Stabilization of second order evolution equations by a class of unbounded feedbacks

URL:http://www.emath.fr/cocv/

STABILIZATION OF SECOND ORDER EVOLUTION EQUATIONS BY A CLASS OF UNBOUNDED FEEDBACKS

Kais Ammari

and Marius Tucsnak

Abstract. In this paper we consider second order evolution equations with unbounded feedbacks.

Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.

Mathematics Subject Classification. 93B52, 93D15, 93B07.

Received October 19, 2000. Revised February 19, 2001.

1. Introduction

In recent years an important literature was devoted to the controllability and stabilizability of second order infinite dimensional systems comming from elasticity (see for instance Lions [16] and references therein). Ac- cording to the classical principle of Russell (see [23]) if a system is uniformly stabilizable by using colocated actuators and sensors then it is exactly controllable by using the same actuators (i.e., the same input oper- ator). As far as we know the converse of this assertion was not proved in a general framework. The only result available in the literature supposes that the input operator is bounded (see Haraux [9]) in the energy space or they are based on non local feed-backs (see for instance Komornik [12] and references therein). In the applications for PDE’s systems this situation leads to non-local feedbacks given, in particular, by Riccati type operators. However in many PDE systems the exponential stabilizability with colocated actuators and sensors was proved by direct methods (see Lagnese [14], Komornik and Zuazua [13]) by using multiplier techniques.

The aim of this paper is to give a class of unbounded input operators for which exact controllability implies uniform stabilizability by colocated actuators and sensors.

More precisely, letX be a complex Hilbert space with norm and inner product denoted respectively by||.||X

and h., .i^X. Let A be a linear unbounded self-adjoint and strictly positive operator in X. LetD(A¹²) be the domain ofA¹². Denote by (D(A¹²))⁰ the dual space ofD(A¹²) obtained by means of the inner product in X.

Further, letU be a complex Hilbert space (which will be identified to its dual space) with norm and inner product respectively denoted by||.||^U andh., .i^U and letB ∈ L(U,(D(A¹²))⁰).

Most of the linear control problems comming from elasticity can be written as x⁰⁰(t) +Ax(t) +Bu(t) = 0,

x(0) =z⁰, x⁰(0) =z¹, (1.1)

Keywords and phrases:Stabilization, observability inequality, second order evolution equations, unbounded feedbacks.

1 Institut Elie Cartan, D´epartement de Math´ematiques, Universit´e de Nancy I, 54506 Vandœuvre-l`es-Nancy Cedex, France;

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

c EDP Sciences, SMAI 2001

where x : [0, T] → X is the state of the system, u ∈ L²(0, T;U) is the input function and we denoted the differentiation with respect to time by “0”.

We define the energy ofx(t) at instanttby E(x(t)) = 1

2

n||x⁰(t)||²X+||A¹²x(t)||²X

o·

Simple formal calculations give

E(x(0))−E(x(t)) =−hBu(t), x⁰(t)i_D(A¹_2),(D(A¹₂₎₎₀, ∀t≥0. (1.2) This is why, in many problems, coming in particular from elasticity, the inputuis given in the feedback form u(t) =B^∗x⁰(t), which obviously gives a nonincreasing energy and which corresponds to colocated actuators and sensors. The aim of this paper is to give sufficient conditions making the corresponding closed loop system

x⁰⁰(t) +Ax(t) +BB^∗x⁰(t) = 0, (1.3)

x(0) =x⁰, x⁰(0) =x¹, (1.4)

uniformly stable in the energy spaceD(A¹²)×X.In the case of non uniform stability we give sufficient conditions for weaker decay properties.

In order to obtain the characterization of decay properties of the damped problemviaobservability inequali- ties for the conservative problem we will use the assumption below. This assumption is less restrictive than the boundedness ofB which was the basic hypothesis in [9].

(H)

Ifβ >0 is fixed andCβ=n

λ∈C|Reλ=βo

, the function λ∈C⁺=

n

λ∈C|Reλ >0

o→H(λ) =λ B^∗(λ²I+A)⁻¹B∈ L(U) (1.5)

is bounded onCβ.

An equivalent statement of (H) is given at Section 3. Under this alternative form this assumption can be verified for PDE systems (such as the systems in the examples below), by proving results called (in the PDE community) “hidden regularity results”.

The main novelties brought in by this paper are the following:

(a) we give a sufficient and necessary condition for the exponential stability of all finite energy solutions of (1.3, 1.4) by using only the undamped problem (i.e. corresponding toB= 0 in (1.3));

(b) in the case of non exponential stability in the energy space we give an explicit decay rate for all initial data lying in a more regular space.

