CONTROLLABILITY RESULTS FOR SEMILINEAR FUNCTIONAL AND NEUTRAL FUNCTIONAL EVOLUTION ... ·...

Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)Volume 4 (2009), 15 – 39

CONTROLLABILITY RESULTS FOR SEMILINEARFUNCTIONAL 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 functionaldifferential 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(.) isgiven in L2(J ;E), the Banach space of admissible control function with E is a realseparable Banach space with the norm |·|, C is a bounded linear operator from E intoE and {A(t)}0≤t≤T is a family of linear closed (not necessarily bounded) operatorsfrom E into E that generate an evolution system of operators {U(t, s)}(t,s)∈J×J for0 ≤ s ≤ t ≤ T . To study the system (1.1) − (1.2), we assume that the historiesyt : (−∞, 0] → E, yt(θ) = y(t + θ) belong to some abstract phase space B, to bespecified later. We consider in Section 4, the neutral functional differential evolutionequation 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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16 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 givenfunction. Finally in Section 5, an example is given to demonstrate the results.

Partial functional and neutral functional differential equations arise in manyareas of applied mathematics, we refer the reader to the book by Hale and VerduynLunel [31], Kolmanovskii and Myshkis [37] and Wu [49].

In the literature devoted to equations with finite delay, the phase space is muchof time the space of all continuous functions on [−r, 0], r > 0, endowed with theuniform norm topology. When the delay is infinite, the notion of the phase spaceB 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 introducedby 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 ODEsin finite dimensional space has been extensively studied. Several authors have ex-tended the controllability concept to infinite dimensional systems in Banach spacewith unbounded operators (see [19, 43, 44]). More details and results can be foundin the monographs [18, 21, 42, 51]. Triggiani [48] established sufficient conditions forcontrollability 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 neutralfunctional differential equations. Fu and Ezzinbi [27] considered the existence of mildand classical solutions for a class of neutral partial functional differential equationswith nonlocal conditions, Balachandran and Dauer have considered various classesof 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 andinclusions and proposed some controllability results in [6, 12, 13, 14, 15, 16, 17]. Seealso the works by Gatsori [28] and Li et al. [40, 41, 42]. Adimy et al [1, 2, 3] studiedpartial 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 mildsolutions for functional partial differential equations with infinite delay. Henriquez[32] and Hernandez [33, 34] considered the existence and regularity of solutions tofunctional and neutral functional differential equations with unbounded delay.

Recently Baghli and Benchohra considered in [7] a class of partial functionalevolution equation and in [8] a class of neutral functional evolution equations on

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Controllability Results for Evolution Equations with Infinite Delay 17

a semiinfinite real interval and with a bounded delay. Extension of these works isgiven in [9] when the delay is infinite.

In this paper, we investigate the controllability of mild solutions of the previousevolution 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 . Sufficientconditions are establish here to get the controllability of mild solutions which arefixed points of appropriate corresponding operators using the nonlinear alternativeof Leray-Schauder type (see [29]).

2 Preliminaries

We introduce notations, definitions and theorems which are used throughout thispaper.

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| isLebesgue 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 areBochner integrable normed by

‖y‖L1 =∫ T

0|y(t)| dt.

Consider the following space

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

where y|J is 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 thefollowing conditions hold :(i) yt ∈ B ;(ii) There exists a positive constant H such that |y(t)| ≤ H‖yt‖B ;






(iii) There exist two functions K(·),M(·) : R+ → R+ independent of y(t) withK 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), yt is 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‖B for every φ ∈ B.

2. Since ‖ · ‖B is a seminorm, two elements φ, ψ ∈ B can verify ‖φ − ψ‖B = 0without 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, forinstance 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;

C∞ :={φ ∈ BC : lim

θ→−∞φ(θ) exist in E

};

C0 :={φ ∈ BC : lim

θ→−∞φ(θ) = 0

}, endowed with the uniform norm

‖φ‖ = sup{|φ(θ)| : θ ≤ 0}.

We have that the spaces BUC, C∞ and C0 satisfy conditions (A1)− (A3). BCsatisfies (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]}

;






C0g :=

{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.

(g1) For all a > 0, sup0≤t≤a

sup{g(t+ θ)g(θ)

: −∞ < θ ≤ −t}<∞.

