Stabilization of second order evolution equations with unbounded feedback with delay

DOI:10.1051/cocv/2009007 www.esaim-cocv.org

STABILIZATION OF SECOND ORDER EVOLUTION EQUATIONS WITH UNBOUNDED FEEDBACK WITH DELAY

Serge Nicaise

and Julie Valein

Abstract. We consider abstract second order evolution equations with unbounded feedback with de- lay. 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.

Mathematics Subject Classification.93D15, 93C20.

Received April 16, 2008. Revised September 19, 2008 and December 1st, 2008.

Published online April 21, 2009.

1. Introduction

Time-delay often appears in many biological, electrical engineering systems and mechanical applications [1,11,21], and in many cases, in particular for distributed parameter systems, even arbitrarily small delays in the feedback may destabilize the system, see e.g. [8–10,12,15–17,20,23]. 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. For a similar approach, we refer to the paper in preparation [4]. Without delay such an approach was developed in [3].

Before going on, let us present our abstract framework. Let H be a real Hilbert space with norm and inner product denoted respectively by ._H and (., .)_H. LetA : D(A)→H be a self-adjoint positive operator with a compact inverse in H. LetV :=D(A¹²) be the domain of A¹². Denote byD(A¹²) the dual space of D(A¹²) obtained by means of the inner product in H.

Further, fori= 1,2, letU_i be a real Hilbert space (which will be identified to its dual space) with norm and inner product denoted respectively by._Ui and (., .)_Ui and letB_i∈ L(Ui, D(A¹²)).

Keywords and phrases. Second order evolution equations, wave equations, delay, stabilization functional.

1 Universit´e de Valenciennes et du Hainaut Cambr´esis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, 59313 Valenciennes Cedex 9, France. [email protected];

[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2009

We consider the system described by

⎧⎪

⎪⎨

⎪⎪

⎩

¨

ω(t) +Aω(t) +B₁u₁(t) +B₂u₂(t−τ) = 0, t >0 ω(0) =ω₀,ω(0) =˙ ω₁,

u₂(t−τ) =f⁰(t−τ),0< t < τ,

(1.1)

wheret∈[0,∞) represents the time,τ is a positive constant which represents the delay,ω : [0,∞)→H is the state of the system and u₁∈L²([0,∞), U1),u₂∈L²([−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 (1.1), whereω stands for the displacement field.

In many problems, coming in particular from elasticity, the input u_i are given in the feedback formu_i(t) = B^∗_iω(t), which corresponds to colocated actuators and sensors. We obtain in this way the closed loop system˙

⎧⎪

⎪⎨

⎪⎪

⎩

¨

ω(t) +Aω(t) +B₁B₁^∗ω(t) +˙ B₂B₂^∗ω(t˙ −τ) = 0, t >0 ω(0) =ω₀,ω(0) =˙ ω₁,

B₂^∗ω(t˙ −τ) =f⁰(t−τ), 0< t < τ.

(1.2)

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 (1.2) 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

∃0< α <1,∀u∈V,B₂^∗u²_U2 ≤αB₁^∗u²_U1 (1.3) 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 (1.3), our system is strongly stable if and only if the same system without delay is strongly stable. Note further that if (1.3) 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 (1.3) and a certain boundedness assumption from [3] between the resolvent operator ofAand of the operatorsB₁andB₂, see condition (4.1), we prove that the exponential decay of the system (1.2) follows from a certain observability estimate. Again this observability estimate is independent of the delay term B₂B₂^∗ω(t˙ −τ) and therefore, under the conditions (1.3) and (4.1), the exponential decay of the system (1.2) follows from the exponential decay of the same system without delay. Nevertheless we give the dependence of the decay rate with respect to the delay, in particular we show that if the delay increases the decay rate 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 rate decreases. In view 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 operatorB₁^∗.

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 (1.2) is well-posed. For that purpose, we use semi-group theory and an idea from [16] (see also [17]). Let us introduce the auxiliary variablez(ρ, t) =B₂^∗ω(t˙ −τ ρ) forρ∈(0, 1) andt >0.

