N° d’ordre
:REPUBLIQUE ALGERIENNE DEMOCRATIQUE & POPULAIRE MINISTERE DE L’ENSEIGNEMENT SUPERIEUR & DE LA RECHERCHE
SCIENTIFIQUE
UNIVERSITE DJILLALI LIABES FACULTE DES SCIENCES EXACTES
SIDI BEL ABBÈS
THESE
DE DOCTORAT
Présentée par
Cheheb Farida
Spécialité : MATHEMATIQUES
Option : EQUATIONS AUX DERIVEES PARTIELLES
Intitulée
« ……… »
Soutenue le 30/07/2019
Devant le jury composé de :
Président
: MECHAB MUSTAPHA
Professeur à l’Université de Sidi Bel Abbes.
Examinatrices : BOUDAOUD FATIMA
Professeur à l’Université d’Oran 1.ABDELLI Mama
MCA à l’Université de MascaraDirecteur de thèse : Abbes BENAISSA
Professeur à l’Université de Sidi Bel Abbes.Etude de l’existence Globale et de la
stabilité de certains systèmes
ﻭ
REMERCIEMENTS
Après plus d’une décennie de coupure, Le professeur Mr Benaissa Abbès m’a encouragée à renouer avec la recherche et pour cette première raison je le remercie vivement, tout comme je le remercie aussi de m’avoir encadrée et permis de réaliser un rêve d’enfance et surtout celui de mon défunt père: Avoir un doctorat en mathématiques. Mr Benaisse: Grand merci pour tout.
Mes remerciements iront aussi au Professeur Mechab Mustapha pour m’avoir honorée de sa présence et pour avoir accepté de présider ce jury. Comme ils s’adresseront aussi à mesdames les examinatrices Boudaoud Fatima de L’université D’oran1 et Abdelli Mama de l’université de Mascara .
Par ailleurs aucune expression ne traduira ma reconnaissance et ma gratitude en vers mon amie, soeur et collègue Mme Benyamina Naima.
Mes remerciements ne prendront fin sans les adresser à Mme Arbaoui, Mr Amroun, mon chef de Département Mr Bouhend et à Mr Benchohra, chacun pour l’aide qu’il m’a apportée.
Enfin que tous ceux et celles qui m’ont aidée d’une manière ou d’une autre, de près ou de loin trouveront ici l’expression de mes chaleureux remerciements.
Etude de l’existence globale et de stabilisation de certains systèmes hyperboliques
Comme le titre l’indique, le but de cette thèse est de se pencher sur certains problèmes de type hyperbolique, en l’occurrence étude de l’existence globale et de stabilisation de certains systèmes hyperboliques . Commençons alors par définir ce que c’est qu’un problème de type hyperbolique. C’est une classe d’équations aux dérivées partielles (EDP) modélisant des phénomènes de propagation aboutissant naturellement à l’étude du mouvement, des déformations ou des états d’équilibres des systèmes physiques dont un type est l’équation des ondes. Les solutions de ces problèmes possèdent les propriétés ondulatoires. Cela veut dire que si une perturbarion localisée est faite sur la donnée initiale d’un problème hyperbolique, alors les points de l’espace éloignés du support de la perturbation ne ressentiront pas ses effets immédiatement. Relativement à un point espace temps fixe, les perturbations ont une vitesse de propagation finie et se déplacent le long des caractéristiques de l’équation. Cette propriété permet de distinguer les problèmes hyperboliques des problèmes elliptiques ou paraboliques où les effets des conditions initiales ou des bords auront des effet instantannés sur tous les points du domaine.
Cette thèse se compse de quatres chapitres dont:
le premier consacré aux rappels de cetaines notions utilisées tout au long de cette thèse. le second Intitulé: "Global existence and energy decay of solutions to a non dissipative Timo-shenko beam system with delay terms" dans lequel nous considérons un système non dissipatif de Timochenko avec retard dans un domaine borné. Ce retard est un phénomène universel qui existe presque dans tous les domaines (mécaniques, électricité, biologie..) et rend le sys-tème moins productif, moins stable et moins optimal et parfois il y joue un rôle positif. Nous prouverons que dans les espaces de Sobolev l’existence des solutions est globale.Et moyen-nant la théorie des semi-groupes, Nous étudierons aussi le comportement assymptotique de l’energie. Le troisième Intitulé"Global existence and energy decay of solutions to a Bress system with delay terms.". Dans le même context et moyennant les mêmes méthodes nous prouverons l’existence globale et étudierons la stabilité du système.
le quatrième Intitiulé "A general result of a wave equation with dynamic boundary control of diffuse type" dans lequel nous ètablissons l’unicité et l’existence globale d’une solution faible et forte dans des espaces de sobolev appropriés ainsi que la stabilité dépendant de la forme de la mesure de diffusion.
Study of the global existence and stabilization of some hyperbolic systems
A hyperbolic problem is a class of partial differential equations modeling propagation phenomena that naturally leads to the study of motions, deformations or equilibrium states of physical systems, one type of which is the wave equation. The solutions to these problems have wave properties. This means that if a localized perturbation is made on the initial data of a hyperbolic problem, then the points in space far from the support of this perturbation will not feel its effects immediately. Relative to a fixed time point,perturbations have a finite propagation rate and move along the characteristics of the equation, which distinguishes hy-perbolic problem from elliptical or parabolic problems where the effects of initial conditions or edges will have instantaneous effects on all points in the domain. This thesis is composed of 4 chapters including:
First one: a reminder of some notions used in this thesis
the second chapter : Global existence and energy decay of solutions to a non dissipative Ty-moshenko beam system with delay terms in which we consider a non dissipative TiTy-moshenko beam system with delay in a bounded domain.This terms delay is are universal phenomenon that exists in almost fields of mechanics, electricity, biology,.... less stable and less optimal. However, it sometimes plays positive role in it. By means of the theory of semi-groups, we will prove that in the presence of this delay the global solution of the said system exist in the Sobolev space and we will study the asymptotic behaviour of these solutions using the multiplier method. In this case, exponential stability. The third entitled: Global existence and energy decay of solutions to a Bresse system with delay terms. In the same context , we will prove the existence of the solutions and examine their asymptotic behaviour, which is in this case exponential
In fourth We study a wave equation with a dynamic boundary control of diffusive type. We establish optimal and explicit energy decay formula by using resolvent estimates. Our new result generalizes and improve the earlier related results in the literature.
Keywords: Timochenko system, Bresse system, Global existence, Exponential decay, mul-tiplier method, beam equation, linear feedback, linear dissipation, delay term.
Contents
I Preliminaries 11
1 Sobolev spaces . . . 11
1.1 Definition of Sobolev Spaces . . . 14
1.2 Some inequalities. . . 17
2 Weak convergence . . . 18
2.1 Weak and strong convergence . . . 19
2.2 Bounded and unbounded linear operators . . . 19
3 Semigroups, Existence and uniqueness of solution . . . 22
4 Stability of semigroup . . . 25
5 Lax-Milgrame Theorem: . . . 27
6 Fractional Derivative Control . . . 27
6.1 Some history of fractional calculus: . . . 28
6.2 Various approaches of fractional derivatives . . . 29
II Global existence and energy decay of solutions to a nondissipative Timo-shenko beam system with delay terms 33 1 Introduction . . . 33
2 Preliminaries and main results . . . 35
3 Global existence . . . 39
4 Asymptotic Behavior . . . 42
III Global existence and energy decay of solutions to a Bresse system with delay terms 47 1 Introduction . . . 47
2 Preliminaries and main results . . . 50
3 Global existence . . . 54
IV A general decay result of a wave equation with a dynamic boundary control
of diffusive type 65
1 Introduction . . . 65
2 Preliminaries results . . . 67
3 Well-posedness . . . 69
4 Lack of exponential stability . . . 73
5 Optimality of energy decay η 6= 0 . . . . 77
Introduction
The subject of this thesis is the study of global existence, asymptotic behavior in time of solutions to linear evolutions equations of hyperbolic type. The decreasing of classical energy plays a crucial role in the study of global existence and in stabilization of various dis-tributed systems. This work consists for a problems of Timoshenko beam and Bress systems.
