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Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)

Volume 4 (2009), 15 – 39

CONTROLLABILITY RESULTS FOR SEMILINEAR

FUNCTIONAL AND NEUTRAL FUNCTIONAL

EVOLUTION EQUATIONS WITH INFINITE DELAY

Selma Baghli, Mouffak Benchohra and Khalil Ezzinbi

Abstract. In this paper sufficient conditions are given ensuring the controllability of mild

solutions defined on a bounded interval for two classes of first order semilinear functional and

neutral functional differential equations involving evolution operators when the delay is infinite

using the nonlinear alternative of Leray-Schauder type.

1 Introduction

Controllability of mild solutions defined on a bounded interval J := [0,T] is con-

sidered, in this paper, for two classes of first order partial and neutral functional

differential evolution equations with infinite delay in a real Banach space (E,| · |).

Firstly, in Section 3, we study the partial functional differential evolution equa-

tion with infinite delay of the form

y?(t) = A(t)y(t) + Cu(t) + f(t,yt),

a.e. t ∈ J

(1.1)

y0= φ ∈ B,

(1.2)

where f : J × B → E and φ ∈ B are given functions, the control function u(.) is

given in L2(J;E), the Banach space of admissible control function with E is a real

separable Banach space with the norm |·|, C is a bounded linear operator from E into

E and {A(t)}0≤t≤T is a family of linear closed (not necessarily bounded) operators

from E into E that generate an evolution system of operators {U(t,s)}(t,s)∈J×Jfor

0 ≤ s ≤ t ≤ T. To study the system (1.1) − (1.2), we assume that the histories

yt: (−∞,0] → E, yt(θ) = y(t + θ) belong to some abstract phase space B, to be

specified later. We consider in Section 4, the neutral functional differential evolution

equation with infinite delay of the form

d

dt[y(t) − g(t,yt)] = A(t)y(t) + Cu(t) + f(t,yt), a.e. t ∈ J

(1.3)

2000 Mathematics Subject Classification: 34G20; 34K40; 93B05.

Keywords: controllability; existence; semilinear functional; neutral functional differential evolu-

tion equations; mild solution; fixed-point; evolution system; infinite delay.

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S. Baghli, M. Benchohra and K. Ezzinbi

y0= φ ∈ B,

(1.4)

where A(·), f, u, C and φ are as in problem (1.1)-(1.2) and g : J ×B → E is a given

function. Finally in Section 5, an example is given to demonstrate the results.

Partial functional and neutral functional differential equations arise in many

areas of applied mathematics, we refer the reader to the book by Hale and Verduyn

Lunel [31], Kolmanovskii and Myshkis [37] and Wu [49].

In the literature devoted to equations with finite delay, the phase space is much

of time the space of all continuous functions on [−r,0], r > 0, endowed with the

uniform norm topology. When the delay is infinite, the notion of the phase space

B plays an important role in the study of both qualitative and quantitative theory,

a usual choice is a normed space satisfying suitable axioms, which was introduced

by Hale and Kato [30], see also Kappel and Schappacher [36] and Schumacher [47].

For detailed discussion on this topic, we refer the reader to the book by Hino et al.

[35], and the paper by Corduneanu and Lakshmikantham [20].

Controllability problem of linear and nonlinear systems represented by ODEs

in finite dimensional space has been extensively studied. Several authors have ex-

tended the controllability concept to infinite dimensional systems in Banach space

with unbounded operators (see [19, 43, 44]). More details and results can be found

in the monographs [18, 21, 42, 51]. Triggiani [48] established sufficient conditions for

controllability of linear and nonlinear systems in Banach space. Exact controllabil-

ity of abstract semilinear equations has been studied by Lasiecka and Triggiani [39].

Quinn and Carmichael [46] have shown that the controllability problem can be con-

verted into a fixed point problem. By means of a fixed point theorem Kwun et al [38]

considered the controllability and observability of a class of delay Volterra systems.