Then we have that the spaces Cg and C0g satisfy conditions (A3). They satisfy

conditions (A1) and (A2) if g1 holds.

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

Cγ :={φ ∈ C((−∞, 0];E) : lim

θ→−∞eγθφ(θ) exist in E

}endowed with the following norm

‖φ‖ = sup{eγθ|φ(θ)| : θ ≤ 0}.

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

Definition 5. A function f : J×B → E is said to be an L1-Caratheodory functionif 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 → Efor (t, s) ∈ ∆ := {(t, s) ∈ J × J : 0 ≤ s ≤ t ≤ T} is called an evolution system if thefollowing properties are satisfied :

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






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

Our results will be based on the following well known nonlinear alternative ofLeray-Schauder type.

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

y(t) = U(t, 0)φ(0) +∫ t

0U(t, s)Cu(s)ds+

∫ t

0U(t, s)f(s, ys)ds for each t ∈ J. (3.1)

Definition 9. The problem (1.1)-(1.2) is said to be controllable on the interval J iffor every initial function φ ∈ B and y1 ∈ E there exists a control u ∈ L2(J ;E) suchthat 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:

(H1) U(t, s) is compact for t− s > 0 and there exists a constant M ≥ 1 such that :

‖U(t, s)‖B(E) ≤ M for every (t, s) ∈ ∆.

(H2) There exists a function p ∈ L1(J ; R+) and a continuous nondecreasing functionψ : R+ → (0,∞) and such that :

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






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

Wu =∫ T

0U(T, s)Cu(s)ds,

has an induced invertible operator W−1 which takes values in L2(J ;E)/ kerWand there exists positive constants M and M1 such that :

‖C‖ ≤ M and ‖W−1‖ ≤ M1.

Remark 10. For the construction of W and W−1 see the paper by Carmichael andQuinn [46].

Theorem 11. Suppose that hypotheses (H1)-(H3) are satisfied and moreover thereexists a constant M? > 0 such that

M?

β +KT M(MMM1T + 1

)ψ(M?) ‖p‖L1

> 1, (3.2)

with

β = β(φ, y1) =(KT MH

[1 + MMM1T

]+MT

)‖φ‖B +KT MMM1T |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. Considerthe operator N : BT → BT defined by :

N(y)(t) =

φ(t), if t ∈ (−∞, 0];

U(t, 0) φ(0) +∫ t

0U(t, s) C uy(s) ds

+∫ t

0U(t, s) f(s, ys) ds, if t ∈ J.

(3.3)

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

uy(t) = W−1

[y1 − U(T, 0) φ(0)−

∫ T

0U(T, s) f(s, ys) ds

](t).

Noting that, we have

|uy(t)| ≤ ‖W−1‖[|y1|+ ‖U(t, 0)‖B(E)|φ(0)|+

∫ T

0‖U(T, τ)‖B(E)|f(τ, yτ )|dτ

].






From (H2), we get

|uy(t)| ≤ M1

[|y1|+ MH‖φ‖B + M

∫ T

0|f(τ, yτ )|dτ

]≤ M1

[|y1|+ MH‖φ‖B + M

∫ T

0p(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

x(t) =

{φ(t), if t ∈ (−∞, 0];

U(t, 0) φ(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

0U(t, s) C uz(s) ds+

∫ t

0U(t, s) f(s, zs + xs) ds for t ∈ J.

Let

B0T = {z ∈ BT : z0 = 0} .

For any z ∈ B0T we have

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

Thus (B0T , ‖ · ‖T ) is a Banach space.

Define the operator F : B0T → B0

T by :

F (z)(t) =∫ t


∫ t

0U(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 turnsto 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.






Let (zn)n be a sequence in B0T such that zn → z in B0

T . Then using (3.4), we get

|F (zn)(t)− F (z)(t)| ≤∫ t

0‖U(t, s)‖B(E) ‖C‖ |uzn(s)− uz(s)| ds

+∫ t

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

≤ MM

∫ t

0M1M

∫ T

0|f(τ, znτ + xτ )− f(τ, zτ + xτ )| dτ ds

+ M

∫ t

0|f(s, zns + xs)− f(s, zs + xs)| ds

≤ M2MM1T

∫ T

0|f(s, zns + xs)− f(s, zs + xs)| ds

+ M

∫ T

0|f(s, zns + xs)− f(s, zs + xs)| ds.