Note thatzverifies the transport equation for 0< ρ <1 andt >0

⎧⎪

⎪⎨

⎪⎪

⎩

τ^∂z_∂t+^∂z_∂ρ = 0 z(0, t) =B^∗₂ω(t)˙

z(ρ,0) =B₂^∗ω(−τ ρ) =˙ f⁰(−τ ρ).

(2.1)

Therefore, the system (1.2) is equivalent to

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

¨

ω(t) +Aω(t) +B₁B₁^∗ω(t) +˙ B₂z(1, t) = 0, t >0 τ^∂z_∂t+^∂z_∂ρ = 0, t >0, 0< ρ <1

ω(0) =ω₀, ω(0) =˙ ω₁, z(ρ,0) =f⁰(−τ ρ),0< ρ <1 z(0, t) =B^∗₂ω(t), t >˙ 0.

(2.2)

If we introduce

U := (ω,ω, z)˙ ^T, thenU satisfies

U= ( ˙ω,ω,¨ z)˙ ^T =

˙

ω,−Aω(t)−B₁B₁^∗ω(t)˙ −B₂z(1, t),−1 τ

∂z

∂ρ _T

· Consequently the system (1.2) may be rewritten as the first order evolution equation

U=AU

U(0) = (ω₀, ω₁, f⁰(−τ.)), (2.3)

where the operatorAis defined by

A

⎛

⎜⎝ ω u z

⎞

⎟⎠=

⎛

⎜⎝

u

−Aω−B₁B^∗₁u−B₂z(1)

−_τ¹_∂ρ^∂z

⎞

⎟⎠,

with domain

D(A) :={(ω, u, z)∈V ×V ×H¹((0,1), U₂);z(0) =B₂^∗u, Aω+B₁B₁^∗u+B₂z(1)∈H}·

Now, we introduce the Hilbert space

H=V ×H×L²((0,1), U₂) equipped with the usual inner product

⎛

⎝ ω u z

⎞

⎠,

⎛

⎝ ω˜

˜ u

˜ z

⎞

⎠

=

A¹²ω, A¹²ω˜

H+ (u,u)˜ _H+ ₁

0

(z(ρ),z(ρ))˜ _U2dρ. (2.4)

Let us suppose now that

∃0< α≤1,∀u∈V,B₂^∗u²_U2≤αB₁^∗u²_U1. (2.5) Under this condition, we will show that the operatorAgenerates aC₀-semigroup inH.

For that purpose, we choose a positive real numberξsuch that 1≤ξ≤ 2

α−1. (2.6)

This constant exists because 0< α≤1.

We now introduce the following inner product onH

⎛

⎝ ω u z

⎞

⎠,

⎛

⎝ ω˜

˜ u

˜ z

⎞

⎠

H

=

A¹²ω, A¹²ω˜

H+ (u,u)˜ _H+τ ξ ₁

0

(z(ρ),z(ρ))˜ _U2dρ.

This new inner product is clearly equivalent to the usual inner product (2.4) onH.

Theorem 2.1. Under the assumption (2.5), for an initial datum U₀ ∈ H, there exists a unique solution U ∈C([0, +∞),H)to system (2.3). Moreover, ifU₀∈D(A), then

U ∈C([0,+∞), D(A))∩C¹([0, +∞),H).

Proof. By Lumer-Phillips’ theorem, it suffices to show thatA is m-dissipative (see Def. 3.3.1 and Thms. 1.4.3 and 1.4.6 of [18]).

We first prove thatAis dissipative. TakeU = (ω, u, z)∈D(A). Then

AU, U_H =

⎛⎜⎝

u

−Aω−B₁B₁^∗u−B₂z(1)

−¹_τ^∂z_∂ρ

⎞

⎟⎠,

⎛

⎜⎝ ω u z

⎞

⎟⎠

H

=

A¹²u, A¹²ω

H−(Aω+B₁B₁^∗u+B₂z(1), u)_H−ξ₁

0

∂z

∂ρ(ρ), z(ρ)

U2

dρ.