Fu in [25, 26] studies the controllability on a bounded interval of a class of neutral

functional differential equations. Fu and Ezzinbi [27] considered the existence of mild

and classical solutions for a class of neutral partial functional differential equations

with nonlocal conditions, Balachandran and Dauer have considered various classes

of first and second order semilinear ordinary, functional and neutral functional dif-

ferential equations on Banach spaces in [10]. By means of fixed point arguments,

Benchohra et al. have studied many classes of functional differential equations and

inclusions and proposed some controllability results in [6, 12, 13, 14, 15, 16, 17]. See

also the works by Gatsori [28] and Li et al. [40, 41, 42]. Adimy et al [1, 2, 3] studied

partial functional and neutral functional differential equations with infinite delay.

Belmekki et al [11] studied partial perturbed functional and neutral functional dif-

ferential equations with infinite delay. Ezzinbi [23] studied the existence of mild

solutions for functional partial differential equations with infinite delay. Henriquez

[32] and Hernandez [33, 34] considered the existence and regularity of solutions to

functional and neutral functional differential equations with unbounded delay.

Recently Baghli and Benchohra considered in [7] a class of partial functional

evolution equation and in [8] a class of neutral functional evolution equations on

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a semiinfinite real interval and with a bounded delay. Extension of these works is

given in [9] when the delay is infinite.

In this paper, we investigate the controllability of mild solutions of the previous

evolution problems studied in [7, 8, 9] for the functional differential evolution prob-

lem (1.1)-(1.2) and the neutral case (1.3)-(1.4) on the finite interval J. Sufficient

conditions are establish here to get the controllability of mild solutions which are

fixed points of appropriate corresponding operators using the nonlinear alternative

of Leray-Schauder type (see [29]).

2 Preliminaries

We introduce notations, definitions and theorems which are used throughout this

paper.

Let C(J;E) be the Banach space of continuous functions with the norm

?y?∞= sup{|y(t)| : 0 ≤ t ≤ T}.

and B(E) be the space of all bounded linear operators from E into E, with the norm

?N?B(E)= sup { |N(y)| : |y| = 1 }.

A measurable function y : J → E is Bochner integrable if and only if |y| is

Lebesgue integrable. (For the Bochner integral properties, see Yosida [50] for in-

stance).

Let L1(J;E) be the Banach space of measurable functions y : J → E which are

Bochner integrable normed by

?T

Consider the following space

?y?L1 =

0

|y(t)| dt.

BT= {y : (−∞,T] → E : y|J∈ C(J;E), y0∈ B},

where y|Jis the restriction of y to J.

In this paper, we will employ an axiomatic definition of the phase space B in-

troduced by Hale and Kato in [30] and follow the terminology used in [35]. Thus,

(B,? · ?B) will be a seminormed linear space of functions mapping (−∞,0] into E,

and satisfying the following axioms :

(A1) If y : (−∞,T] → E, is continuous on J and y0∈ B, then for every t ∈ J the

following conditions hold :

(i) yt∈ B ;

(ii) There exists a positive constant H such that |y(t)| ≤ H?yt?B;

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S. Baghli, M. Benchohra and K. Ezzinbi

(iii) There exist two functions K(·),M(·) : R+→ R+independent of y(t) with

K continuous and M locally bounded such that :

?yt?B≤ K(t)sup{ |y(s)| : 0 ≤ s ≤ t} + M(t)?y0?B.

Denote KT= sup{K(t) : t ∈ J} and MT= sup{M(t) : t ∈ J}.

(A2) For the function y(.) in (A1), ytis a B−valued continuous function on J.

(A3) The space B is complete.

Remark 1.

1. Condition (ii) in (A1) is equivalent to |φ(0)| ≤ H?φ?Bfor every φ ∈ B.

2. Since ? · ?B is a seminorm, two elements φ,ψ ∈ B can verify ?φ − ψ?B= 0

without necessarily φ(θ) = ψ(θ) for all θ ≤ 0.

3. From the equivalence of (ii), we can see that for all φ,ψ ∈ B such that ?φ −

ψ?B= 0. This implies necessarily that φ(0) = ψ(0).

Hereafter are some examples of phase spaces. For other details we refer, for

instance to the book by Hino et al [35].

Example 2. Let the spaces

BC

the space of bounded continuous functions defined from (−∞,0] to E;

BUC the space of bounded uniformly continuous funct. defined from (−∞,0] to E;

?