Since f is L1-Caratheodory, we obtain by the Lebesgue dominated convergencetheorem

|F (zn)(t)− F (z)(t)| → 0 as n→ +∞.

Thus F is continuous.

Step 2 : F maps bounded sets of B0T into bounded sets. For any d > 0, there

exists a positive constant ` such that for each z ∈ Bd = {z ∈ B0T : ‖z‖T ≤ d} one

has ‖F (z)‖T ≤ `.Let z ∈ Bd. By (3.4), we have for each t ∈ J

|F (z)(t)| ≤∫ t

0‖U(t, s)‖B(E) ‖C‖ |uz(s)| ds+

∫ t

0‖U(t, s)‖B(E) |f(s, zs + xs)| ds

≤ MM

∫ t

0|uz(s)| ds+ M

∫ t

0|f(s, zs + xs)| ds

≤ MM

∫ t

0M1

[|y1|+ MH‖φ‖B + M

∫ T

0p(τ) ψ(‖zτ + xτ‖B) dτ

]ds

+ M

∫ t

0|f(s, zs + xs)| ds

≤ MMM1T

[|y1|+ MH‖φ‖B + M

∫ T

0p(s) ψ(‖zs + xs‖B) ds

]+ M

∫ t


≤ MMM1T[|y1|+ MH‖φ‖B

]+ M

(MMM1T + 1

) ∫ T

0p(s) ψ(‖zs + xs)‖B) ds.






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≤ KT |z(s)|+ (KT MH +MT )‖φ‖B.

Set α := (KT MH +MT )‖φ‖B and 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

|F (z)(t)| ≤ MMM1T[|y1|+ MH‖φ‖B

]+ M

(MMM1T + 1

)ψ(δ) ‖p‖L1 := `.

Thus there exists a positive number ` such that

‖F (z)‖T ≤ `.

Hence F (Bd) ⊂ Bd.

Step 3 : F maps bounded sets into equicontinuous sets of B0T . We consider Bd as

in Step 2 and we show that F (Bd) is equicontinuous.Let τ1, τ2 ∈ J with τ2 > τ1 and z ∈ Bd. Then

|F (z)(τ2)− F (z)(τ1)| ≤∫ τ1

0‖U(τ2, s)− U(τ1, s)‖B(E) ‖C‖ |uz(s)| ds

+∫ τ1

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

+∫ τ2

τ1

‖U(τ2, s)‖B(E) ‖C‖ |uz(s)| ds

+∫ τ2

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, weget

|uz(t)| ≤ M1

[|y1|+ MH‖φ‖B + M ψ(δ) ‖p‖L1

]:= ω. (3.8)






Then

|F (z)(τ2)− F (z)(τ1)| ≤ ‖C‖B(E) ω

∫ τ1

0‖U(τ2, s)− U(τ1, s)‖B(E) ds

+ ψ(δ)∫ τ1

0‖U(τ2, s)− U(τ1, s)‖B(E) p(s) ds

+ ‖C‖B(E) ω

∫ τ2

τ1

‖U(τ2, s)‖B(E) ds

+ ψ(δ)∫ τ2

τ1

‖U(τ2, s)‖B(E) p(s) ds.

Noting that |F (z)(τ2) − F (z)(τ1)| tends to zero as τ2 − τ1 → 0 independently ofz ∈ 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) fort > s implies the continuity in the uniform operator topology (see [5, 45]). As aconsequence of Steps 1 to 3 together with the Arzela-Ascoli theorem it suffices toshow that the operator F maps Bd into a precompact set in E.Let t ∈ J be fixed and let ε be a real number satisfying 0 < ε < t. For z ∈ Bd wedefine

Fε(z)(t) = U(t, t− ε)∫ t−ε

0U(t− ε, s) C uz(s) ds

+ U(t, t− ε)∫ t−ε

0U(t− ε, s) f(s, zs + xs) ds.

Since U(t, s) is a compact operator, the set Zε(t) = {Fε(z)(t) : z ∈ Bd} is pre-compact in E for every ε sufficiently small, 0 < ε < t. Moreover using (3.8), wehave

|F (z)(t)− Fε(z)(t)| ≤∫ t

t−ε‖U(t, s)‖B(E) ‖C‖ |uz(s)| ds

+∫ t

t−ε‖U(t, s)‖B(E) |f(s, zs + xs)| ds

≤ ‖C‖B(E) ω

∫ t

t−ε‖U(t, s)‖B(E) ds

+ ψ(δ)∫ t

t−ε‖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, 2and 3 that F is a compact operator.