Our approach has common points with the result obtained in [9] for feedbacks which are bounded in the energy space. The main difference is that we replace the assumption of boundedness of B by the assumption (H).

Moreover our methods are related to those proposed in [28] for a general class of first order systems (see the comming paper [26] for a description of the connections between our results and those in [28]).

The paper is organized as follows. In the second section we give precise statements of the main results. Some regularity results implied by (H) are given in Section 3. Section 4 contains the proof of the main results. The last section is devoted to some applications.

2. Statement of the main results

Let x(t) be a solution of (1.3, 1.4). Simple formal calculations show that a sufficiently smooth solution of (1.3, 1.4) satisfies the energy estimate

E(x(0))−E(x(t)) = Z t

0

||B^∗x⁰(s)||²Uds, ∀t≥0. (2.1) In particular (2.1) implies that

E(x(t))≤E(x(0)), ∀t≥0.

Estimate above suggests that the natural well-posedness space for (1.3, 1.4) isV ×X where V =D(A¹²) and

||x||^V =||A¹²x||^X,∀x∈V.

The existence and uniqueness of finite energy solutions of (1.3, 1.4) can be obtained by standard semigroup methods. This why the results below are given without proofs.

Proposition 2.1. Suppose that(x⁰, x¹)∈V ×X. Then the problem (1.3, 1.4) admits a unique solution x(t)∈C(0, T;V)∩C¹(0, T;X)

such that B^∗x(.)∈H¹(0, T;U)and

k(B^∗x)⁰(·)k²L²(0,T;U)≤Ck(x⁰, x¹)k²V×X, (2.2) where the constantC >0is independent of(x⁰, x¹). Moreoverx(t)satisfies the energy estimate (2.1).

Let us now consider the problem

φ⁰⁰(t) +Aφ(t) = 0, (2.3)

φ(0) =x⁰, φ⁰(0) =x¹. (2.4)

It is well known that (2.3, 2.4) is well-posed inD(A)×V and inV ×X. The following theorem is a direct generalisation of the result in [9].

Theorem 2.2. Assume that the hypothesis (H) is verified. Then, the system described by (1.3, 1.4) is expo- nentially stable inV ×X if and only if there existsT >0, C >0such that

Z T 0

||(B^∗φ)⁰(t)||²Udt≥C||(x⁰, x¹)||²V×X, ∀(x⁰, x¹)∈ D(A)×V. (2.5) Remark 2.3. Assumption (H) is not necessary for the implication: uniform exponential stability⇒(2.5). The latter follows from (indirectly) Russell’s principle [23].

The statement of our second main result requires some notations.

Consider the unbounded linear operator

A^d:D(A^d)→V ×X,A^d=

0 I

−A −BB^∗

, (2.6)

where

D(A^d) = n

(u, v)∈V ×X, Au+BB^∗v∈X, v∈V o·

LetX1, X2, Y1, Y2be four Banach spaces such that

V ×X ⊂X1×X2,D(A^d)⊂Y1×Y2⊂V ×X,

∀z∈ D(A^d),||z||D(Ad)∼ ||z||^Y1×Y2

and

[Y1×Y2, X1×X2]θ=V ×X, (2.7)

for a fixed real numberθ∈]0,1[, where [., .]θ denotes the interpolation space (see for instance Triebel [24]).

Let G : R⁺ → R⁺ such that G is continuous, invertible, increasing onR⁺ and suppose that the function x→ ¹

x1−θ^θ G(x) increasing on (0,1).

In the case of non exponential decay in the energy space we give explicit decay estimates valid for regular initial data, as stated in the result below:

Theorem 2.4. Assume that assumption (H) is verified and that the function G satisfies assumptions above.

Then the following assertions hold true:

1. If for all (x⁰, x¹)∈ D(A)×V we have Z T

0

||(B^∗φ)⁰(t)||²Udt≥C||(x⁰, x¹)||²V×XG

||(x⁰, x¹)||²X1×X2

||(x⁰, x¹)||²V×X

, (2.8)

for some constantC >0 then there exists a constantC1>0 such that for allt >0 and for all (x⁰, x¹)

∈ D(A^d) we have

E(x(t))≤C1

G⁻¹

1 1 +t

_1−θ^θ

||(x⁰, x¹)||²_D(Ad). (2.9) 2. If for all (x⁰, x¹)∈ D(A)×V we have

Z T 0

||(B^∗φ)⁰(t)||²Udt≥C||(x⁰, x¹)||²X1×X2, (2.10) for some constantC >0 then there exists a constantC2>0 such that for allt >0 and for all (x⁰, x¹)

∈ D(A^d) we have

E(x(t))≤ C2

(1 +t)^1−θθ ||(x⁰, x¹)||²_D(Ad). (2.11) Remark 2.5. 1. Estimates similar to (2.11) were first given by Russell [22] in the case of bounded feedback

controls. Russell’s method cannot be directly extended to unbounded feedbacks.