?

?φ? = sup{|φ(θ)| : θ ≤ 0}.

C∞

:=

φ ∈ BC : lim

θ→−∞φ(θ) exist in E

?

;

C0

:=

φ ∈ BC : lim

θ→−∞φ(θ) = 0

?

, endowed with the uniform norm

We have that the spaces BUC, C∞and C0satisfy conditions (A1) − (A3). BC

satisfies (A1),(A3) but (A2) is not satisfied.

Example 3. Let g be a positive continuous function on (−∞,0]. We define :

?

Cg :=

φ ∈ C((−∞,0];E) :φ(θ)

g(θ)is bounded on (−∞,0]

?

;

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C0

g:=

?

φ ∈ Cg: lim

θ→−∞

φ(θ)

g(θ)= 0

?

, endowed with the uniform norm

?φ? = sup

?|φ(θ)|

g(θ)

: θ ≤ 0

?

.

We consider the following condition on the function g.

?g(t + θ)

(g1) For all a > 0,

sup

0≤t≤asup

g(θ)

: −∞ < θ ≤ −t

?

< ∞.

Then we have that the spaces Cg and C0

conditions (A1) and (A2) if g1holds.

gsatisfy conditions (A3). They satisfy

Example 4. For any real constant γ, we define the functional space Cγby

?

endowed with the following norm

Cγ:=

φ ∈ C((−∞,0];E) : lim

θ→−∞eγθφ(θ) exist in E

?

?φ? = sup{eγθ|φ(θ)| : θ ≤ 0}.

Then in the space Cγthe axioms (A1)-(A3) are satisfied.

Definition 5. A function f : J ×B → E is said to be an L1-Carath´ eodory function

if it satisfies :

(i) for each t ∈ J the function f(t,.) : B → E is continuous ;

(ii) for each y ∈ B the function f(.,y) : J → E is measurable ;

(iii) for every positive integer k there exists hk∈ L1(J;R+) such that

|f(t,y)| ≤ hk(t) for all ?y?B≤ k

and almost each t ∈ J.

In what follows, we assume that {A(t)}t≥0 is a family of closed densely de-

fined linear unbounded operators on the Banach space E and with domain D(A(t))

independent of t.

Definition 6. A family of bounded linear operators {U(t,s)}(t,s)∈∆: U(t,s) : E → E

for (t,s) ∈ ∆ := {(t,s) ∈ J × J : 0 ≤ s ≤ t ≤ T} is called an evolution system if the

following properties are satisfied :

1. U(t,t) = I where I is the identity operator in E,

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S. Baghli, M. Benchohra and K. Ezzinbi

2. U(t,s) U(s,τ) = U(t,τ) for 0 ≤ τ ≤ s ≤ t ≤ T,

3. U(t,s) ∈ B(E) the space of bounded linear operators on E, where for every

(t,s) ∈ ∆ and for each y ∈ E, the mapping (t,s) → U(t,s) y is continuous.

More details on evolution systems and their properties could be found on the

books of Ahmed [4], Engel and Nagel [22] and Pazy [45].

Our results will be based on the following well known nonlinear alternative of

Leray-Schauder type.

Theorem 7. (Nonlinear Alternative of Leray-Schauder Type, [29]). Let X be a

Banach space, Y a closed, convex subset of E, U an open subset of Y and 0 ∈ X.

Suppose that N : U → Y is a continuous, compact map. Then either,

(C1) N has a fixed point in U; or

(C2) There exists λ ∈ (0,1) and x ∈ ∂U (the boundary of U in Y ) with x = λ N(x).

3 Semilinear Evolution Equations

Before stating and proving the main result, we give first the definition of mild

solution of problem (1.1)-(1.2).

Definition 8. We say that the function y(·) : R → E is a mild solution of (1.1)-

(1.2) if y(t) = φ(t) for all t ∈ (−∞,0] and y satisfies the following integral equation

?t

Definition 9. The problem (1.1)-(1.2) is said to be controllable on the interval J if

for every initial function φ ∈ B and y1∈ E there exists a control u ∈ L2(J;E) such

that the mild solution y(·) of (1.1)-(1.2) satisfies y(T) = y1.