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

E ={z ∈ B0

T : z = λ F (z) for some 0 < λ < 1}

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

|z(t)| ≤∫ t

0‖U(t, s)‖B(E) ‖C‖ |uz(s)| ds+

∫ t

0‖U(t, s)‖B(E) |f(s, zs + xs)| ds

≤ MM

∫ t

0M1

[|y1|+ MH‖φ‖B + M

∫ T

0p(τ) ψ(‖zτ + xτ‖B) dτ

]ds

+ M

∫ t


≤ MMM1T

[|y1|+ MH‖φ‖B + M

∫ T


]+ M

∫ t

0p(s) ψ(‖zs + xs‖B) ds.

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

|z(t)| ≤ MMM1T

[|y1|+ MH‖φ‖B + M

∫ T

0p(s) ψ(KT |z(s)|+ α) ds

]+ M

∫ t

0p(s) ψ(KT |z(s)|+ α) ds.

Then

KT |z(t)|+ α ≤ α+KT MMM1T[|y1|+ MH‖φ‖B

+ M

∫ T


]+ KT M

∫ t


Set β := α+KT MMM1T[|y1|+ MH‖φ‖B

], thus

KT |z(t)|+ α ≤ β +KT M2MM1T

∫ T


+ KT M

∫ t


We consider the function µ defined by

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






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

µ(t) ≤ β +KT M2MM1T

∫ T

0p(s) ψ(µ(s)) ds+KT M

∫ t

0p(s) ψ(µ(s)) ds.

Then, we have

µ(t) ≤ β +KT M(MMM1T + 1

) ∫ T

0p(s) ψ(µ(s)) ds.

Consequently,

‖z‖T

β +KT M(MMM1T + 1

)ψ(‖z‖T ) ‖p‖L1

≤ 1.

Then by (3.2), there exists a constant M? such that ‖z‖T 6= M?. Set

U = { z ∈ B0T : ‖z‖T ≤ M? + 1 }.

Clearly, U is a closed subset of B0T . From the choice of U there is no z ∈ ∂U

such that z = λ F (z) for some λ ∈ (0, 1). Then the statement (C2) in Theorem7 does not hold. As a consequence of the nonlinear alternative of Leray-Schaudertype ([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 isa mild solution of the problem (1.1)-(1.2). Thus the evolution system (1.1)-(1.2) iscontrollable on (−∞, T ].

4 Semilinear Neutral Evolution Equations

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

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

0U(t, s)A(s)g(s, ys)ds

+∫ t

0U(t, s)Cu(s)ds+

∫ t

0U(t, s)f(s, ys) ds for each t ∈ J.

(4.1)

Definition 13. The neutral evolution problem (1.3)-(1.4) is said to be controllableon the interval J if for every initial function φ ∈ B and y1 ∈ E there exists a controlu ∈ L2(J ;E) such that the mild solution y(·) of (1.3)-(1.4) satisfies y(T ) = y1.






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

Theorem 14. Suppose that hypotheses (H1)-(H7) are satisfied and moreover thereexists a constant M?? > 0 with

M??

β + MKTMMM1T + 11−M0LKT

[M?? + ψ(M??)] ‖ζ‖L1

> 1, (4.2)

where ζ(t) = max(L; p(t)) and

β = β(φ, y1) = (KT MH +MT )‖φ‖B +KT

1−M0LKT

×

×{[(

M + 1)M0L+ MLT

] (1 + MMM1T

)+M

[M0L

(1 + MMM1T

)+ MM1T

(MH +M0LMT

)]‖φ‖B

+ M0L(KT MH +MT )‖φ‖B + MMM1T(1 +M0LKT

)|y1|

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

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

N(y)(t) =

φ(t), if t ∈ (−∞, 0];

U(t, 0) [φ(0)− g(0, φ)] + g(t, yt)

+∫ t

0U(t, s)A(s)g(s, ys)ds

+∫ t

0U(t, s)Cuy(s)ds+

∫ t

0U(t, s)f(s, ys)ds, if t ∈ J.

