Stabilization of second order evolution equations with unbounded feedback with delay

Description

Stabilization of second order evolution equations with unbounded feedback with delay Serge Nicaise?, Julie Valein† December 1, 2008 Abstract We consider abstract second order evolution equations with unbounded feedback with delay. Existence results are obtained under some realistic assumptions. Sufficient and explicit conditions are derived that guarantee the exponential or polynomial stability. Some new examples that enter into our abstract framework are presented. Keywords second order evolution equations, wave equations, delay, stabi- lization functional. 1 Introduction Time-delay often appears in many biological, electrical engineering systems and mechanical applications [11, 21, 1], and in many cases, in particular for dis- tributed parameter systems, even arbitrarily small delays in the feedback may destabilize the system, see e.g. [8, 9, 10, 12, 15, 16, 17, 20, 23]. The stability issue of systems with delay is, therefore, of theoretical and practical importance. We further remark that some techniques developed recently [16, 17] in order to obtain some existence results and decay rates have some similarities. We therefore propose to consider an abstract setting as large as possible in order to contain a quite large class of problems with time delay feedbacks. In a second step we prove existence and stability results in this setting under realistic assumptions. Finally in order to show the usefulness of our approach, we give some examples where our abstract framework can be applied.

• valenciennes


• delay increases


• datum u0 ?


• evolution equations


• ?université de valenciennes et du hainaut cambrésis


• b?2 ?˙


• self-adjoint positive operator

Sujets

• Saint Nicaise
• Valenciennes

Informations

• Publié par: pefav
• Nombre de lectures: 18
• Langue: English

Extrait

Stabilization of second order evolution equations
with unbounded feedback with delay

∗ †
Serge Nicaise, Julie Valein

December 1, 2008

Abstract
We consider abstract second order evolution equations with unbounded
feedback with delay.Existence results are obtained under some realistic
assumptions. Sufficientand explicit conditions are derived that guarantee
the exponential or polynomial stability.Some new examples that enter
into our abstract framework are presented.
Keywordssecond order evolution equations, wave equations, delay,
stabilization functional.

1

Introduction

Time-delay often appears in many biological, electrical engineering systems and
mechanical applications [11, 21, 1], and in many cases, in particular for
distributed parameter systems, even arbitrarily small delays in the feedback may
destabilize the system, see e.g.[8, 9, 10, 12, 15, 16, 17, 20, 23].The stability
issue of systems with delay is, therefore, of theoretical and practical importance.
We further remark that some techniques developed recently [16, 17] in order
to obtain some existence results and decay rates have some similarities.We
therefore propose to consider an abstract setting as large as possible in order
to contain a quite large class of problems with time delay feedbacks.In a
second step we prove existence and stability results in this setting under realistic
assumptions. Finallyin order to show the usefulness of our approach, we give
some examples where our abstract framework can be applied.For a similar
approach, we refer to the paper in preparation [2].Without delay such an
approach was developed in [4].

∗
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956,
Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9 France,
[e-mail]
†
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956,
Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9 France,
[e-mail]

1

Before going on, let us present our abstract framework.LetHbe a real
Hilbert space with norm and inner product denoted respectively byk.kand
H
(., .)H.LetA:D(A)→Hbe a self-adjoint positive operator with a compact
1 11
′
inverse inH.LetV:=D(A)be the domain ofA .Denote byD(A)the
2 22
1
dual space ofD(A)obtained by means of the inner product inH.
2
Further, fori= 1,2, letUibe a real Hilbert space (which will be identified
to its dual space) with norm and inner product denoted respectively byk.k
Ui
1
′
2
and(., .)Uiand letBi∈ L(Ui, D(A) ).
We consider the system described by

ω¨(t) +Aω(t) +B1u1(t) +B2u2(t−τ) = 0, t >0
(1)ω(0) =ω0,˙ω(0) =ω1,
0
u2(t−τ) =f(t−τ),0< t < τ,

wheret∈[0,∞)represents the time,τis a positive constant which represents
2
the delay,ω: [0,∞)→His the state of the system andu1∈L([0,∞), U1),
2
u2∈L([−τ,∞), U2)are the input functions.Most of the linear equations
modelling the vibrations of elastic structures with distributed control with delay
can be written in the form (1), whereωstands for the displacement field.
In many problems, coming in particular from elasticity, the inputuiare given
∗
in the feω(t), which co
edback formui(t) =Bi˙rresponds to colocated actuators
and sensors.We obtain in this way the closed loop system

∗ ∗
¨ω(t) +Aω(t) +B B˙ω(t)ω˙ (t−τ) = 0, t >0
1 1+B2B2
(2)ω(0) =ω0,˙ω(0) =ω1,
∗0
B ω˙ (t−τ) =f(t−τ),0< t < τ.
2

The first natural question is the well posedness of this system.In section 2 we
will give a sufficient condition that guarantees that this system (2) is well-posed,
where we closely follow the approach developed in [16] for the wave equation.
Secondly, we may ask if this system is dissipative.We show in section 3 that
the condition