We will need to introduce the following hypotheses which are assumed hereafter:

y(t) = U(t,0)φ(0)+

0

U(t,s)Cu(s)ds+

?t

0

U(t,s)f(s,ys)ds for each t ∈ J. (3.1)

(H1) U(t,s) is compact for t − s > 0 and there exists a constant?

?U(t,s)?B(E)≤?

(H2) There exists a function p ∈ L1(J;R+) and a continuous nondecreasing function

ψ : R+→ (0,∞) and such that :

|f(t,u)| ≤ p(t) ψ(?u?B) for a.e. t ∈ J and each u ∈ B.

M ≥ 1 such that :

M

for every (t,s) ∈ ∆.

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21

(H3) The linear operator W : L2(J;E) → E is defined by

Wu =

?T

0

U(T,s)Cu(s)ds,

has an induced invertible operator˜ W−1which takes values in L2(J;E)/kerW

and there exists positive constants?

?C? ≤?

Remark 10. For the construction of W and˜ W−1see the paper by Carmichael and

Quinn [46].

M and?

and

M1such that :

M

?˜ W−1? ≤?

M1.

Theorem 11. Suppose that hypotheses (H1)-(H3) are satisfied and moreover there

exists a constant M?> 0 such that

M?

β + KT?

?

M

??

?

M?

M?

M1T + 1

?

ψ(M?) ?p?L1

> 1,

(3.2)

with

β = β(φ,y1) =

KT?

MH

1 +?

M?

M?

M1T

?

+ MT

?

?φ?B+ KT?

M?

M?

M1T|y1|.

Then the problem (1.1)-(1.2) is controllable on (−∞,T].

Proof. Transform the problem (1.1)-(1.2) into a fixed-point problem. Consider

the operator N : BT→ BT defined by :

Using assumption (H3), for arbitrary function y(·), we define the control

?

Noting that, we have

N(y)(t) =

φ(t),

if t ∈ (−∞,0];

U(t,0) φ(0) +

?t

?t

0

U(t,s) C uy(s) ds

+

0

U(t,s) f(s,ys) ds,

if t ∈ J.

(3.3)

uy(t) =˜ W−1

y1− U(T,0) φ(0) −

?T

0

U(T,s) f(s,ys) ds

?

(t).

|uy(t)|≤?˜ W−1?

?

|y1| + ?U(t,0)?B(E)|φ(0)| +

?T

0

?U(T,τ)?B(E)|f(τ,yτ)|dτ

?

.

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S. Baghli, M. Benchohra and K. Ezzinbi

From (H2), we get

|uy(t)|≤

?

?

M1

?

?

|y1| +?

|y1| +?

MH?φ?B+?

MH?φ?B+?

M

?T

?T

0

|f(τ,yτ)|dτ

?

≤

M1

M

0

p(τ) ψ(?yτ?B) dτ

?

.

(3.4)

Clearly, fixed points of the operator N are mild solutions of the problem (1.1)-(1.2).

For φ ∈ B, we will define the function x(.) : R → E by

?φ(t),

U(t,0) φ(0),

x(t) =

if t ∈ (−∞,0];

if t ∈ J.

Then x0= φ. For each function z ∈ BT, set

y(t) = z(t) + x(t).

(3.5)

It is obvious that y satisfies (3.1) if and only if z satisfies z0= 0 and

z(t) =

?t

0

U(t,s) C uz(s) ds +

?t

0

U(t,s) f(s,zs+ xs) ds

for t ∈ J.

Let

B0

T= {z ∈ BT: z0= 0}.

For any z ∈ B0

Twe have

?z?T= sup{ |z(t)| : t ∈ J } + ?z0?B= sup{ |z(t)| : t ∈ J }.

Thus (B0

Define the operator F : B0

T,? · ?T) is a Banach space.

T→ B0

Tby :

F(z)(t) =

?t

0

U(t,s) C uz(s) ds +

?t

0

U(t,s) f(s,zs+ xs) ds

for t ∈ J.