(4.3)






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

uy(t) = W−1 [y1 − U(T, 0) (φ(0)− g(0, φ))− g(T, yT )

−∫ T

0U(T, s)A(s)g(s, ys)ds−

∫ T

0U(T, s)f(s, ys)ds

](t).

Noting that

|uy(t)| ≤ ‖W−1‖[|y1|+ ‖U(t, 0)‖B(E)

(|φ(0)|+ ‖A−1(0)‖|A(0)g(0, φ)|

)+‖A−1(T )‖|A(T )g(T, yT )|+

∫ T

0‖U(T, τ)‖B(E)|A(τ)g(τ, yτ )|dτ

+∫ T

0‖U(T, τ)‖B(E)|f(τ, yτ )|dτ

]≤ M1

[|y1|+ MH‖φ‖B + MM0L(‖φ‖B + 1) +M0L(‖yT ‖B + 1)

]+ M1ML

∫ T

0(‖yτ‖B + 1)dτ + M1M

∫ T

0|f(τ, yτ )|dτ.

From (H2), we get

|uy(t)| ≤ M1

[|y1|+ M

(H +M0L

)‖φ‖B +

(M + 1

)M0L+ MLT

]+ M1M0L‖yT ‖B + M1ML

∫ T

0‖yτ‖B dτ + M1M

∫ T

0|f(τ, yτ )| dτ

≤ M1

[|y1|+ M

(H +M0L

)‖φ‖B +

(M + 1

)M0L+ MLT

]+ M1M0L‖yT ‖B + M1ML

∫ T

0‖yτ‖Bdτ + M1M

∫ T

0p(τ)ψ(‖yτ‖B)dτ.

(4.4)We shall show that using this control the operator N has a fixed point y(·), whichis a mild solution of the neutral evolution system (1.3)-(1.4).

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

x(t) =

{φ(t), if t ∈ (−∞, 0];

U(t, 0) φ(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 , weget

z(t) = g(t, zt + xt)− U(t, 0)g(0, φ) +∫ t

0U(t, s)A(s)g(s, zs + xs)ds

+∫ t

0U(t, s)Cuz(s)ds+

∫ t

0U(t, s)f(s, zs + xs)ds.






Define the operator F : B0T → B0

T by :

F (z)(t) = g(t, zt + xt)− U(t, 0) g(0, φ) +∫ t

0U(t, s) A(s) g(s, zs + xs) ds

+∫ t


∫ t

0U(t, s) f(s, zs + xs) ds.

(4.6)

Obviously the operator N has a fixed point is equivalent to F has one, so it turns toprove that F has a fixed point. We can show as in Section 3 that the operator F iscontinuous and compact. To apply Theorem 7, we must check (C2), i.e., it remainsto show that the set

E ={z ∈ BT

0 : z = λ F (z) for some 0 < λ < 1}

is bounded.Let z ∈ E . By (H1) to (H5) and (4.4), we have for each t ∈ J

|z(t)| ≤ ‖A−1(t)‖ |A(t)g(t, zt + xt)|+ ‖U(t, 0)‖B(E) ‖A−1(0)‖ |A(0)g(0, φ)|

+∫ t

0‖U(t, s)‖B(E)|A(s)g(s, zs + xs)| ds+

∫ t

0‖U(t, s)‖B(E) ‖C‖ |uz(s)| ds

+∫ t

0‖U(t, s)‖B(E) |f(s, zs + xs)| ds

≤ M0L (‖zt + xt‖B + 1) + MM0L (‖φ‖B + 1) + ML

∫ t

0(‖zs + xs‖B + 1) ds

+ MM

∫ t

0M1

[|y1|+ M

(H +M0L

)‖φ‖B +

(M + 1

)M0L+ MLT

+ M0L‖zT + xT ‖B + ML

∫ T

0‖zτ + xτ‖Bdτ + M

∫ T

0p(τ)ψ(‖zτ + xτ‖B)dτ

]ds

+ M

∫ t


≤[(M + 1

)M0L+ MLT

] (1 + MMM1T

)+ MMM1T |y1|

+ M[M0L

(1 + MMM1T

)+ MMM1TH

]‖φ‖B + MMM1M0LT‖zT + xT ‖B

+ M0L‖zt + xt‖B + ML

∫ t

0‖zs + xs‖B ds+ M2MM1LT

∫ T

0‖zs + xs‖B ds

+ M2MM1T

∫ T

0p(s)ψ(‖zs + xs‖B) ds+ M

∫ t

0p(s)ψ(‖zs + xs‖B) ds.