SinceAω+B₁B₁^∗u+B₂z(1)∈H,we obtain AU, U_H =

A¹²u, A¹²ω

H− Aω, u_V, V − B1B₁^∗u, u_V, V − B2z(1), u_V, V

−ξ ₁

0

∂z

∂ρ(ρ), z(ρ)

U2

dρ

= Aω, u_V, V − Aω, u_V, V − B₁^∗u²_U1−(z(1), B^∗₂u)_U2

−ξ ₁

0

∂z

∂ρ(ρ), z(ρ)

U2

dρ, by duality. By integrating by parts, we obtain

₁

0

∂z

∂ρ(ρ), z(ρ)

U2

dρ=− ₁

0

z(ρ), ∂z

∂ρ(ρ)

U2

dρ+ (z(1)²_U2− z(0)²_U2),

and thus ₁

0

∂z

∂ρ(ρ), z(ρ)

U2

dρ= 1

2(z(1)²_U2− B₂^∗u²_U2).

Therefore, by Cauchy-Schwarz’s inequality, we find

AU, U_H = − B₁^∗u²_U1−(z(1), B₂^∗u)_U2−ξ

2z(1)²_U2+ξ

2B₂^∗u²_U2

≤ − B₁^∗u²_U1+ 1

2 −ξ 2

z(1)²_U2+ 1

2+ξ 2

B^∗₂u²_U2.

By (2.5), we obtain

AU, U_H≤ α

2 +ξα 2 −1

B₁^∗u²_U1+ 1

2 −ξ 2

z(1)²_U2

with ^α₂+^ξα₂ −1≤0 and ¹₂−^ξ₂ ≤0 becauseξsatisfies condition (2.6). This shows thatAU, U_H≤0 and then the dissipativeness ofA.

Let us now prove thatλI− Ais surjective for someλ >0.

Let (f, g, h)^T ∈ H.We look forU = (ω, u, z)^T ∈D(A) solution of

(λI− A)

⎛

⎝ ω u z

⎞

⎠=

⎛

⎝ f g h

⎞

⎠

or equivalently

⎧⎨

⎩

λω−u=f

λu+Aω+B₁B₁^∗u+B₂z(1) =g λz+¹_τ^∂z_∂ρ =h.

(2.7) Suppose that we have found ωwith the appropriate regularity. Then, we have

u=−f+λω∈V.

We can then determinez,indeedz satisfies the differential equation λz+1

τ

∂z

∂ρ =h

and the boundary conditionz(0) =B₂^∗u=−B₂^∗f +λB^∗₂ω.Thereforez is explicitly given by z(ρ) =λB₂^∗ωe^−λτρ−B₂^∗fe^−λτρ+τe^−λτρ

_ρ

0

e^λτσh(σ)dσ.

This means that once ωis found with the appropriate properties, we can findz andu.In particular, we have z(1) =λB₂^∗ωe^−λτ−B₂^∗fe^−λτ+τe^−λτ

₁

0

e^λτσh(σ)dσ=λB₂^∗ωe^−λτ+z⁰, (2.8) wherez⁰=−B^∗₂fe^−λτ+τe^−λτ₁

0 e^λτσh(σ)dσ is a fixed element ofU₂depending only on f andh.

It remains to findω.By (2.7),ω must satisfy

λ²ω+Aω+λB₁B₁^∗ω+B₂z(1) =g+B₁B₁^∗f+λf, and thus by (2.8),

λ²ω+Aω+λB₁B₁^∗ω+λe^−λτB₂B^∗₂ω=g+B₁B₁^∗f+λf−B₂z⁰=:q, whereq∈V. We take then the duality brackets., ._V, V withφ∈V:

λ²ω+Aω+λB₁B^∗₁ω+λe^−λτB₂B₂^∗ω, φ

V, V =q, φ_V, V.