A simple model describing the transverse vibration of a beam, which was developed in
[36], is given by a system of coupled hyperbolic equations of the form
(
ρutt(x, t) = (K(ux− φ))x in ]0, L[×]0, +∞[, ˜
ρφtt(x, t) = (EIψx)x+ K(ux− φ) in ]0, L[×]0, +∞[,
where t denotes the time variable, x is the space variable along the beam of length L, in its equilibrium configuration, u is the transverse displacement of the beam and φ is the rotation
angle of the filament of the beam. The coefficients ρ, ˜ρ, E, I and K are respectively the
density (the mass per unit length), the polar moment of inertia of a cross section, Young’s modulus of elasticity, the moment of inertia of a cross section, and the shear modulus.
There are a number of publications concerning the stabilization of Timoshenko system with different kinds of damping (see [19], [25], [26] and [29]). Raposo et al. [32] proved the exponential decay of the solution for the following linear system of Timoshenko-type beam equations with linear frictional dissipative terms:
ρ1ϕtt− Gh(ϕx+ ψ)x+ µ1ϕt= 0
ρ2ψtt− EIψxx+ Gh(ϕx+ ψ) +µf1ψt= 0.
Messaoudi and Mustafa [25] (see also [29]) considered the stabilization for the following Timoshenko system with nonlinear internal feedbacks:
ρ1ϕtt− Gh(ϕx+ ψ)x+ g1(ψt) = 0
Recently, Park and Kang [29] considered the stabilization of the Timoshenko system with weakly nonlinear internal feedbacks.
The original Bresse system is given by the following equations (see [10]) :
ρ1ϕtt = Qx+ lN + F1, ρ2ψtt = Mx− Q + F2, ρ1ωtt = Nx− lQ + F3,
where we use N, Q and M to denote the axial force, the shear force and the bending moment respectively. These forces are stress-strain relations for elastic behavior and given by
N = Eh(ωx− lϕ), Q = Gh(ϕx+ ψ + lω), and M = EIψx,
where G, E, I and h are positive constants. Finally, by the terms Fiwe are denoting external
forces.
The Bresse system without delay (i.e µ2 =µf2 =µff2 = 0), is more general than the
well-known Timoshenko system where the longitudinal displacement ω is not considered l = 0.
Chapter 1: Global existence and energy decay of solutions to a nondissipative Timoshenko beam system with delay terms
In this chapter we investigate the existence and decay properties of solutions for the initial boundary value problem of the linear Timochenko system of the type
(
ρ1ϕtt− K(ϕx+ ψ)x+ µ1ϕt+ µ2ϕt(x, t − τ1) = 0
ρ2ψtt− bψxx+ ˜K(ϕx+ ψ) +µf1ψt+µf2ψt(x, t − τ2) = 0
(P )
where (x, t) ∈ (0, L) × (0, +∞), τi > 0 (i = 1, 2, 3) is a time delay, µ1, µ2,µf1,µf2, are positive
real numbers. This system is subject to the Dirichlet boundary conditions
and to the initial conditions ϕ(x, 0) = ϕ0(x), ϕt(x, 0) = ϕ1(x), ψ(x, 0) = ψ0(x), ψt(x, 0) = ψ1(x), x ∈ (0, L) ϕt(x, t − τ1) = f0(x, t − τ1), in (0, L) × [0, τ1] ψt(x, t − τ2) = ˜f0(x, t − τ2), in (0, L) × [0, τ2]
where the initial data (ϕ0, ϕ1, ψ0, ψ1, f0, ˜f0) belong to a suitable Sobolev space.
We prove the global solvability in Sobolev spaces and energy decay estimates of the solutions
to the problem (P ) for linear damping and delay terms and in the case when k 6= ˜k. To
obtain global solutions to the problem (P ), we use the argument combining the semigroup theory (see [28] and [11]) with the energy estimate method. To prove decay estimates, we use a multiplier method.
Chapter 2 : Global existence and energy decay of solutions to a Bresse system with delay terms
In this chapter we investigate the existence and decay properties of solutions for the initial boundary value problem of the linear Bresse system of the type
ρ1ϕtt− K(ϕx+ ψ + lω)x− lK0(ωx− lϕ) + µ1ϕt+ µ2ϕt(x, t − τ1) = 0 ρ2ψtt− bψxx+ K(ϕx+ ψ + lω) +µf1ψt+µf2ψt(x, t − τ2) = 0 ρ1ωtt− K0(ωx− lϕ)x+ lK(ϕx+ ψ + lω) +f f µ1ωt+f f µ2ωt(x, t − τ3) = 0 (P )
where (x, t) ∈ (0, L) × (0, +∞), τi > 0 (i = 1, 2, 3) is a time delay, µ1, µ2,µf1,µf2,µff1,µff2 are
positive real numbers. This system is subject to the Dirichlet boundary conditions
ϕ(0, t) = ϕ(L, t) = ψ(0, t) = ψ(L, t) = ω(0, t) = ω(L, t) = 0, t > 0 and to the initial conditions
ϕ(x, 0) = ϕ0(x), ϕt(x, 0) = ϕ1(x), ψ(x, 0) = ψ0(x), ψt(x, 0) = ψ1(x), ω(x, 0) = ω0(x), ωt(x, 0) = ω1(x), x ∈ (0, L) ϕt(x, t − τ1) = f0(x, t − τ1), in (0, L) × [0, τ1] ψt(x, t − τ2) = ˜f0(x, t − τ2), in (0, L) × [0, τ2] ωt(x, t − τ3) = fe˜0(x, t − τ3), in (0, L) × [0, τ3]
solutions to the problem (P ) for linear damping and delay terms. To obtain global solutions to the problem (P ), we use the argument combining the semigroup theory (see [28] and [11]) with the energy estimate method. To prove decay estimates, we use a multiplier method.
Chapter 3: A general decay result of a wave equation with a dynamic bound-ary control of diffusive type
In this chapter we investigate the existence and decay properties of solutions for the initial boundary value problem of the wave equation of the type
(P ) ytt(x, t) − yxx(x, t) = 0 in (0, L) × (0, +∞) y(0, t) = 0 in (0, +∞) mytt(L, t) + yx(L, t) = −ζ Z +∞ −∞ µ(ξ)φ(ξ, t) dξ in (0, +∞) ∂tφ(ξ, t) + (ξ2+ η)φ(ξ, t) − yt(L, t)µ(ξ) = 0 in (−∞, ∞) × (0, +∞) y(x, 0) = y0(x), yt(x, 0) = y1(x) in (0, L) φ(ξ, 0) = φ0 in (−∞, ∞) where (x, t) ∈ (0, L) × (0, +∞), m > 0, ζ > 0 and η ≥ 0.
we give a global solvability in Sobolev spaces and optimal energy decay estimates of the solutions to the problem (P ). We think that the interaction of the tip mass term and the control of fractional derivative type have an effect on the result of [6]. Moreover we obtain a precise and optimal energy decay estimate for a general control of diffusive type, from which the usual control of fractional derivative type is a special case.
Chapter I
Preliminaries
1
Sobolev spaces
We denote by Ω an open domain in Rn, n ≥ 1, with a smooth boundary Γ = ∂Ω. In general,
some regularity of Ω will be assumed. We will suppose that either Ω is Lipschitz, a.e., the boundary Γ is locally the graph of a Lipschitz function, or
Ω is of class Cr, r ≥ 1,
a.e., the boundary Γ is a manifold of dimension n ≥ 1 of class Cr. In both cases we assume
that Ω is totally on one side of Γ. These definition mean that locally the domain Ω is below the graph of some function ψ, the boundary Γ is represented by the graph of ψ and its regularity is determined by that of the function ψ. Moreover, it is necessary to note that a domain with a continuous boundary is never on both sides of its boundary at any point of this boundary and that a Lipschitz boundary has almost everywhere a unit normal vector ν. We will also use the following multi-index notation for partial differential derivatives of a function: ∂k iu = ∂ku ∂xk i
for all k ∈ N and i = 1, ..., n,
Dαu = ∂α1 1 ∂ α2 2 . . . ∂nαnu = ∂α1+...+αnu ∂xα1 1 . . . ∂xαnn , α = (α1, α2, . . . , αn) ∈ Nn, |α| = α1+ . . . + αn.