2. Ifθ∈(0,¹₂) then the identity function satisfies the assumptions onG in Theorem 2.4. In this case (2.11) is a consequence of (2.9). However ifθ∈(¹₂,1) then the identity function does not satisfy the assumptions onG in the Theorem 2.4. In this second case (2.11) is not a consequence of (2.9).

3. Some regularity results

Consider the evolution problem

y⁰⁰(t) +Ay(t) =Bv(t), (3.1)

y(0) =y⁰(0) = 0. (3.2)

A natural question is the regularity ofy whenv∈L²(0, T;U). By applying standard energy estimates we can easily check thaty∈C(0, T;X)∩C¹(0, T;V⁰). However ifB satisfies a certain admissibility condition thenyis more regular. More precisley the following result, which is a version of the general transposition method (see, for instance, Lions and Magenes [17]), holds true.

Lemma 3.1. Suppose that v ∈ L²(0, T;U) and that the solutions φ of (2.3, 2.4) are such that B^∗φ(.)

∈H¹(0, T;U)and there exists a constantC >0such that

k(B^∗φ)⁰(·)kL²(0,T;U)≤Ck(x⁰, x¹)k^V×X, ∀(x⁰, x¹)∈V ×X. (3.3) Then the problem (3.1, 3.2) admits a unique solution having the regularity

y ∈C(0, T;V)∩C¹(0, T;X). (3.4)

Proof. Let

D(A) =D(A)×V

and denote by [D(A)]⁰ the dual space ofD(A) with respect to the pivot spaceV ×X.

If we putZ= y

y⁰

it is clear that (3.1, 3.2) can be written as Z⁰+AZ(t) =Bv(t), Z(0) = 0, where

A=

0 −I

A 0

:V ×X →[D(A)]⁰, B=

0 B

:U →[D(A)]⁰.

It well known thatAis a skew adjoint operator so it generates a group of isometries in [D(A)]⁰, denoted byS(t).

After simple calculations we get that the operatorB^∗:D(A)→U is given by B^∗

u v

=B^∗v,∀(u, v)∈ D(A).

This implies that

B^∗S^∗(t) x⁰

x¹

=B^∗φ⁰,∀(x⁰, x¹)∈ D(A),

withφsatisfying (2.3, 2.4). From the inequality above and (3.3) we deduce that there exists a constantC >0 such that

Z T 0

B^∗S^∗(t) x⁰

x¹

²_U dt≤C||(x⁰, x¹)||²V×X,∀(x⁰, x¹)∈ D(A).

According to Theorem 3.1 in [6] (p. 187) the inequality above implies the interior regularity (3.5).

Proposition 3.2. Suppose thatv∈L²(0, T;U)and that the problem (3.1, 3.2) admits a unique solution having the regularity

y∈C(0, T;V)∩C¹(0, T;X). (3.5)

Then hypothesis (H) holds if and only ifB^∗y(.)∈H¹(0, T;U)and there exists a constantC >0such that k(B^∗y)⁰(·)kL²(0,T;U)≤CkvkL²(0,T;U), ∀v∈L²(0, T;U). (3.6) Proof. As equation (3.1) is time reversible, after extending v by zero fort∈R\[0, T], we can solve (3.1, 3.2), fort∈R. By this way, we obtain a function, denoted also byy, such that

y∈C(R;V)∩C¹(R;X)∩L²(R;V),

y(t) = 0, ∀t≤0, (3.7)

andysatisfies (3.1, 3.2) for allt∈R.

Lety(λ), whereb λ=γ+iη, γ >0 and η ∈R, be the Laplace (with respect to t) transform of y. Since y satisfies (3.7), estimate (3.6) is equivalent to the fact that the functiont →e^{−γ t}B^∗y(t) belongs toH¹(R;U) and that there exists a constantM1>0 such that

ke^{−γ .}B^∗y(·)k²H¹(R;U)≤M1kv(·)k²L²(R;U).