Moreover:

λ²ω+Aω+λB₁B₁^∗ω+λe^−λτB₂B^∗₂ω, φ

V, V

=λ²ω, φ_V, V +Aω, φ_V, V +λ(B1B^∗₁ω, φ_V, V + e^−λτB2B₂^∗ω, φ_V, V)

=λ²(ω, φ)_H+

A¹²ω, A¹²φ

H+λ((B₁^∗ω, B₁^∗φ)_U1+ e^−λτ(B^∗₂ω, B^∗₂φ)_U2) becauseω∈V ⊂H. Consequently, we arrive at the problem

λ²(ω, φ)_H+

A¹²ω, A¹²φ

H+λ((B^∗₁ω, B^∗₁φ)_U

1+ e^−λτ(B₂^∗ω, B₂^∗φ)_U

2) =q, φ_V, V . (2.9) The left hand side of (2.9) is continuous and coercive on V.Indeed, we have

λ²(ω, φ)_H+

A¹²ω, A¹²φ

H+λ((B^∗₁ω, B₁^∗φ)_U

1+ e^−λτ(B^∗₂ω, B₂^∗φ)_U

2)

≤ λ²ω_Hφ_H+A¹²ω

H

A¹²φ

H+λ(B₁^∗ω_U1B₁^∗φ_U1+ e^−λτB^∗₂ω_U2B^∗₂φ_U2)

≤Cλ²ω_Vφ_H+A¹²²ω_V φ_V +λ(B^∗₁²_{L(V, U1)}ω_V φ_V + e^−λτB^∗₂²_{L(V, U2)}ω_V φ_V)

≤Cω_V φ_V , and forφ=ω∈V

λ²ω²_H+

A¹²ω, A¹²ω

H+λ(B₁^∗ω²_U1+ e^−λτB₂^∗ω²_U2)≥A¹²ω²

H≥Cω²_V .

Therefore, this problem (2.9) has a unique solutionω∈V by Lax-Milgram’s lemma. MoreoverAω+B₁B₁^∗u+ B₂z(1) =g+λf−λ²ω∈H.In summary, we have found (ω, u, z)^T ∈D(A) satisfying (2.7).

Remark 2.2. We deduce from Theorem2.1thatD(A) is dense inH(see [18]).

Remark 2.3. For initial data (ω₀, ω₁, f⁰(−τ.))^T inD(A), we easily show that the solution (ω(t), u(t), z(t))^T = T(t)(ω₀, ω₁, f⁰(−τ.))^T, whereT(t) is the semigroup generated byA, is indeed solution of (1.2) in the sense that

u(t) = ˙ω(t), and

z(ρ, t) =B₂^∗(t−τ ρ), andω satisfies (1.2).

3. The energy

We now restrict hypothesis (2.5) to obtain the decay of the energy. Namely, we suppose that (1.3) holds, namely

∃0< α <1,∀u∈V,B₂^∗u²_U2≤αB₁^∗u²_U1.

For an initial datum (ω₀, ω₁, f⁰(−τ.))^T ∈ H, Theorem 2.1 guarantees the existence of a weak solution (ω(t), u(t), z(t))^T = T(t)(ω₀, ω₁, f⁰(−τ.))^T. Hence the associated energy (which corresponds to the inner product onH) is defined by

E(t) := 1 2

A¹²ω(t)²

H+u(t)²_H+τ ξ ₁

0

z(ρ, t)²_U2dρ

,

whereξ is a positive constant satisfying

1< ξ < 2

α−1, (3.1)

that exists because 0< α <1.