We denote by C(D) (respectively Ck(D), k ∈ N or k = +∞) the space of real continuous
functions on D (respectively the space of k times continuously differentiable functions on D), where D plays the role of Ω or its closure Ω. The space of real C∞ functions on Ω
of Schwartz.The distributions space on Ω is denoted by D0(Ω), a.e., the space of continuous
linear form over D(Ω).
Definition 1.1 We define Lp(Ω) as: If p ∈ [1, +∞[, Lp(Ω) = f : Ω → R Z Ω |f (x)|pdx < ∞ We define on Lp(Ω) the norm :
kf kLp(Ω) = Z Ω |f (x)|pdx 1 p If p = +∞, L∞(Ω) = {f : Ω −→ R /∃c ∈ R |f (x)|6c a.e in Ω}.
We define in L∞(Ω) the norm :
||f ||L∞(Ω) = inf {c ∈ R : |f |6c a.e in Ω}.
For 1 ≤ p ≤ ∞, we call Lp(Ω) the space of measurable functions f on Ω such that
kf kLp(Ω) = Z Ω |f (x)|pdx1/p < +∞ for p < +∞ kf kL∞(Ω) = sup Ω |f (x)| < +∞ for p = +∞
The space Lp(Ω) equipped with the norm f −→ kf kLp is a Banach space: it is reflexive and
separable for 1 < p < ∞ (its dual is Lp−1p (Ω)), separable but not reflexive for p = 1 (its dual
is L∞(Ω)), and not separable, not reflexive for p = ∞ (its dual contains strictly L1(Ω)). In
particular the space L2(Ω) is a Hilbert space equipped with the scalar product defined by
(f, g)L2(Ω)=
Z
Ω
f (x)g(x)dx.
We denote by Lploc(Ω) the space of functions which are Lp on any bounded sub-domain of Ω.
Similar space can be defined on any open set other than Ω, in particular, on the cylinder set Ω×]a, b[ or on the set Γ×]a, b[, where a, b ∈ R and a < b. Let X the space of Hilbert and
]a, b[ an open interval in Rn. we refer to the measure of Lebesgue as dt in ]a, b[ and as k.k
X The norm in X.
Lp(a, b; X) = {f :]a, b[−→ X mesurable} We provide the space Lp(a, b; X) of the norm
kf kLp(a,b;X) = Z b a kf (t)kpX !1 p
- We define L∞(a, b; X), as being the espace of functions defined from ]a, b[→ X, measurable et bounded presume all over in ]a, b[, provided of the norm
kf kL∞(a,b;X) = sup
t∈[a,b]
esskf (t)kX
- Lp(a, b; X) (16p6 + ∞) is space of Banach for the norm defined above.
a- The space L2(a, b; X) is space of Hilbert for the inner product
(f, g)L2(a,b;X) =
Z b
a
(f (t), g(t))xdt
or (., .)X is the inner product in X.
b- For 16p6 + ∞, Lp(a, b; X) is an space separable.
The injection of Lp(a, b; X) in D0(]a, b[; X) is strongly and weakly sequentializing continuous a.e: If : fj → f strongly (resp weakly) in Lp(a, b; X), the fj tends strongly (resp weakly) towards f in the meaning of D0(]a, b[; X).
kf kLp(a,b;X) = Z b a kf (x)kpX dt !1/p < +∞ for p < +∞
and for the norm
kf kL∞(a,b;X) = sup t∈(a,b)
kf (x)kU < +∞ for p = +∞
Similarly, for a Banach space X, k ∈ N and −∞ < a < b < +∞, we denote by C([a, b]; X)
(respectively Ck([a, b]; X)) the space of continuous functions (respectively the space of k
respectively, for the norms kf kC(a,b;U ) = sup t∈(a,b) kf (x)kX, kf kCk(a,b;X) = k X i=0 ∂if ∂ti C(a,b;X)
1.1
Definition of Sobolev Spaces
Now, we will introduce the Sobolev spaces: The Sobolev space Wk,p(Ω) is defined to be the
subset of Lp such that function f and its weak derivatives up to some order k have a finite
Lp norm, for given p ≥ 1.
Wk,p(Ω) = {f ∈ Lp(Ω); Dαf ∈ Lp(Ω). ∀α; |α| ≤ k} ,
With this definition, the Sobolev spaces admit a natural norm,
f −→ kf kWk,p(Ω) = X |α|≤m kDαf kp Lp(Ω) 1/p , for p < +∞ and f −→ kf kWk,∞(Ω) = X |α|≤m kDαf k L∞(Ω) , for p = +∞
Space Wk,p(Ω) equipped with the norm k . k
Wk,p is a Banach space. Moreover is a reflexive
space for 1 < p < ∞ and a separable space for 1 ≤ p < ∞. Sobolev spaces with p = 2 are especially important because of their connection with Fourier series and because they form a Hilbert space. A special notation has arisen to cover this case:
Wk,2(Ω) = Hk(Ω)
the Hk inner product is defined in terms of the L2 inner product:
(f, g)Hk(Ω)=
X
|α|≤k
(Dαf, Dαg)L2(Ω) .
The space Hm(Ω) and Wk,p(Ω) contain C∞(Ω) and Cm(Ω). The closure of D(Ω) for the
Hm(Ω) norm (respectively Wm,p(Ω) norm) is denoted by H0m(Ω) (respectively W0k,p(Ω)).
Now, we introduce a space of functions with values in a space X (a separable Hilbert space).
The space L2(a, b; X) is a Hilbert space for the inner product
(f, g)L2(a,b;X) =
Z b
a
(f (t), g(t))Xdt
We note that L∞(a, b; X) = (L1(a, b; X))0.
Now, we define the Sobolev spaces with values in a Hilbert space X For k ∈ N, p ∈ [1, ∞], we set: Wk,p(a, b; X) = ( v ∈ Lp(a, b; X); ∂v ∂xi ∈ Lp(a, b; X). ∀i ≤ k ) ,
The Sobolev space Wk,p(a, b; X) is a Banach space with the norm
kf kWk,p(a,b;X) = k X i=0 ∂f ∂xi p Lp(a,b;X) 1/p , for p < +∞ kf kWk,∞(a,b;X) = k X i=0 ∂v ∂xi L∞(a,b;X) , for p = +∞
The spaces Wk,2(a, b; X) form a Hilbert space and it is noted Hk(0, T ; X). The Hk(0, T ; X)
inner product is defined by:
(u, v)Hk(a,b;X) = k X i=0 Z b a ∂u ∂xi, ∂v ∂xi ! X dt .
Theorem 1.3 Let 1 ≤ p ≤ n, then
W1,p(Rn) ⊂ Lp∗(Rn) where p∗ is given by 1 p∗ = 1 p − 1 n (where p = n, p
∗ = ∞). Moreover there exists a constant
C = C(p, n) such that
kukLp∗ ≤ Ck∇ukLp(Rn)∀u ∈ W1,p(Rn).
Corolary 1.4 Let 1 ≤ p < n, then
W1,p(Rn) ⊂ Lq(Rn) ∀q ∈ [p, p∗]
For the case p = n, we have
W1,n(Rn) ⊂ Lq(Rn) ∀q ∈ [n, +∞[
Theorem 1.5 Let p > n, then
W1,p(Rn) ⊂ L∞(Rn)
with continuous imbedding.
Corolary 1.6 Let Ω a bounded domain in Rn of C1 class with Γ = ∂Ω and 1 ≤ p ≤ ∞. We have if 1 ≤ p < ∞, then W1,p(Ω) ⊂ Lp∗(Ω) where 1 p∗ = 1 p − 1 n. if p = n, then W1,p(Ω) ⊂ Lq(Ω), ∀q ∈ [p, +∞[. if p > n, then W1,p(Ω) ⊂ L∞(Ω)
with continuous imbedding.