Noting that we have ‖zT + xT ‖B ≤ KT |y1|+MT ‖φ‖B and using the first inequality‖zt + xt‖B ≤ KT |z(t)| + α in (3.7), then by the nondecreasing character of ψ, we






obtain

|z(t)| ≤[(M + 1

)M0L+ MLT

] (1 + MMM1T

)+ M

[M0L

(1 + MMM1T

)+ MM1T

(MH +M0LMT

)]‖φ‖B

+ MMM1T(1 +M0LKT

)|y1|+M0L (KT |z(t)|+ α)

+ ML

∫ t

0(KT |z(s)|+ α) ds+ M2MM1LT

∫ T

0(KT |z(s)|+ α) ds

+ M2MM1T

∫ T

0p(s)ψ (KT |z(s)|+ α) ds+ M

∫ t

0p(s)ψ (KT |z(s)|+ α) ds.

Then(1−M0LKT

)|z(t)| ≤

[(M + 1

)M0L+ MLT

] (1 + MMM1T

)+M0Lα

+ M[M0L

(1 + MMM1T

)+ MM1T

(MH +M0LMT

)]‖φ‖B

+ MMM1T(1 +M0LKT

)|y1|+ ML

∫ t

0(KT |z(s)|+ α) ds

+ M2MM1LT

∫ T

0(KT |z(s)|+ α) ds

+ M2MM1T

∫ T

0p(s)ψ (KT |z(s)|+ α) ds

+ M

∫ t

0p(s)ψ (KT |z(s)|+ α) ds.

Set

β := α+KT

1−M0LKT

{[(M + 1

)M0L+ MLT

] (1 + MMM1T

)+M0Lα

+M[M0L

(1 + MMM1T

)+ MM1T

(MH +M0LMT

)]‖φ‖B

+ MMM1T(1 +M0LKT

)|y1|

}thus

KT |z(t)|+ α ≤ β +MKT

1−M0LKT

×

×[L

∫ t

0(KT |z(s)|+ α) ds+ MMM1LT

∫ T

0(KT |z(s)|+ α) ds

+MMM1T

∫ T

0p(s)ψ (KT |z(s)|+ α) ds+

∫ t

0p(s)ψ (KT |z(s)|+ α) ds

].






We consider the function µ defined by

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

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

µ(t) ≤ β +MKT

1−M0LKT

[L

∫ t

0µ(s) ds+ MMM1LT

∫ T

0µ(s) ds

+ MMM1T

∫ T

0p(s)ψ(µ(s)) ds+

∫ t

0p(s)ψ(µ(s)) ds

].

Then, we have

µ(t) ≤ β + MKTMMM1T + 11−M0LKT

[L

∫ T

0µ(s) ds+

∫ T

0p(s)ψ(µ(s)) ds

].

Set ζ(t) := max(L; p(t)) for t ∈ J

µ(t) ≤ β + MKTMMM1T + 11−M0LKT

∫ T

0ζ(s) [µ(s) + ψ(µ(s))] ds.

Consequently,

‖z‖T

β + MKTMMM1T + 11−M0LKT

[‖z‖T + ψ(‖z‖T )] ‖ζ‖L1

≤ 1.

Then by (4.2), there exists a constant M?? such that ‖z‖T 6= M??. Set

U = { z ∈ B0T : ‖z‖T ≤ M?? + 1 }.

Clearly, U is a closed subset of B0T . From the choice of U there is no z ∈ ∂U

such that z = λ F (z) for some λ ∈ (0, 1). Then the statement (C2) in Theorem7 does not hold. As a consequence of the nonlinear alternative of Leray-Schaudertype ([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 isa mild solution of the problem (1.3)-(1.4). Thus the evolution system (1.3)-(1.4) iscontrollable on (−∞, T ].