Note that by Remark2.3for initial data inD(A), this energy takes the form

E(t) := 1 2

A¹²ω²

H+ω˙ ²_H+τ ξ ₁

0

B^∗₂ω(t˙ −τ ρ)²_U2dρ

. (3.2)

3.1. Decay of the energy

Proposition 3.1. If(1.3)holds, then for all(ω₀, ω₁, f⁰(−τ.))^T ∈D(A), the energy of the corresponding regular solution of (1.2)(i.e. (ω,ω, B˙ ₂ω(t˙ −τ ρ))^T ∈C([0,+∞), D(A))∩C¹([0,+∞),H)) is non-increasing and there exist two positive constantsC₁ andC₂ depending only onαandξ such that

−C₂

B₁^∗ω(t)˙ ²_U1+B₂^∗ω(t˙ −τ)²_U2

≤E(t)≤ −C₁

B₁^∗ω(t)˙ ²_U1+B₂^∗ω(t˙ −τ)²_U2

. (3.3)

Proof. Deriving (3.2), we obtain

E(t) =

A¹²ω, A¹²ω˙

H+ ( ˙ω,ω)¨ _H+τ ξ ₁

0

(B₂^∗ω(t˙ −τ ρ), B₂^∗ω(t¨ −τ ρ))_U

2dρ

= Aω,ω˙ _V,V −( ˙ω, Aω+B₁B₁^∗ω˙ +B₂B₂^∗ω(t˙ −τ))_H +ξτ

₁

0

(B₂^∗ω(t˙ −τ ρ), B₂^∗ω(t¨ −τ ρ))_U2dρ

= Aω,ω˙ _V,V − ω, Aω˙ +B₁B₁^∗ω˙ +B₂B₂^∗ω(t˙ −τ)_{V, V}

+ξτ ₁

0

(B₂^∗ω(t˙ −τ ρ), B₂^∗ω(t¨ −τ ρ))_U2dρ,

becauseAω+B₁B₁^∗ω˙ +B₂B₂^∗ω(t˙ −τ)∈H.Then

E(t) = Aω,ω˙ _V,V − ω, Aω˙ _{V, V}− ω, B˙ ₁B^∗₁ω˙ _{V, V}− ω, B˙ ₂B₂^∗ω(t˙ −τ)_{V, V}

+ξτ ₁

0

(B₂^∗ω(t˙ −τ ρ), B₂^∗ω(t¨ −τ ρ))_U2dρ

= − B₁^∗ω˙ ²_U1−(B₂^∗ω, B˙ ₂^∗ω(t˙ −τ))_U2 +ξτ

₁

0

(B₂^∗ω(t˙ −τ ρ), B₂^∗ω(t¨ −τ ρ))_U2dρ.

Moreover, recalling thatz(ρ, t) =B₂^∗ω(t˙ −τ ρ), we see that ₁

0

(B₂^∗ω(t˙ −τ ρ), B₂^∗ω(t¨ −τ ρ))_U

2dρ =

₁

0

z(ρ, t), ∂z

∂t(ρ, t)

U2

dρ

= −1 τ

₁

0

z(ρ, t), ∂z

∂ρ(ρ, t)

U2

dρ,

because ^∂z_∂ρ(ρ, t) =−τ^∂z_∂t(ρ, t).Then, we have ₁

0

(B₂^∗ω(t˙ −τ ρ), B₂^∗ω(t¨ −τ ρ))_U2dρ = − 1 2τ

₁

0

∂

∂ρz(ρ, t)²_U2dρ

= − 1

2τ(z(1, t)²_U2− z(0, t)²_U2)

= − 1

2τ(B₂^∗ω(t˙ −τ)²_U2− B₂^∗ω(t)˙ ²_U2).