Moreover, if p > n, we have: ∀u ∈ W1,p(Ω),
|u(x) − u(y)| ≤ C|x − y|αkukW1,p(Ω) a.e x, y ∈ Ω
with α = 1 − n
p > 0 and C is a constant which depend on p, n and Ω. In particular
W1,p(Ω) ⊂ C(Ω).
Corolary 1.7 Let Ω a bounded domain in Rn of C1 class with Γ = ∂Ω and 1 ≤ p ≤ ∞. We
have if p < n, then W1,p(Ω) ⊂ Lq(Ω)∀q ∈ [1, p∗[ where 1 p∗ = 1 p− 1 n. if p = n, then W1,p(Ω) ⊂ Lq(Ω), ∀q ∈ [p, +∞[. if p > n, then W1,p(Ω) ⊂ C(Ω) with compact imbedding.
Remark 1.8 We remark in particular that
W1,p(Ω) ⊂ Lq(Ω)
Corolary 1.9 if 1 p − m n > 0, then W m,p (Rn) ⊂ Lq(Rn) where 1 q = 1 p − m n. if 1 p − m n = 0, then W m,p (Rn) ⊂ Lq(Rn), ∀q ∈ [p, +∞[. if 1 p − m n < 0, then W m,p (Rn) ⊂ L∞(Rn)
with continuous imbedding.
1.2
Some inequalities.
Formula of Green: Let u, v ∈ W (a, b, V, V0) with a, b finished. Then we have the formula of the Green: Z b a * du dt(x), v(t) + V ×V0 dt + Z b a * dv dt(x), u(t) + V ×V0
dt = (u(b), v(b)) − (u(a), v(a)).
Proposition 1.10 For u ∈ W (a, b, V, V0) et v ∈ V , we have:
* du dt(.), v + V ×V0 = d dt(u(.), v), in D 0 (]a, b[).
Young Inequality : For all a, b ∈ R, (or C) and for all p, q ∈ [1, +∞[ with 1 q + 1 p = 1, we have : |ab|61 p|a| p+1 q|b| q.
Hölder Inequality : Let 1 < p, q < +∞, with 1 p +
1
q = 1. Let f the function de L
p(Ω)
et g one function de Lq(Ω). Then Hölder l’inequality writes:
kf gkL1(Ω) = kf kLp(Ω) kgkLq(Ω)· a.e Z Ω|f (x)g(x)| dx6 Z Ω |f (x)p| dx 1pZ Ω |g(x)q| dx 1q , if p, q ∈ [1, +∞[, Z Ω|f (x)g(x)|dx6kgkL ∞ Z Ω |f (x)| dx, if p = 1, and q = +∞.
Green Formula: Let Ω an open bounded of frontiers regulars ∂Ω and v(x) the normal
exteriors the point x. Let u a function of H2(Ω) and v a function de H1(Ω). then the Green
formula write : Z Ω (∆u)vdx = Z ∂Ω ∂u ∂nvds − Z Ω ∇u ∇vdx Z Ω (u4v − v4u)dx = Z ∂Ω u∂u ∂n − v ∂u ∂n !
2
Weak convergence
Let (E; k.kE) a Banach space and E0 its dual space, a.e., the Banach space of all continuous
linear forms on E endowed with the norm k.k0E defined by
kf kE0 =: sup
x6=0
|hf, xi| kxk
; where hf, xi; denotes the action of f onx, i.e.hf, xi := f (x). In the same way, we can define
the dual space of E0 that we denote by E00. (The Banach space E00 is also called the bi-dual
space of E.) An element x of E can be seen as a continuous linear form on E0 by setting
x(f ) := hx, f i, which means that E ⊂ E00:
Definition 2.1 The Banach space E is said to be reflexive if E = E00.
Definition 2.2 The Banach space E is said to be separable if there exists a countable subset D of E which is dense in E, a.e. D = E.
Theorem 2.3 (Riesz). If (H; h., .i) is a Hilbert space, h., .i being a scalar product on H, then H0 = H in the following sense: to each f ∈ H0 there corresponds a unique x ∈ H such that f = hx, .i and kf k0H = kxkH
Remark : From this theorem we deduce that H00 = H. This means that a Hilbert space is
reflexive.
Proposition 2.4 If E is reflexive and if F is a closed vector subspace of E, then F is reflexive.
Corolary 2.5 The following two assertions are equivalent: (i) E is reflexive; (ii) E0 is reflexive.
2.1
Weak and strong convergence
Definition 2.6 (Weak convergence in E). Let x ∈ E and let {xn} ⊂ E. We say that {xn} weakly converges to x in E, and we write xn* x in E, if hf, xni → hf, xi, for all f ∈ E0.
Definition 2.7 (weak convergence in E0). Let f ∈ E0 and let {fn} ⊂ E0. We say that {fn} weakly converges to f in E0, and we write fn* f in E0, if hfn, xi → hf, xi, for all x ∈ E00.
Definition 2.8 (strong convergence). Let x ∈ E(resp. f ∈ E0) and let {xn} ⊂ E (resp
{fn} ⊂ E0). We say that {xn} (resp. {fn}) strongly converges to x (resp. f), and we write
xn→ x in E (resp. fn → f in E0), if
lim
n kxn− xkE = 0; (resp. limn kfn− f k
0
E = 0)
Proposition 2.9 Let x ∈ E, let {xn} ⊂ E, let f ∈ E0 and let {fn} ⊂ E0.
i. If xn → x in E then xn* x in E.
ii. If xn * x in E then {xn} is bounded.
iii. If xn * x in E then lim infn→∞ kxnkE ≥ kxkE
iv. If fn→ f in E0 then fn* f inE0 (and so fn
∗
* f in E0).
v. If fn* f in E0 then {fn} is bounded. vi. If fn* f in E0 then then lim inf
n→∞ kfnk
0
E ≥ kf k
0
E.
Proposition 2.10 (finite dimension). If dim E < ∞ then strong, weak and weak star convergence are equivalent.
2.2
Bounded and unbounded linear operators
Let (E, k.kE) and (F, k.kF) be two Banach spaces over C, and H will always denote a Hilbert
space equipped with the scalar product < ., . >H and the corresponding norm k.kH. A linear
operator T : E −→ F is a transformation which maps linearly E in F , that is
Definition 2.11 A linear operator T : E −→ F is said to be bounded if there exists c ≥ 0 such that
kT ukF ≤ ckukE ∀u ∈ E.
The set of all bounded linear operators from E into F is denoted by L(E, F ). Moreover, the set of all bounded linear operators from E into E is denoted by L(E).
Definition 2.12 A bounded operator T ∈L(E, F ) is said to be compact if for each sequence
(xn)n∈N ∈ E with kxnkE = 1 for each n ∈ N, the sequence (T xn)n∈N has a subsequence which
converges in F .
The set of all compact operators from E into F is denoted by K(E, F ). For simplicity one writes K(E) = K(E, F ).
Definition 2.13 Let T ∈L(E, F ) we define
• Range of T by
R(T ) = {T u : u ∈ E} ⊂ F. • Kernel of T by
ker(T ) = {u ∈ E : T u = 0} ⊂ E. Theorem 2.14 (Fredholm alternative)
If T ∈K(E), then
• ker(I − T ) is finite dimension, (I is the identity operator on E) .
• R(I − T ) is closed.
• ker(I − T ) = 0 ⇔R(I − T ) = E.
Definition 2.15 An unbounded linear operator T from E into F is a pair (T, D(T )), con-sisting of a subspace D(T ) ⊂ E (called the domain of T ) and a linear transformation.
T : D(T ) ⊂ E 7→ F.
In the case when E = F then we say (T, D(T )) is an unbounded linear operator on E. If D(T ) = E then T ∈L(E, F ).
Definition 2.16 Let T : D(T ) ⊂ E 7→ F be an unbounded linear operator.
