Operators and

Appli-cations to Partial Differential Equations, Springer, New York, (1983).

[23] Y. Heno, S. Murakami and T. Naito, Functional differ-ential equations with infinite delay, Lecturer Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. [24] E. Hernandez, R. Sakthivel and S. Tanaka, Existence

results for impulsive evolution equations with state-dependent delay, Electro. J. Differential Equations, 28 (2008), 1-11.

[25] A. E. Bashirov and N. I. Mahmudov, On concept of con-trollability for linear deterministic and stochastic sys-tems, SIAM, J. Control Optim., 37 (1999), 1808-1821. [26] E. Hernandez, A. Prokopczyk and L. A. Ladeira, A note

on partial functional differential equations with state-dependent delay, Nonlinear Anal. RWA, 7 (2006), 510-519.