Consequently,

E(t) =− B^∗₁ω˙ ²_U1−(B^∗₂ω, B˙ ^∗₂ω(t˙ −τ))_U2−ξ

2B₂^∗ω(t˙ −τ)²_U2+ξ

2B^∗₂ω(t)˙ ²_U2. Cauchy-Schwarz’s inequality yields

E(t)≤ − B₁^∗ω˙²_U1+ 1

2+ξ 2

B₂^∗ω(t)˙ ²_U2 + 1

2 −ξ 2

B₂^∗ω(t˙ −τ)²_U2 and

E(t)≥ − B₁^∗ω˙²_U1+

−1 2 +ξ

2

B₂^∗ω(t)˙ ²_U2− 1

2+ξ 2

B₂^∗ω(t˙ −τ)²_U2. Therefore, by (1.3), these estimates leads to

E(t)≤ −C1

B^∗₁ω(t)˙ ²_U1+B^∗₂ω(t˙ −τ)²_U2

with

C₁= min

1−ξα 2 −α

2

, ξ

2 −1 2

which is positive according to the assumption (3.1), and to E(t)≥ −C2

B₁^∗ω˙ ²_U1+B₂^∗ω(t˙ −τ)²_U2

with

C₂= max

1, ξ+ 1 2

which is also positive.

Remark 3.2. Integrating the expression (3.3) between 0 andT,we obtain _T

0

B₁^∗ω(t)˙ ²_U1+B^∗₂ω(t˙ −τ)²_U2 dt≤ 1

C₁(E(0)−E(T))≤ 1 C₁E(0).

This estimate implies thatB^∗₁ω(.)˙ ∈L²((0, T), U₁) andB₂^∗ω(.˙ −τ)∈L²((0, T), U₂).

Remark 3.3. If (1.3) is not satisfied, there exist cases where instabilities may appear, see [16,17,23] for the wave equation. Hence this condition appears to be quite realistic.

3.2. Decay of the energy to 0

We give a necessary and sufficient condition that guarantees the decay to 0 of the energy.

Proposition 3.4. Assume that (1.3)holds. Then, for all initial data inH, we have

t→∞limE(t) = 0 (3.4)

if and only if for any (non zero) eigenvectorϕ∈D(A)of A, we have

B^∗₁ϕ= 0. (3.5)

Remark 3.5. Notice that this necessary and sufficient condition is the same than in the case without delay (see [22]) and therefore, the system (1.2) with delay is strongly stable (i.e.the energy tends to zero) if and only if the system without delay (i.e.forB₂= 0) is strongly stable.

Proof. ⇐ Let us show that (3.5) implies (3.4). For that purpose we closely follow [22].

First, we show that Ahas no eigenvalue on the imaginary axis. If it is not the case, let iω be an eigenvalue ofAwhereω∈R. Letϕbe an eigenvector associated with iω. Thenϕis of the form

ϕ=

⎛

⎝ z iωz iωe^−iωτρB₂^∗z

⎞

⎠,

with

−ω²z+Az+ iωB₁B₁^∗z+ iωe^−iωτB₂B^∗₂z= 0. (3.6) It is an immediate consequence of the identity (iωI− A)ϕ= 0.

First we notice that ω = 0 since forω = 0, the above identity reduces to Az = 0 with z∈D(A). Since by hypothesisAis invertible, we getz= 0 and therefore 0 is not an eigenvalue ofA.

We now take the duality bracket., ._V, V between (3.6) andz∈V:

0 =

−ω²z+Az+ iωB₁B^∗₁z+ iωe^−iωτB₂B₂^∗z, z

V, V

=

(−ω²I+A)z, z

V, V + iωB^∗₁z²_U1+ iωe^−iωτB₂^∗z²_U2. We look at the imaginary part of this expression to obtain

ω

B₁^∗z²_U1+ cos(ωτ)B₂^∗z²_U2

= 0, which implies, becauseω= 0,

B₁^∗z²_U1+ cos(ωτ)B₂^∗z²_U2= 0.

We deduce that

0 =B^∗₁z²_U1 + cos(ωτ)B₂^∗z²_U2 ≥ B^∗₁z²_U1− B₂^∗z²_U2 ≥(1−α)B^∗₁z²_U1 ≥0, by (1.3) withα <1.Consequently

B^∗₁z_U1 = 0 which implies

B₁^∗z= 0. (3.7)

Moreover, by (3.6), (3.7) and (1.3), we have

Az=ω²z.