• The range of T is defined by
• The Kernel of T is defined by
ker(T ) = {u ∈ D(T ) : T u = 0} ⊂ E. • The graph of T is defined by
G(T ) = {(u, T u) : u ∈ D(T )} ⊂ E × F.
Definition 2.17 A map T is said to be closed if G(T ) is closed in E × F . The closedness of an unbounded linear operator T can be characterize as following if un ∈ D(T ) such that
un→ u in E and T un → v in F , then u ∈ D(T ) and T u = v.
Definition 2.18 Let T : D(T ) ⊂ E 7→ F be a closed unbounded linear operator.
• The resolvent set of T is defined by
ρ(T ) = {λ ∈ C : λI − T is bijective from D(T ) onto F }. • The resolvent of T is defined by
R(λ, T ) = {(λI − T )−1
: λ ∈ ρ(T )}
• The spectrum set of T is the complement of the resolvent set in C , denoted by
σ(T ) = C/ρ(T )
Definition 2.19 Let T : D(T ) ⊂ E 7→ F be a closed unbounded linear operator. we can split the spectrum σ(T ) of T into three disjoint sets, given by
• The punctual spectrum of T is define by
σp(T ) = {λ ∈ C : ker(λI − T ) 6= {0}} in this case λ is called an eigenvalue of T .
• The continuous spectrum of T is define by:
• The residual spectrum of T is define by
σr(T ) = {λ ∈ C : ker (λI − T ) = 0, and R(λI − T ) is not dense in F }
Definition 2.20 Let T : D(T ) ⊂ E −→ F be a closed unbounded linear operator and let λ be an eigevalue of A. non-zero element e ∈ E is called a generalized eigenvector of T associated with the eigenvalue value λ, if there exists n ∈ N∗ such that
(λI − T )ne = 0 and (λI − T )n−1e 6= 0.
if n = 1, then e is called an eigenvector.
Definition 2.21 Let T : D(T ) ⊂ E −→ F be a closed unbounded linear operator. We say that T has compact resolvent, if there exist λ0 ∈ ρ(T ) such that (λ0I − T )−1 is compact.
Theorem 2.22 Let (T, D(T )) be a closed unbounded linear operator on H then the space
(D(T ), k.kD(T )) where kukD(T ) = kT ukH + kukH ∀u ∈ D(T ) is Banach space .
Theorem 2.23 Let (T, D(T )) be a closed unbounded linear operator on H then, ρ(T ) is an open set of C.
3
Semigroups, Existence and uniqueness of solution
The vast majority of the evolution equations can be reduced to the form
(
Ut(t) =AU(t), t > 0,
U (0) = U0,
(I.1)
where A is the infinitesimal generator of a C0-semigroup S(t) over a Hilbert space H.
Lets start by basic definitions and theorems.
Let (X, k.kX) be a Banach space, and H be a Hilbert space equipped with the inner product
< ., . >H and the induced norm k.kH .
Definition 3.1 Let X be a Banach space and let I : X → X its identity operator.
1. A one parameter family (S(t))t≥0, of bounded linear operators from X into X is a
(i) S(0) = I;
(ii) S(t + s) = S(t)S(s) for every s, t ≥ 0.
2. A semigroup of bounded linear operators, (S(t))t≥0, is uniformly continuous if
lim
t→0kS(t) − Ik = 0.
3. A semigroup (S(t))t≥0 of bounded linear operators on X is a strongly continuous
semi-group of bounded linear operators or a C0-semigroup if
lim
t→0S(t)x = x
4. The linear operator A defined by Ax = lim t→0 S(t)x − x t , ∀x ∈ D(A) where D(A) =nx ∈ X; lim t→0 S(t)x − x t exists o
is the infinitesimal generator of the semigroup (S(t))t≥0.
Some properties of semigroup and its generator operator A are given in the following
theo-rems:
Theorem 3.2 (Pazy) Let A be the infinitesimal generator of a C0- semigroup of
contrac-tions (S(t))t≥0. Then, the resolvent (λI −A)−1 of A contains the open right half-plane, a.e.,
ρ(A) ⊂ {λ : R(λ) > 0} and for such λ we have
k(λI −A)−1kL(H)≤ 1
R(λ).
Theorem 3.3 (Kato) Let A be a closed operator in a Banach space X such that the resol-vent (I −A)−1 of A exists and is compact. Then the spectrum σ(A) of A consists entirely of isolated eigenvalues with finite multiplicities.
Theorem 3.4 (Pazy) Let (S(t))t≥0 be a C0-semigroup on a Hilbert space H. Then there
exist two constants ω ≥ 0 and M ≥ 1 such that
If ω = 0, the semigroup (S(t))t≥0 is called uniformly bounded and if moreover M = 1, then
it is called a C0-semigroup of contractions. For the existence of solution of problem (I.1), we
typically use the following Lumer-Phillips and Hille-Yosida theorems :
Theorem 3.5 (Lumer-Phillips) LetA be a linear operator with dense domain D(A) in a Hilbert space H. If
(i) A is dissipative, i.e., < R(< Ax, x >H) ≤ 0, ∀x ∈ D(A)
and if
(ii) there exists a λ0 > 0 such that the range R(λ0I −A) = H,
then A generates a C0-semigroup of contractions on H.
Theorem 3.6 (Hille-Yosida) Let A be a linear operator on a Banach space X and let ω ∈ R, M ≥ 1 be two constants. Then the following properties are equivalent
(i) A generates a C0-semigroup (S(t))t≥0, satisfying
kS(t)kL(H)≤ M eωt, ∀t ≥ 0.
(ii) A is closed, densely defined, and for every λ > ω one has λ ∈ ρ(A) and
k(λ − ω)n(λ −A)−n
k ≤ M, ∀n ∈ N.
(iii) A is closed, densely defined, and for every λ ∈ C with R > ω, one has λ ∈ ρ(A) and
k(λ −A)−nk ≤ M
(R(λ) − ω)n, ∀n ∈ N.
Consequently, A is maximal dissipative operator on a Hilbert space H if and only if it
generates a C0-semigroup of contractions (S(t))t≥0 on H. Thus, the existence of solution is
justified by the following corollary which follows from Lumer-Phillips theorem.
Corolary 3.7 Let H be a Hilbert space and letA be a linear operator defined from D(A) ⊂ H into H. If A is maximal dissipative operator then the initial value problem (I.1) has a unique solution U (t) = SA(t)U 0 such that U ∈ C([0, +1), H), for each initial datum U0 ∈ H.
Moreover, if U0 ∈ D(A), then
Finally, we also recall the following theorem concerning a perturbations by a bounded linear operators
Theorem 3.8 Let X be a Banach space and let A be the infinitesimal generator of a C0
-semigroup (S(t))t≥0 on X, satisfying kSA(t)kL(H) ≤ M eωt for all t ≥ 0. If B is a bounded linear operator on X , then the operator A + B becomes the infinitesimal generator of a C0-semigroup (SA+B(t))t≥0 on X, satisfying kSA+B(t)kL(H)≤ M e(ω+M kBk)t for all t ≥ 0 .
4
Stability of semigroup
In this section we start by introducing some definition about strong, exponential and
poly-nomial stability of a C0-semigroup. Then we collect some results about the stability of
C0-semigroup. Let (X, k.kX) be a Banach space, and H be a Hilbert space equipped with
the inner product < ., . >H and the induced norm k.kH.
Definition 4.1 Assume that A is the generator of a strongly continuous semigroup of con-tractions (S(t))t≥0 on X. We say that the C0-semigroup (S(t))t≥0 is
1. Strongly stable if lim t→+∞kS(t)ukX = 0, ∀u ∈ X. 2. Uniformly stable if lim t→+∞kS(t)kL(X) = 0.
3. Exponentially stable if there exist two positive constants M and such that
kS(t)ukX ≤ M e−tkukX, ∀t > 0, ∀u ∈ X.
4. Polynomially stable if there exist two positive constants C and α such that
kS(t)ukX ≤ Ct−αkukX, ∀t > 0, ∀u ∈ X.