(3.6)

Obviously the operator N has a fixed point is equivalent to F has one, so it turns

to prove that F has a fixed point. The proof will be given in several steps.

Let us first show that the operator F is continuous and compact.

Step 1 : F is continuous.

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Let (zn)nbe a sequence in B0

Tsuch that zn→ z in B0

?U(t,s)?B(E)?C? |uzn(s) − uz(s)| ds

T. Then using (3.4), we get

|F(zn)(t) − F(z)(t)|≤

?t

?t

?

?

?

?

0

+

0

?U(t,s)?B(E)|f(s,zns+ xs) − f(s,zs+ xs)| ds

?t

?t

M2?

M

0

≤

M?

M

M

0

?

M1?

|f(s,zns+ xs) − f(s,zs+ xs)| ds

?T

|f(s,zns+ xs) − f(s,zs+ xs)| ds.

M

?T

0

|f(τ,znτ+ xτ) − f(τ,zτ+ xτ)| dτ ds

+

0

≤

M?

M1T

0

|f(s,zns+ xs) − f(s,zs+ xs)| ds

+

?T

Since f is L1-Carath´ eodory, we obtain by the Lebesgue dominated convergence

theorem

|F(zn)(t) − F(z)(t)| → 0

Thus F is continuous.

as n → +∞.

Step 2 : F maps bounded sets of B0

exists a positive constant ? such that for each z ∈ Bd= {z ∈ B0

has ?F(z)?T≤ ?.

Let z ∈ Bd. By (3.4), we have for each t ∈ J

|F(z)(t)|

0

?t

≤

0

?t

≤

?t

≤

??

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Tinto bounded sets. For any d > 0, there

T: ?z?T ≤ d} one

≤

?t

?

?

?

?

?

?

?

?U(t,s)?B(E)?C? |uz(s)| ds +

?t

0

?U(t,s)?B(E)|f(s,zs+ xs)| ds

≤

M?

M?

M

M

0

|uz(s)| ds +?

?

|f(s,zs+ xs)| ds

?

p(s) ψ(?zs+ xs?B) ds

M?

M?

M

?t

0

|f(s,zs+ xs)| ds

?T

M

?t

M1

?

|y1| +?

MH?φ?B+?

M

0

p(τ) ψ(?zτ+ xτ?B) dτ

?

ds

+

0

M?

M

M?

M1T

|y1| +?

MH?φ?B+?

M

?T

0

p(s) ψ(?zs+ xs?B) ds

?

+

0

M?

M

M1T

?

|y1| +?

M1T + 1

MH?φ?B

??T

?

+

M?

0

p(s) ψ(?zs+ xs)?B) ds.

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S. Baghli, M. Benchohra and K. Ezzinbi

Using the assumption (A1), we get

?zs+ xs?B

≤

≤

≤

≤

≤

?zs?B+ ?xs?B

K(s)|z(s)| + M(s)?z0?B+ K(s)|x(s)| + M(s)?x0?B

KT|z(s)| + KT?U(s,0)?B(E)|φ(0)| + MT?φ?B

KT|z(s)| + KT?

MH?φ?B+ MT?φ?B

MH + MT)?φ?B.KT|z(s)| + (KT?

Set α := (KT?

MH + MT)?φ?Band δ := KTd + α. Then,

?zs+ xs?B≤ KT|z(s)| + α ≤ δ.

(3.7)

Using the nondecreasing character of ψ, we get for each t ∈ J

?

Thus there exists a positive number ? such that

|F(z)(t)| ≤?

M?

M?

M1T

|y1| +?

MH?φ?B

?

+?

M

??

M?

M?

M1T + 1

?

ψ(δ) ?p?L1 := ?.

?F(z)?T≤ ?.

Hence F(Bd) ⊂ Bd.

Step 3 : F maps bounded sets into equicontinuous sets of B0

in Step 2 and we show that F(Bd) is equicontinuous.