Therefore,z is an eigenvector ofAof associated eigenvalueω²such that B₁^∗z= 0,

which contradicts (3.5). Thus,Ahas no eigenvalue on the imaginary axis.

Now, we can apply the main theorem of Arendt and Batty [5]: Asσ(A)∩iRis empty (because the surjectivity of (iωI− A) is equivalent to the injectivity of−ω²+A−iωB₁B^∗₁−iωe^iωτB₂B₂^∗), we obtain (3.4).

⇒ Let us show that (3.4) implies (3.5). For that purpose we use a contradiction argument. Suppose that there exists an eigenvectorϕofAof associated eigenvalueλ² such that

B^∗₁ϕ= 0.

Let us set

ω(., t) =ϕcos(λt).

Thenω is solution of (1.2) and satisfies

E(t) =E(0) because

B₁^∗ω(t)˙ ²_U1 =λ²sin²(λt)B₁^∗ϕ²_U1 = 0 and

B₂^∗ω(t˙ −τ)²_U2 = λ²sin²(λ(t−τ))B₂^∗ϕ²_U2

≤ αλ²sin²(λ(t−τ))B^∗₁ϕ²_U1= 0,

by (1.3). This means that we have obtained a solution of system (1.2) with a constant energy, which

contradicts (3.4).

4. The exponential stability 4.1. A priori estimate

In order to obtain the characterization of decay properties of the damped systemviaobservability inequalities for the conservative system we will use the following assumption from [3]:

Ifβ >0 is fixed andC_β ={λ∈C|λ=β}, the function

λ∈C_β→H(λ) =λB^∗(λ²I+A)⁻¹B∈ L(U) is bounded, (4.1) whereB ∈ L(U, V) withU a Hilbert space.

We consider the evolution system

¨

y(t) +Ay(t) =Bv(t)

y(0) = ˙y(0) = 0, (4.2)

and the following conservative system

φ(t) +¨ Aφ(t) = 0

φ(0) =ω₀,φ(0) =˙ ω₁. (4.3)

Let us recall the two following results proved in [3]:

Lemma 4.1. Suppose thatv∈L²(0, T;U)and that the solutionsφof(4.3)are such thatB^∗φ(.)∈H¹(0, T;U) and there exists a constant C >0such that

(B^∗φ)(.)_L2(0,T;U)≤C(ω0, ω₁)_V×H ∀(ω0, ω₁)∈V ×H.

Then the system (4.2)admits a unique solution having the regularity y∈C(0, T;V)∩C¹(0, T;H).

Proposition 4.2. Suppose thatv ∈L²(0, T;U)and that the system (4.2)admits a unique solution having the regularity

y∈C(0, T;V)∩C¹(0, T;H).

Then hypothesis (4.1)holds if and only if B^∗y(.)∈H¹(0, T;U)and there exists a constant C >0independent of T such that

(B^∗y)(.)_L2(0,T;U)≤Ce^βTv_L2(0,T;U).

Let ω∈ C(0, T;V)∩C¹(0, T;H) be the solution of (1.2) with (ω₀, ω₁, f⁰(−τ.))^T ∈D(A). Then it can be split up in the form

ω=φ+ψ,

whereφis solution of the system without damping (4.3), andψsatisfies ψ(t) +¨ Aψ(t) =−B1B₁^∗ω(t)˙ −B₂B^∗₂ω(t˙ −τ)

ψ(0) = 0,ψ(0) = 0.˙ (4.4)

We now set B = (B₁B₂)∈ L(U, V) whereU =U₁×U₂. It is easy to verify thatB^∗ = B₁^∗

B₂^∗

∈ L(V, U).

Thereforeψis solution of

ψ(t) +¨ Aψ(t) =Bv(t)

ψ(0) = 0,ψ(0) = 0,˙ (4.5)

wherev(t) =

−B₁^∗ω(t)˙

−B^∗₂ω(t˙ −τ)

. In other words,ψis solution of system (4.2) with B = (B₁B₂)

and by Remark3.2

v=

−B₁^∗ω(·)˙

−B^∗₂ω(· −˙ τ)

∈L²((0, T), U).