Proposition 4.2 Assume that A is the generator of a strongly continuous semigroup of contractions (S(t))t≥0 on X. The following statements are equivalent
• (S(t))t≥0 is uniformly stable.
First, we look for the necessary conditions of strong stability of a C0-semigroup. The result
was obtained by Arendt and Batty.
Theorem 4.3 (Arendt and Batty) Assume that A is the generator of a strongly continuous semigroup of contractions (S(t))t≥0 on a reflexive Banach space X. If
(i) A has no pure imaginary eigenvalues.
(ii) σ(A) ∩ iR is countable. Then S(t) is strongly stable.
Remark 4.4 If the resolvent (I − T )−1 of T is compact, then σ(T ) = σp(T ). Thus, the state of Theorem (...) lessens to σp(A) ∩ iR = ∅. Next, when the C0-semigroup is strongly
stable, we look for the necessary and sufficient conditions of exponential stability of a C0
-semigroup. In fact, exponential stability results are obtained using different methods like: multipliers method, frequency domain approach, Riesz basis approach, Fourier analysis or a combination of them .
Theorem 4.5 (Huang-Pruss)Assume that A is the generator of a strongly continuous semi-group of contractions (S(t))t≥0 on H. S(t) is uniformly stable if and only if
1. iR ⊂ ρ(A).
2. supβ∈Rk(iβI −A)−1k
L(H) < +∞.
The second one, is a classical method based on the spectrum analysis of the operator A
In the case when the C0-semigroup is not exponentially stable we look for a polynomial
one. In general, polynomial stability results also are obtained using different methods like : multipliers method, frequency domain approach, Riesz basis approach, Fourier analysis or a combination of them .
Theorem 4.6 (Batty , A.Borichev and Y.Tomilov, Z. Liu and B. Rao.)Assume thatA is the generator of a strongly continuous semigroup of contractions (S(t))t≥0 on H. If iR ⊂ ρ(A), then for a fixed l > 0 the following conditions are equivalent
1. lim|λ|→+∞supλ1lk(λI −A) −1k
L(H) < +∞.
5
Lax-Milgrame Theorem:
Let H be a Hilbert space equipped with the inner product (., .)H and the induced norm k.kH.
Definition 5.1 A bilinear form
a : H × H → R is said to be
• (i) continuous if there is a constant C such that
|a(u, v)| ≤ Ckukkvk, ∀u, v ∈ H
• (ii) coercive if there is a constant α > 0 such that
|a(u, u)| ≥ αkuk2, ∀u ∈ H
Theorem 5.2 (Lax-Milgrame Theorem) Assume that a(., .) is a continuous coercive bilinear form on H. Then, given any L ∈ L(H, C), there exists a unique element u ∈ H such that
a(u, v) = L(v), ∀v ∈ H
6
Fractional Derivative Control
In this part, we introduce the necessary elements for the good understanding of this manuscript. It includes a brief reminder of the basic elements of the theory of fractional computation. The concept of fractional computation is a generalization of ordinary derivation and inte-gration to an arbitrary order. Derivatives of non-integer order are now widely applied in many domains, for example in economics, electronics, mechanics, biology, probability and viscoelasticity. A particular interest for fractional derivation is related to the mechanical modeling of gums and rubbers. In short, all kinds of materials that preserve the memory of previous deformations in particular viscoelastic. Indeed, the fractional derivation is intro-duced naturally.
The fractional calculus is an important developing field in both pure and applied mathemat-ics. Many real world problems have been investigated within the fractional derivatives, par-ticularly Caputo fractional derivative is extensively and successfully used in many branches of sciences and engineering.
6.1
Some history of fractional calculus:
In a letter dated September 30th, 1695 L’Hospital wrote to Leibniz asking him about the
meaning of dny/dxn if n = 1/2, that is « what if n is fractional? ». Leibniz response:« An
apparent paradox, from which one day useful consequences will be drawn »
In 1819 S. F. Lacroix [100] was the first to mention in some two pages a derivative of
arbitrary order.Thus for y = xa, a ∈ R
+, he showed that d1/2y dx1/2 = Γ(a + 1) Γ(1 + 1/2)x a−1/2 . In particular he had (d/dx)1/2x = 2qx/π.
In 1822 J. B. J. Fourier derived an integral representation for f(x),
f (x) = 1 2π Z R f (α)dα Z R cos p(x − α)dp,
obtained (formally) the derivative version dν dxνf (x) = 1 2π Z R f (α)dα Z R pνcos[p(x − α) +νπ 2 ]dp
where "the number v will be regarded as any quantity whatever, positive ornegative". In 1823 Abel resolved the integral equation arising from the brachistochrone problem, namely 1 Γ(α) Z x 0 g(u) (x − u)1−αdu = f (x), 0 < α < 1
with the solution
g(x) = 1 Γ(1 − α) d dx Z x 0 f (u) (x − u)αdu
Abel never solved the problem by fractional calculus but, in 1832 Liouville [103], did solve this integral equation.
Perhaps the first serious attempt to give a logical definition of a fractional derivative is due to Liouville; he published nine papers on the subject between 1832 and 1837, the last in the field in 1855. They grew out of Liouville’s early work on electromagnetism. There is further work of George Peacock (1833), D. F. Gregory (1841), Augustus de Morgan (1842), P. Kelland (1846), William Center (1848). Especially basic is Riemann’s student paper of
1847 [139].
mathematicians spearheading research in the broad area of fractional calculus until 1941 were S.F. Lacroix, J.B.J. Fourier, N.H. Abel, J. Liouville, A. De Morgan, B. Riemann, Hj. Holmgren, K. Griinwald, A.V. Letnikov, N.Ya. Sonine, J. Hadamard, G.H. Hardy, H. Weyl, M. Riesz, H.T. Davis, A. Marchaud, J.E. Littlewood, E.L. Post, E.R. Love, B.Sz.-Nagy, A. Erdelyi and H. Kober.
Fractional calculus has developed especially intensively since 1974 when the first interna-tional conference in the field took place.It was organized by Bertram Ross [144].
Samko et al in their encyclopedic volume [153, p. xxxvi] state and we cite: "We pay tribute to investigators of recent decades by citing the names of mathematicians who have made a valu-able scientific contribution to fractional calculus development from 1941 until the present [1990]. These are M.A. Al- Bassam, L.S. Bosanquet, P.L. Butzer, M.M. Dzherbashyan, A. Erdelyi, T.M. Flett, Ch. Fox, S.G. Gindikin, S.L. Kalla, LA. Kipriyanov, H. Kober, P.I. Lizorkin, E.R. Love, A.C. McBride, M. Mikolas, S.M. Nikol’skii, K. Nishimoto, LI. Ogievet-skii, R.O. O’Neil, T.J. Osier, S. Owa, B. Ross, M. Saigo, I.N. Sneddon, H.M. Srivastava, A.F. Timan, U. Westphal, A. Zygmund and others". To this list must of course be added the names of the authors of Samko et al [153] and many other mathematicians, particularly those of the younger generation. Books especially devoted to fractional calculus include K.B. Oldham and J. Spanier [133], S.G. Samko, A.A. Kilbas and O.I. Marichev [153], V.S. Kiryakova [91], K.S. Miller and B. Ross [121], B. Rubin [147]. Books containing a chapter or sections dealing with certain aspects of fractional calculus include H.T. Davis [37], A. Zygmund [181], M.M.Dzherbashyan [45], I.N. Sneddon [159], P.L. Butzer and R.J. Nessel [25], P.L. Butzer and W. Trebels [28], G.O. Okikiolu [132], S. Fenyo and H.W. Stolle [55], H.M. Srivastava and H.L. Manocha [162], R. Gorenfio and S. Vessella [65].
6.2
Various approaches of fractional derivatives
There exists a many mathematical definitions of fractional order integration and derivation. These definitions do not always lead to identical results but are equivalent for a wide large of functions. We introduce the fractional integration operator as well as the two most defini-tions of fractional derivatives, used, namely that Riemann-Liouville, Caputo and Hadamard.