Let τ1,τ2∈ J with τ2> τ1and z ∈ Bd. Then

T. We consider Bdas

|F(z)(τ2) − F(z)(τ1)|≤

?τ1

?τ1

?τ2

?τ2

0

?U(τ2,s) − U(τ1,s)?B(E)?C? |uz(s)| ds

+

0

?U(τ2,s) − U(τ1,s)?B(E)|f(s,zs+ xs)| ds

+

τ1

?U(τ2,s)?B(E)?C? |uz(s)| ds

+

τ1

?U(τ2,s)?B(E)|f(s,zs+ xs)| ds.

By the inequalities (3.4) and (3.7) and using the nondecreasing character of ψ, we

get

?

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|uz(t)| ≤?

M1

|y1| +?

MH?φ?B+?

M ψ(δ) ?p?L1

?

:= ω.

(3.8)

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Controllability Results for Evolution Equations with Infinite Delay

25

Then

|F(z)(τ2) − F(z)(τ1)|≤?C?B(E)ω

?τ1

?C?B(E)ω

?τ2

?τ1

?U(τ2,s) − U(τ1,s)?B(E)p(s) ds

?τ2

?U(τ2,s)?B(E)p(s) ds.

0

?U(τ2,s) − U(τ1,s)?B(E)ds

+

ψ(δ)

0

+

τ1

?U(τ2,s)?B(E)ds

+

ψ(δ)

τ1

Noting that |F(z)(τ2) − F(z)(τ1)| tends to zero as τ2− τ1 → 0 independently of

z ∈ Bd. The right-hand side of the above inequality tends to zero as τ2− τ1→ 0.

Since U(t,s) is a strongly continuous operator and the compactness of U(t,s) for

t > s implies the continuity in the uniform operator topology (see [5, 45]). As a

consequence of Steps 1 to 3 together with the Arzel´ a-Ascoli theorem it suffices to

show that the operator F maps Bdinto a precompact set in E.

Let t ∈ J be fixed and let ? be a real number satisfying 0 < ? < t. For z ∈ Bdwe

define

?t−?

+

U(t,t − ?)

0

F?(z)(t)=

U(t,t − ?)

0

U(t − ?,s) C uz(s) ds

?t−?

U(t − ?,s) f(s,zs+ xs) ds.

Since U(t,s) is a compact operator, the set Z?(t) = {F?(z)(t) : z ∈ Bd} is pre-

compact in E for every ? sufficiently small, 0 < ? < t. Moreover using (3.8), we

have

?t

+

t−?

≤

|F(z)(t) − F?(z)(t)|≤

t−?

?t

?C?B(E)ω

?U(t,s)?B(E)?C? |uz(s)| ds

?U(t,s)?B(E)|f(s,zs+ xs)| ds

?t

?t

t−?

?U(t,s)?B(E)ds

+

ψ(δ)

t−?

?U(t,s)?B(E)p(s) ds.

Therefore there are precompact sets arbitrary close to the set {F(z)(t) : z ∈ Bd}.

Hence the set {F(z)(t) : z ∈ Bd} is precompact in E. So we deduce from Steps 1, 2

and 3 that F is a compact operator.

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Page 12

26

S. Baghli, M. Benchohra and K. Ezzinbi

Step 4 : For applying Theorem 7, we must check (C2) : i.e. it remains to show

that the set

E =?z ∈ B0

is bounded.

Let z ∈ E. By (3.4), we have for each t ∈ J

?t

≤

0

?t

≤

?t

Using the first inequality in (3.7) and the nondecreasing character of ψ, we get

?

+

?

Then

T: z = λ F(z) for some 0 < λ < 1?

|z(t)|≤

0

?U(t,s)?B(E)?C? |uz(s)| ds +

?t

p(s) ψ(?zs+ xs?B) ds

?

p(s) ψ(?zs+ xs?B) ds.

?t

?T

0

?U(t,s)?B(E)|f(s,zs+ xs)| ds

?

?

?

?

M?

M

M

?

M1

?

|y1| +?

MH?φ?B+?

M

0

p(τ) ψ(?zτ+ xτ?B) dτ

?

ds

+

0

M?

M

M?

M1T

|y1| +?

MH?φ?B+?

M

?T

0

p(s) ψ(?zs+ xs?B) ds

?

+

0

|z(t)|≤

?

M

M?

M?

M1T

?t

|y1| +?

p(s) ψ(KT|z(s)| + α) ds.