Then ψ = ω−φ ∈ C(0, T;V)∩C¹(0, T;H). Suppose that hypothesis (4.1) is satisfied for B = (B₁B₂) and U =U₁×U₂. By applying Proposition4.2, we obtain

_T

0

(B^∗ψ)²_Udt≤Ce^2βT _T

0

v(t)²_Udt, which is equivalent to

_T

0

((B₁^∗ψ)²_U1+(B₂^∗ψ)²_U2)dt≤Ce^2βT _T

0

(B₁^∗ω(t)˙ ²_U1+B₂^∗ω(t˙ −τ)²_U2)dt.

In particular, we have _T

0

(B^∗₁ψ)²_U1dt≤Ce^2βT _T

0

(B₁^∗ω(t)˙ ²_U1+B₂^∗ω(t˙ −τ)²_U2)dt.

Therefore, since ω=φ+ψ, we have _T

0

(B₁^∗φ)(t)²_U1dt ≤ 2 _T

0

(B^∗₁ω)(t)²_U1dt+ _T

0

(B₁^∗ψ)(t)²_U1dt

≤ Ce^2βT _T

0

(B₁^∗ω)(t)˙ ²_U1+(B₂^∗ω)(t˙ −τ)²_U2 dt.

Thus, we have proved the following result:

Lemma 4.3. Suppose that assumption (4.1)is satisfied for B= (B₁B₂), U =U₁×U₂. Then the solutionsω of (1.2)andφof (4.3)satisfy

_T

0

(B^∗₁φ)(t)²_U1dt≤Ce^2βT _T

0

(B₁^∗ω(t)˙ ²_U1+B₂^∗ω(t˙ −τ)²_U2)dt, with C >0 independent ofT.

4.2. The stability result

Theorem 4.4. Assume that hypotheses (1.3) and (4.1) are verified forB = (B₁B₂), U =U₁×U₂. If there exist a time T >0and a constant C >0 such that the observability estimate

A¹²ω₀²

H+ω1²_H ≤C _T

0

(B₁^∗φ)(t)²_U1dt (4.6)

holds, where φ is solution of (4.3), then system (1.2) is exponentially stable in the energy space: there exist C >0 independent ofτ andν >0 such that, for all initial data in H,

E(t)≤CE(0)e^−νt ∀t >0. (4.7)

Proof. Letω be a solution of (1.2) with initial data (ω₀, ω₁, f⁰(−τ·))∈D(A).

Without loss of generality, we can always assume that (4.6) holds withT > τ andCindependent ofτ.

Integrating inequality (3.3) of Proposition3.1between 0 andT, we obtain E(0)−E(T) ≥C

_T

0

B₁^∗ω(t)˙ ²_U1+B₂^∗ω(t˙ −τ)²_U2 dt

≥ C 2

_T

0

B₁^∗ω(t)˙ ²_U1+B₂^∗ω(t˙ −τ)²_U2 dt+C

2 _T

0

B₂^∗ω(t˙ −τ)²_U2dt

≥Ce^−2βT _T

0

(B₁^∗φ)(t)²_U1dt+ _T

0

B₂^∗ω(t˙ −τ)²_U2dt

by Lemma4.3.

By assumption (4.6), we obtain

E(0)−E(T)≥Ce^−2βT

A¹²ω₀²

H+ω₁²_H+ _T

0

B^∗₂ω(t˙ −τ)²_U2dt

,

withC independent ofτ. AsT > τ, by change of variables, we have _T

0

B₂^∗ω(t˙ −τ)²_U2dt =

_T−τ

−τ B₂^∗ω(t)˙ ²_U2dt

≥ ₀

−τB₂^∗ω(t)˙ ²_U2dt=τ ₁

0

B^∗₂ω(−τ ρ)˙ ²_U2dρ.