5 An Example

To illustrate the previous results, we consider in this section the following model

∂

∂t

[v(t, ξ)−

∫ 0

−∞T (θ)w(t, v(t+ θ, ξ))dθ

]= a(t, ξ)

∂2v

∂ξ2(t, ξ)+

+d(ξ)u(t) +∫ 0

−∞P (θ)r(t, v(t+ θ, ξ))dθ t ∈ [0, T ], ξ ∈ [0, π]

v(t, 0) = v(t, π) = 0 t ∈ [0, T ]

v(θ, ξ) = v0(θ, ξ) −∞ < θ ≤ 0, ξ ∈ [0, π]

(5.1)

where a(t, ξ) is a continuous function and is uniformly Holder continuous in t ;T, P : (−∞, 0] → R ; w, r : [0, T ] × R → R ; v0 : (−∞, 0] × [0, π] → R andd : [0, π] → R are continuous functions. u(·) : [0, π] → R is a given control.

Consider E = L2([0, π],R) and define A(t) by A(t)w = a(t, ξ)w′′ with domain

D(A) = { w ∈ E : w, w′ are absolutely continuous, w′′ ∈ E, w(0) = w(π) = 0 }

Then A(t) generates an evolution system U(t, s) satisfying assumption (H1) and(H4) (see [24]).

For the phase space B, we choose the well known space BUC(R−, E) : the spaceof uniformly bounded continuous functions endowed with the following norm

‖ϕ‖ = supθ≤0

|ϕ(θ)| for ϕ ∈ B.

If we put for ϕ ∈ BUC(R−, E) and ξ ∈ [0, π]

y(t)(ξ) = v(t, ξ), t ∈ [0, T ], ξ ∈ [0, π],

φ(θ)(ξ) = v0(θ, ξ), −∞ < θ ≤ 0, ξ ∈ [0, π],

g(t, ϕ)(ξ) =∫ 0

−∞T (θ)w(t, ϕ(θ)(ξ))dθ, −∞ < θ ≤ 0, ξ ∈ [0, π],

and

f(t, ϕ)(ξ) =∫ 0

−∞P (θ)r(t, ϕ(θ)(ξ))dθ, −∞ < θ ≤ 0, ξ ∈ [0, π]

Finally let C ∈ B(R;E) be defined as

Cu(t)(ξ) = d(ξ)u(t), t ∈ [0, T ], ξ ∈ [0, π].

Then, problem (5.1) takes the abstract neutral evolution form (1.3)-(1.4). Inorder to show the controllability of mild solutions of system (5.1), we suppose thefollowing assumptions :






- w is Lipschitz with respect to its second argument. Let lip(w) denotes theLipschitz constant of w.

- There exist a continuous function p ∈ L1(J,R+) and a nondecreasing contin-uous function ψ : [0,∞) → [0,∞) such that

|r(t, u)| ≤ p(t)ψ(|u|), for t ∈ J, and u ∈ R.

- T and P are integrable on (−∞, 0].

By the dominated convergence theorem, one can show that f is a continuousfunction from B to E. Moreover the mapping g is Lipschitz continuous in its secondargument, in fact, we have

|g(t, ϕ1)− g(t, ϕ2)| ≤M0L?lip(w)∫ 0

−∞|T (θ)| dθ |ϕ1 − ϕ2| , for ϕ1, ϕ2 ∈ B.

On the other hand, we have for ϕ ∈ B and ξ ∈ [0, π]

|f(t, ϕ)(ξ)| ≤∫ 0

−∞|p(t)P (θ)|ψ(|(ϕ(θ))(ξ)|)dθ.

Since the function ψ is nondecreasing, it follows that

|f(t, ϕ)| ≤ p(t)∫ 0

−∞|P (θ)| dθψ(|ϕ|), for ϕ ∈ B.

Proposition 15. Under the above assumptions, if we assume that condition (4.2)in Theorem 14 is true, ϕ ∈ B, then the problem (5.1) is controllable on (−∞, π].

Acknowledgement 16. This work was completed when the second author was vis-iting the department of Mathematics of university of Marrakech. It is a pleasure forhim to express his gratitude for the provided support and the warm hospitality.

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Selma Baghli Mouffak Benchohra

Laboratoire de Mathematiques, Laboratoire de Mathematiques,

Universite de Sidi Bel-Abbes, Universite de Sidi Bel-Abbes,

BP 89, 22000 Sidi Bel-Abbes, Algerie. BP 89, 22000 Sidi Bel-Abbes, Algerie.

e-mail: selma [email protected] e-mail: [email protected]

Khalil Ezzinbi

Laboratoire de Mathematiques,

Faculte des Sciences de Semlalia,

BP 2390, Marrakech, Morocco.

e-mail: [email protected]





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