MH?φ?B+?

M

?T

0

p(s) ψ(KT|z(s)| + α) ds

?

0

KT|z(t)| + α

≤

α + KT?

?

KT?

|y1| +?

β + KT?

KT?

M?

p(s) ψ(KT|z(s)| + α) ds

?t

?

M2?

p(s) ψ(KT|z(s)| + α) ds.

M?

M1T

?

|y1| +?

MH?φ?B

+

M

?T

M

0

?

+

0

p(s) ψ(KT|z(s)| + α) ds.

Set β := α + KT?

KT|z(t)| + α

M?

M?

≤

M1T

?

MH?φ?B

, thus

M?

M1T

?T

0

p(s) ψ(KT|z(s)| + α) ds

+

M

?t

0

We consider the function µ defined by

µ(t) := sup { KT|z(s)| + α : 0 ≤ s ≤ t },

0 ≤ t ≤ T.

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Controllability Results for Evolution Equations with Infinite Delay

27

Let t?∈ [0,t] be such that µ(t) = KT|z(t?)| + α. If t?∈ J, by the previous

inequality, we have for t ∈ J

?T

Then, we have

µ(t) ≤ β + KT?

M2?

M?

M1T

0

p(s) ψ(µ(s)) ds + KT?

??T

M

?t

0

p(s) ψ(µ(s)) ds.

µ(t) ≤ β + KT?

M

??

M?

M?

M1T + 1

0

p(s) ψ(µ(s)) ds.

Consequently,

?z?T

M1T + 1

β + KT?

M

??

M?

M?

?

ψ(?z?T) ?p?L1

≤ 1.

Then by (3.2), there exists a constant M?such that ?z?T?= M?. Set

U = { z ∈ B0

Clearly, U is a closed subset of B0

such that z = λ F(z) for some λ ∈ (0,1). Then the statement (C2) in Theorem

7 does not hold. As a consequence of the nonlinear alternative of Leray-Schauder

type ([29]), we deduce that (C1) holds : i.e. the operator F has a fixed-point z?.

Then y?(t) = z?(t) + x(t), t ∈ (−∞,T] is a fixed point of the operator N, which is

a mild solution of the problem (1.1)-(1.2). Thus the evolution system (1.1)-(1.2) is

controllable on (−∞,T].

T : ?z?T ≤ M?+ 1 }.

T. From the choice of U there is no z ∈ ∂U

4 Semilinear Neutral Evolution Equations

In this section, we give controllability result for the neutral functional differential

evolution problem with infinite delay (1.3)-(1.4). Firstly we define the mild solution.

Definition 12. We say that the function y(·) : (−∞,T] → E is a mild solution of

(1.3)-(1.4) if y(t) = φ(t) for all t ∈ (−∞,0] and y satisfies the following integral

equation

y(t)= U(t,0)[φ(0) − g(0,φ)] + g(t,yt) +

?t

Definition 13. The neutral evolution problem (1.3)-(1.4) is said to be controllable

on the interval J if for every initial function φ ∈ B and y1∈ E there exists a control

u ∈ L2(J;E) such that the mild solution y(·) of (1.3)-(1.4) satisfies y(T) = y1.

?t

0

U(t,s)A(s)g(s,ys)ds

+

0

U(t,s)Cu(s)ds +

?t

0

U(t,s)f(s,ys) dsfor each t ∈ J.

(4.1)

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28

S. Baghli, M. Benchohra and K. Ezzinbi

We consider the hypotheses (H1)-(H3) and we will need the following assump-

tions :

(H4) There exists a constant M0> 0 such that :

?A−1(t)?B(E)≤ M0

for all t ∈ J.

(H5) There exists a constant 0 < L <

1

M0KT

such that :

|A(t) g(t,φ)| ≤ L (?φ?B+ 1) for all t ∈ J and φ ∈ B.

(H6) There exists a constant L?> 0 such that :

|A(t) g(s,φ) − A(t) g(s,φ)| ≤ L?(|s − s| + ?φ − φ?B)

for all 0 ≤ t,s,s ≤ T and φ,φ ∈ B.