From the classical fractional calculus, we recall
a has the following form (aIαf )(n) = 1 Γ(α) Z x a (x − t)α−1f (t)dt.
Definition 6.2 The right Riemann-Liouville fractional integral of order α > 0 ending at b > a is defined by (Ibαf )(n) = 1 Γ(α) Z b x (x − t)α−1f (t)dt.
Definition 6.3 The left Riemann-Liouville fractional derivative of order α > 0 starting at a is given below
(aDαf )(x) = ( d dx)
n(aIn−αf )(x), n = [α] + 1.
Definition 6.4 The right Riemann-Liouville fractional derivative of order α > 0 ending at b becomes
(Dαbf )(x) = (− d dx)
n(In−α b f )(x).
Definition 6.5 The left Caputo fractional of order α > 0 starting from a has the following form
(aDαf )(x) = (aIn−αf(n))(x), n = [α] + 1.
Definition 6.6 The right Caputo fractional derivative of order α > 0 ending at b becomes
(Dαbf )(x) = (Ibn−α(−1)nf(n))(x).
The Hadamard type fractional integrals and derivatives were introduced in [15] as:
Definition 6.7 The left Hadamard fractional integral of order α > 0 starting from a has the following form
(aIαf )(x) = 1 Γ(α)
Z x
a
(ln x − ln t)α−1f (t)dt
Definition 6.8 The right Hadamard fractional integral of order α > 0 ending at b > a is defined by (Ibαf )(x) = 1 Γ(α) Z b x (ln t − ln x)α−1f (t)dt
Definition 6.9 The left Hadamard fractional derivative of order α > 0 starting at a is given below
(aDαf )(x) = (x d dx)
Definition 6.10 The right Hadamard fractional derivative of order α > 0 ending at b be-comes (Dαbf )(x) = (−x d dx) n (Ibn−αf )(x).
Definition 6.11 The fractional derivative of order α, 0 < α < 1, in sense of Caputo, is defined by Dαf (t) = 1 Γ(1 − α) Z t 0 (t − s)−αdf ds(s)ds.
Definition 6.12 The fractional integral of order α, 0 < α < 1, in sense Riemann-Liouville, is defined by Iαf (t) = 1 Γ(α) Z t 0 (t − s)α−1f (s)ds.
Remark 6.13 From the above definitions, clearly
Dαf = Iα−1Df, 0 < α < 1. Lemma 6.14 IαDαf (t) = f (t) − f (0), 0 < α < 1. Lemma 6.15 If Dβf (0) = 0. then DαDβf = Dα+βf, 0 < α < 1, 0 < β < 1.
Now, we give the definitions of the generalized Caputo’s fractional derivative and the generalized fractional integral. These exponentially modified fractional integro-differential.
Definition 6.16 The generalized Caputo’s fractional derivative is given by Dα,ηf (t) = 1 Γ(1 − α) Z t 0 (t − s)−αe−η(t−s)df ds(s) ds, 0 < α < 1, η ≥ 0. Remark 6.17 The operators Dα and Dα,η differ just by their kernels.
Definition 6.18 The generalized fractional integral is given by Iα,ηf (t) = 1
Γ(α)
Z t
0
Remark 6.19 We have
Chapter II
Global existence and energy decay of
solutions to a nondissipative
Timoshenko beam system with delay
terms
abstract: We consider the nondissipative Timoshenko beam system in bounded domain with delay terms in the internal feedback and prove the global existence of its solutions in Sobolev spaces by means of semigroup theory under a condition between the weight of the delay terms in the feedbacks and the weight of the terms without delay. Furthermore, we study the asymptotic behavior of solutions using multiplier method.
1
Introduction
In this work we investigate the existence and decay properties of solutions for the initial boundary value problem of the linear Timochenko system of the type
(
ρ1ϕtt− K(ϕx+ ψ)x+ µ1ϕt+ µ2ϕt(x, t − τ1) = 0
ρ2ψtt− bψxx+ ˜K(ϕx+ ψ) +µf1ψt+µf2ψt(x, t − τ2) = 0
(P )
where (x, t) ∈ (0, L) × (0, +∞), τi > 0 (i = 1, 2, 3) is a time delay, µ1, µ2,µf1,µf2, are positive
real numbers. This system is subject to the Dirichlet boundary conditions
and to the initial conditions ϕ(x, 0) = ϕ0(x), ϕt(x, 0) = ϕ1(x), ψ(x, 0) = ψ0(x), ψt(x, 0) = ψ1(x), x ∈ (0, L) ϕt(x, t − τ1) = f0(x, t − τ1), in (0, L) × [0, τ1] ψt(x, t − τ2) = ˜f0(x, t − τ2), in (0, L) × [0, τ2]
where the initial data (ϕ0, ϕ1, ψ0, ψ1, f0, ˜f0) belong to a suitable Sobolev space.
A simple model describing the transverse vibration of a beam, which was developed in
[36], is given by a system of coupled hyperbolic equations of the form
(
ρutt(x, t) = (K(ux− φ))x in ]0, L[×]0, +∞[,
˜
ρφtt(x, t) = (EIψx)x+ K(ux− φ) in ]0, L[×]0, +∞[,
where t denotes the time variable, x is the space variable along the beam of length L, in its equilibrium configuration, u is the transverse displacement of the beam and φ is the rotation
angle of the filament of the beam. The coefficients ρ, ˜ρ, E, I and K are respectively the
density (the mass per unit length), the polar moment of inertia of a cross section, Young’s modulus of elasticity, the moment of inertia of a cross section, and the shear modulus.
There are a number of publications concerning the stabilization of Timoshenko system with different kinds of damping (see [19], [25], [26] and [29]). Raposo et al. [32] proved the exponential decay of the solution for the following linear system of Timoshenko-type beam equations with linear frictional dissipative terms:
ρ1ϕtt− Gh(ϕx+ ψ)x+ µ1ϕt= 0
ρ2ψtt− EIψxx+ Gh(ϕx+ ψ) +µf1ψt= 0.
Messaoudi and Mustafa [25] (see also [29]) considered the stabilization for the following Timoshenko system with nonlinear internal feedbacks:
ρ1ϕtt− Gh(ϕx+ ψ)x+ g1(ψt) = 0
ρ2ψtt− EIψxx+ Gh(ϕx+ ψ) + g2(ψt) = 0.
Recently, Park and Kang [29] considered the stabilization of the Timoshenko system with weakly nonlinear internal feedbacks.
Time delay is the property of a physical system by which the response to an applied force is delayed in its effect (see [34]). Whenever material, information or energy is physically transmitted from one place to another, there is a delay associated with the transmission. In
recent years, the PDEs with time delay effects have become an active area of research and arise in many pratical problems (see for example [1], [35]). The presence of delay may be a source of instability. For example, it was proved in [14] that an arbitrarily small delay may destabilize a system which is uniformly asymptotically stable in the absence of delay. To stabilize a hyperbolic system involving input delay terms, additional control terms will be necessary (see [28] and [37]). For instance, in [28] the authors studied the wave equation with a linear internal damping term with constant delay and determined suitable relations
between µ1 and µ2, for which the stability or alternatively instability takes place. More
precisely, they showed that the energy is exponentially stable if µ2 < µ1 and they found a
sequence of delays for which the solution will be instable if µ2 ≥ µ1. The main approach
used in [28], is an observability inequality obtained with a Carleman estimate. The same results were showed if both the damping and the delay acting in the boundary domain. We also recall the result by Xu, Yung and Li [37], where the authors proved the same result as in [28] for the one space dimension by adopting the spectral analysis approach.
Our purpose in this chapter is to give a global solvability in Sobolev spaces and energy decay estimates of the solutions to the problem (P ) for linear damping and delay terms and
in the case when k 6= ˜k. To obtain global solutions to the problem (P ), we use the argument
combining the semigroup theory (see [28] and [11]) with the energy estimate method. To prove decay estimates, we use a multiplier method.
2
Preliminaries and main results
First assume the following hypotheses: (H1)
|µ2| < µ1, |˜µ2| < ˜µ1. (II.1)
and
| ˜K − K| < 1. (II.2)
We first state some lemmas which will be needed later.