∗2∗2
(3)∃0< α <1,∀u∈V,kB uk ≤αkB uk
2U21U1
guarantees the energy is decreasing; under this condition, using a result from [5]
(see also [22]) we pertain a necessary and sufficient condition for the decay to
zero of the energy.Note that this last condition is independent of the delay and
therefore under the condition (3), our system is strongly stable if and only if
the same system without delay is strongly stable.Note further that if (3) is not
satisfied, there exist cases where some instabilities may appear (see [16, 17, 23]
for the wave equation).Hence this assumption seems realistic.
In a third step, again under the condition (3) and a certain boundedness
assumption from [4] between the resolvent operator ofAand of the operators
B1andB2, see condition (20), we prove that the exponential decay of the
system (2) follows from a certain observability estimate.Again this observability
∗
estimate is independent of the delay termB2B˙ω(t−τ)and therefore, under the
2

2

conditions (3) and (20), the exponential decay of the system (2) follows from the
exponential decay of the same system without delay.Nevertheless we give the
dependence of the decay with respect to the delay, in particular we show that
if the delay increases the decay decreases.This is the content of section 4.A
similar analysis for the polynomial decay is performed in section 5 by weakening
the observability estimate.Again we show that if the delay increases the decay
decreases. Inview of some applications, section 6 is devoted to the proof of
these two observability estimates by using a frequency domain method and a
reduction to some conditions between the eigenvectors ofAand the feedback
∗
operatorB.
1
Finally we finish this paper by considering in section 7 different examples
where our abstract framework can be applied.To our knowledge, all the
examples, with the exception of the first one, are new.

2

Well posedness of the system

We aim to show that system (2) is well-posed.For that purpose, we use
semigroup theory and an idea from [16] (see also [17]).Let us introduce the auxiliary
∗
variablez(ρ, t) =B˙ω(t−τ ρ)forρ∈(0,1)andt >0. Notethatzverifies the
2
transport equation for0< ρ <1andt >0

∂z ∂z
τ0+ =
∂t ∂ρ
∗
(4)z(0, t) =B˙ω(t)
2

∗0
z(ρ,0) =B˙ω(−τ ρ) =f(−τ ρ).
2
Therefore, the system (2) is equivalent to

∗
ω¨(t) +Aω(t˙) +ω(, t) = 0, t >0
B1B1t) +B2z(1

∂z ∂z
τ+ =0, t >0,0< ρ <1
∂t ∂ρ
(5)
0
ω(0) =ω0, ω˙ (0) =ω1, z(ρ,0) =f(−τ ρ),0< ρ <1


∗
z(0, t) =B ω˙ (t), t >0.
2
If we introduce
T
U:= (ω, ω˙, z),
thenUsatisfies
µ ¶T
1∂z
′T∗
ω˙.
U= (ω˙,¨ω, z˙) =ω˙,−Aω(t)−B1B1(t)−B2z(1, t),−
τ ∂ρ
Consequently the system (2) may be rewritten as the first order evolution
equation
½
′
U=AU
(6)
0
U(0) = (ω0, ω1, f(−τ.)),
where the operatorAis defined by
  
ω u
∗
 −Aω−B B−B z
Au=1 1u2(1),
1∂z
z−
τ ∂ρ

3

with domain

1∗ ∗
D(A) :={(ω, u, z)z(1)∈H}.
1B1u+B2
∈V×V×H((0,1), U2);z(0) =B2u, Aω+B

Now, we introduce the Hilbert space

2
H=V×H×L((0,1), U2)

equipped with the usual inner product
  
* +
Z
ω˜ω³ ´1
1 1
  2 2
(7)uu ,˜ =AA ω,˜ω+ (u,˜u() +z(ρ), z˜(ρ))dρ.
H U2
H
0
z z˜

(8)

Let us suppose now that

∗2∗2
∃0< α≤1,∀u∈V,kB uk ≤αkB uk.
2U1U1
2

Under this condition, we will show that the operatorAgenerates aC0-semigroup
inH.
For that purpose, we choose a positive real numberξsuch that

2
(9)1≤ξ≤ −1.
α
This constant exists because0< α≤1.
We now introduce the following inner product onH
  
* +
Z
ω ω˜³ ´1
1 1
  2 2
u ,u˜ =AA ω,˜ω+ (u,˜u) +τ ξ(z(ρ), z˜(ρ))dρ.
H U2
H
0
z z˜
H
This new inner product is clearly equivalent to the usual inner product onH
(7).

Theorem 2.1
exists a unique
D(A), then

Under the assumption (8), for an initial datumU0∈ H, there
solutionU∈C([0,+∞),H)to system (6).Moreover, ifU0∈

1
U∈C([0,+∞), D(A))∩C([0,+∞),H).

Proof.By Lumer-Phillips’ theorem, it suffices to show thatAis m-dissipative
(see Definition 3.3.1 and Theorems 1.4.3 and 1.4.6 of [18]).
⊤
We first prove thatAis dissipative.TakeU= (ω, u, z)∈D(A). Then
  
* +
u ω
∗
−Aω−B B u−B z(1) 
hAU, Ui=21 1, u
H
1∂z
−z
τ ∂ρ
³ ´H³ ´
1 1R
1
∗∂z
=uA u,A ω+B z(
−(Aω+B1B1 21), u)−ξ(ρ), z(ρ)dρ.
2 2
H0∂ρ
H U2

4