(H7) The function g is completely continuous and for any bounded set Q ⊆ BTthe

set {t → g(t,xt) : x ∈ Q} is equicontinuous in C(J;E).

Theorem 14. Suppose that hypotheses (H1)-(H7) are satisfied and moreover there

exists a constant M??> 0 with

M??

?

where ζ(t) = max(L;p(t)) and

?β +?

MKT

M?

M?

M1T + 1

1 − M0LKT

[M??+ ψ(M??)] ?ζ?L1

> 1,

(4.2)

?β

=

?β(φ,y1) = (KT?

MH + MT)?φ?B+

????

MM0L

KT

1 − M0LKT

MLT

?

×

M?

M1T

M1T?1 + M0LKT

×

+?

M + 1

?

1 +?

M0L +?

??

+?

1 +?

M?

M1T

??

?

??

MH + MT)?φ?B+?

M?

M?

M1TM?

MH + M0LMT

??

?|y1|

?φ?B

?

+ M0L(KT?

M?

M?

then the neutral evolution problem (1.3)-(1.4) is controllable on (−∞,T].

Proof. Consider the operator? N : BT→ BT defined by :

0

? N(y)(t) =

φ(t),

if t ∈ (−∞,0];

U(t,0) [φ(0) − g(0,φ)] + g(t,yt)

?t

+

U(t,s)Cuy(s)ds +

+

0

U(t,s)A(s)g(s,ys)ds

?t

?t

0

U(t,s)f(s,ys)ds,

if t ∈ J.

(4.3)

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Controllability Results for Evolution Equations with Infinite Delay

29

Using assumption (H3), for arbitrary function y(·), we define the control

uy(t)=

?T

Noting that

?˜ W−1??|y1| + ?U(t,0)?B(E)

+?A−1(T)?|A(T)g(T,yT)| +

?T

≤

?T

From (H2), we get

?

+

?

≤

+

?

We shall show that using this control the operator? N has a fixed point y(·), which

For φ ∈ B, we will define the function x(.) : R → E by

?φ(t),

U(t,0) φ(0),

˜ W−1[y1− U(T,0)(φ(0) − g(0,φ)) − g(T,yT)

U(T,s)A(s)g(s,ys)ds −−

0

?T

0

U(T,s)f(s,ys)ds

?

(t).

|uy(t)|≤

?|φ(0)| + ?A−1(0)?|A(0)g(0,φ)|?

?U(T,τ)?B(E)|A(τ)g(τ,yτ)|dτ

?

MM0L(?φ?B+ 1) + M0L(?yT?B+ 1)

?T

?T

0

+

0

?

ML

?U(T,τ)?B(E)|f(τ,yτ)|dτ

|y1| +?

M1?

M?H + M0L??φ?B+

M1M0L?yT?B+?

M1

?

?

M1

MH?φ?B+?

(?yτ?B+ 1)dτ +?

?

+

0

M1?

M

0

|f(τ,yτ)|dτ.

|uy(t)|≤

?

M1

|y1| +?

??

??

M + 1

?

M0L +?

M

0

M0L +?

M

0

MLT

?

M1?

ML

?T

0

?yτ?Bdτ +?

M1?

?T

|f(τ,yτ)| dτ

MLT

?

?

|y1| +?

M?H + M0L??φ?B+

M1M0L?yT?B+?

M + 1

?

?

M1?

ML

?T

0

?yτ?Bdτ +?

M1?

?T

p(τ)ψ(?yτ?B)dτ.

(4.4)

is a mild solution of the neutral evolution system (1.3)-(1.4).

x(t) =

if t ∈ (−∞,0];

if t ∈ J.

Then x0= φ. For each function z ∈ BT, set

y(t) = z(t) + x(t).

(4.5)

It is obvious that y satisfies (4.1) if and only if z satisfies z0= 0 and for t ∈ J, we

get

z(t)=

g(t,zt+ xt) − U(t,0)g(0,φ) +

?t

?t

0

U(t,s)A(s)g(s,zs+ xs)ds

+

0

U(t,s)Cuz(s)ds +

?t

0

U(t,s)f(s,zs+ xs)ds.

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