Lemma 2.1 (Sobolev-Poincaré’s inequality) Let q be a number with 2 ≤ q < +∞. Then there is a constant c∗ = c∗((0, 1), q) such that
Lemma 2.2 ([18], [20]) Let E : R+ → R+ be a non increasing function and assume that
there are two constants σ > −1 and ω > 0 such that
(Z +∞
S E
1+σ(t) dt ≤ a
1E(S) + a2Ep+1(S) + a3Er+1(T ). 0 ≤ S < +∞,
E0 ≤ λE (II.3)
If a3λ(r + 1) < 1, then there exist two constants c, ω such that
E(t) ≤ ce−ω ∀t ≥ 0, if r = 0 (II.4) E(t) ≤ c(1 + t)−1 r ∀t ≥ 0, if r > 0and λ = 0, (II.5) E(t) ≤ c(1 + t)− 1 r(r+1) ∀t ≥ 0, if r > 0and λ = 0, (II.6)
We introduce, as in [28], the new variables
z1(x, ρ, t) = φt(x, t − τ1ρ), x ∈ (0, L), ρ ∈ (0, 1), t > 0,
z2(x, ρ, t) = ψt(x, t − τ2ρ), x ∈ (0, L), ρ ∈ (0, 1), t > 0,
(II.7)
Then, we have
τizit(x, ρ, t) + ziρ(x, ρ, t) = 0, in (0, L) × (0, 1) × (0, +∞) for i = 1, 2. (II.8)
Therefore, problem (P ) takes the form:
ρ1ϕtt(x, t) − K(ϕx+ ψ)x(x, t) + µ1ϕt(x, t) + µ2z1(x, 1, t) = 0, τ1z1t(x, ρ, t) + z1ρ(x, ρ, t) = 0, ρ2ψtt(x, t) − bψxx(x, t) + ˜K(ϕx+ ψ)(x, t) +µf1ψt(x, t) +µf2z2(x, 1, t) = 0, τ2z2t(x, ρ, t) + z2ρ(x, ρ, t) = 0, (II.9)
The above system subjected to the following initial and boundary conditions
ϕ(0, t) = ϕ(L, t) = ψ(0, t) = ψ(L, t) = 0 t > 0 z1(x, 0, t) = ϕt(x, t), z2(x, 0, t) = ψt(x, t), x ∈ (0, L), t > 0 ϕ(x, 0) = ϕ0, ϕt(x, 0) = ϕ1, ψ(x, 0) = ψ0, ψt(x, 0) = ψ1, z1(x, 1, t) = f1(x, t − τ1), in (0, L) × (0, τ1) z2(x, 1, t) = f2(x, t − τ2), in (0, L) × (0, τ2) (II.10)
Let ξ1, ξ2 and ξ3 be positive constants such that
(
τ1|µ2| < ξ1 < τ1(2µ1− |µ2|),
τ2|˜µ2| < ξ2 < τ2(2˜µ1− |˜µ2|),
(II.11)
thanks to hypothesis (H1). We define the energy associated to the solution of the problem (III.8) by the following formula:
E(t) = ρ1 2kϕtk 2 2+ ρ2 2kψtk 2 2+ b 2kψxk 2 2+ K 2kϕx+ ψk 2 2+ 2 X i=1 ξi 2 Z 1 0 kzi(x, ρ, t)k22dρ. (II.12)
We have the following theorem.
Theorem 2.3 Let (ϕ0, ϕ1, f1(., −.τ1), ψ0, ψ1, f2(., −.τ2)) ∈ (H2(0, L) ∩ H01(0, L) × H01(0, L) ×
L2((0, L) × (0, 1)))3. Assume that the hypotheses (H1) holds. Then problem (P ) admits a
unique solution ϕ ∈ C([0, +∞); H2(0, L) ∩ H1 0(0, L)) ∩ C1([0, +∞); H01(0, L)), ψ ∈ C([0, +∞); H2(0, L) ∩ H1 0(0, L)) ∩ C1([0, +∞); H01(0, L)) z1, z2 ∈ C([0, +∞); L2((0, L) × (0, 1))).
In addition, we have the following decay estimate:
E(t) ≤ cE(0)e−ωt
, ∀ t ≥ 0, (II.13)
where c and ω are positive constants, independent of the initial data.
We finish this section by giving an explicit upper bound for the derivative of the energy.
Lemma 2.4 Let (ϕ, ψ, z1, z2) be a solution of the problem (III.8). Then, the energy
func-tional defined by (III.11) satisfies E0(t) ≤ −µ 1− 2τξ11 − |µ2| 2 kϕtk22− f µ1 −2τξ22 − |µe2| 2 kψtk22 −ξ1 2τ1 − |µ2| 2 kz1(x, 1, t)k22− ξ 2 2τ2 − |µe2| 2 kz2(x, 1, t)k22+ | ˜K − K|kϕtk2k(ϕx− ψ)k2 ≤ 2| ˜√K − K| ρ2K E(t). (II.14)
over (0, L) and using integration by parts, we get 1 2ρ1 d dtkϕtk 2 2− K Z L 0 (ϕx+ ψ)xϕtdx + µ1kϕtk22+ µ2 Z L 0 z1(x, 1, t)ϕtdx = 0, and 1 2ρ2 d dtkψtk 2 2+ b 2kψxk 2 2+K Z L 0 (ϕx+ψ)ψtdx+( ˜K−K) Z L 0 (ϕx+ψ)ψtdx+µf1kψtk 2 2+µf2 Z L 0 z1(x, 1, t)ψtdx = 0. Then d dt ρ1 2 kϕtk 2 2 + ρ2 2kψtk 2 2+ b 2kψxk 2 2+ K 2kϕx+ ψk 2 2 ! + µ1kϕtk22+µf1kψtk 2 2 +µf2 Z L 0 z1(x, 1, t)ψtdx + µ2 Z L 0 z1(x, 1, t)ϕtdx + ( ˜K − K) Z L 0 (ϕx+ ψ)ψtdx = 0 (II.15)
Multiplying the equation in (III.7) by ξizi and integrating over (0, L) × (0, 1), to obtain:
ξi 2 d dt Z L 0 Z 1 0 z2i(x, ρ, t) dρ dx = −ξi τ1 Z L 0 Z 1 0 ziziρdρ dx = ξi 2τi Z L 0 zi2(x, 0, t) − zi2(x, 1, t)dx = ξi 2τi h kz2 i(x, 0, t)k 2 2− kzi(x, 1, t)k22 i , (II.16)
where z1(x, 0, t) = ϕt(x, t) and z2(x, 0, t) = ψt(x, t). From (III.11), (III.14), (III.15) and
using Young inequality we get
E0(t) = − µ1 −2τξ11 kϕtk22− f µ1 −2τξ22 kψtk22− f f µ1− 2τξ33 kωtk22− 2 X i=1 ξi 2τi kzi(x, 1, t)k22 −µ2 Z L 0 z1(x, 1, t)ϕtdx −µf2 Z L 0 z1(x, 1, t)ψtdx + ( ˜K − K) Z L 0 (ϕx+ ψ)ψtdx. (II.17) Due to Young’s inequality, we have
Z L 0 z1(x, 1, t)ϕt(x, t) dx ≤ 1 2kϕt(x, t)k 2 2+ 1 2kz1(x, 1, t)k 2 2 Z L 0 z2(x, 1, t)ϕt(x, t) dx ≤ 1 2kψt(x, t)k 2 2+ 1 2kz2(x, 1, t)k 2 2 (II.18)
Inserting (III.17) into (III.16), we obtain
E0(t) ≤ − µ1− 2τξ1 1 − |µ2| 2 kϕtk22− f µ1− 2τξ2 2 − |µe2| 2 kψtk22 −ξ1 2τ1 − |µ2| 2 kz1(x, 1, t)k22− ξ 2 2τ2 − | e µ2| 2 kz2(x, 1, t)k22+ | ˜K − K|kϕtk2k(ϕx− ψ)k2.