Equivalently, by the Parseval identity (see for instance Doetsch [8], p. 212), it suffices to prove that the function η→(γ+iη)B^∗y(γb +iη)

belongs toL²(R^η;U), for someγ >0, and that there exists a constantM2>0 such that k(γ+iη)B^∗y(γb +iη)k²L²(Rη;U)≤M2

Z +∞

−∞ ||bv(γ+iη)||²Udη. (3.8) It can be easily checked thatby satisfies:

λ²y(λ) +b Aby(λ) =Bbv(λ), ∀Reλ >0. (3.9) Relation above implies, forReλ >0 that

λ B^∗y(λ) =b H(λ)v(λ),b ∀Re λ >0, (3.10) where H(λ) is defined in (1.5). Assumption (H) implies the existence of a constant M2 > 0 such that (3.8) holds true. This ends the proof of the fact that assumption (H) implies that (3.6) holds for all finite energy solution of (3.1, 3.2).

Suppose now that (3.6) holds true. By using the time reversibility and the invariance with respect to translations (in time) of (3.1) we obtain that (3.1, 3.2) is well posed for all inputv ∈L²(R, U), v compactly supported. More precisely, we have

Z

supp(v)

||(B^∗y)⁰(t)||²Udt≤C Z

supp(v)

||v(t)||²Udt, for all compactly supportedv∈L²(R, U), with the same constant as in (3.6).

Using (3.10) it follows that

||H(γ+iη)bv(γ+iη)||²L²(Rη,U)≤C||bv(λ)||²L²(Rη,U), (3.11) for all compactly supportedv∈L²(R, U).

By density it follows that (3.11) holds for allv∈L²(R, U). We have thus proved that (3.6) implies that (H) holds true.

Proposition 3.3. Suppose that hypothesis (H) is satisfied. Then for (x⁰, x¹) ∈ V ×X we have that B^∗φ(.)

∈ H¹(0, T;U)and there exist C, T > 0 such that the solution φ(t) of (2.3, 2.4) satisfies (3.3). In the other words assumption (H) implies (3.3).

Proof. Suppose that hypothesis (H) is satisfied. Let x(t) ∈ C(0, T;V)∩C¹(0, T;X) be the unique solution of (1.3, 1.4). By Proposition 2.1 we know that B^∗x ∈ H¹(0, T;U) and that (2.1) holds true. Let φ be the solution of (2.3, 2.4). We clearly haveψ=x−φ∈C(0, T;V)∩C¹(0, T;X) andψsatisfies

ψ⁰⁰(t) +Aψ(t) =BB^∗x⁰(t),inC(0, T;V⁰), ψ(0) =ψ⁰(0) = 0.

By applying now Proposition 3.2 withv=B^∗x⁰∈L²(0, T;U) we obtain that Z T

0 ||(B^∗ψ)⁰(t)||²Udt≤C Z T

0 ||(B^∗x)⁰(t)||²Udt. (3.12) SinceB^∗φ=B^∗x−B^∗ψrelations (2.1) and (3.12) imply the conclusion of the proposition.

Corollary 3.4. Suppose that assumption (H) is satisfied. Then, for all v ∈ L²(0, T;U), problem (3.1, 3.2) admits a unique solutiony satisfying (3.5) and (3.6).

Proof. Suppose that assumption (H) is satisfied. Then Proposition 3.3 and Lemma 3.1 imply that problem (3.1, 3.2) admits a unique solutiony satisfying (3.5). Finally Proposition 3.2 implies thaty satisfies (3.6).

4. Proof of the main results

Letx(t)∈C(0, T;V)∩C¹(0, T;X) be the solution of (1.3, 1.4). Thenx(t) can be written as

x(t) =φ(t) +ψ(t), (4.1)

whereφ(t) satisfies (2.3, 2.4) andψ(t) satisfies

ψ⁰⁰(t) +Aψ(t) =−BB^∗x⁰(t), (4.2)

ψ(0) =ψ⁰(0) = 0. (4.3)

The main ingredient of the proof of Theorem 2.2 and of the proof of Theorem 2.4 is the following result:

Lemma 4.1. Let (x⁰, x¹)∈V ×X and suppose that (H) is verified. Then the solutionx(t) of (1.3, 1.4) and the solutionφ(t)of (2.3, 2.4) satisfy

C1

Z T 0

||(B^∗φ)⁰(t)||²Udt≤Z T 0

||(B^∗x)⁰(t)||²Udt≤4 Z T

0

||(B^∗φ)⁰(t)||²Udt, (4.4) whereC1>0is a constant independent of(x⁰, x¹).

Remark 4.2. By Proposition 2.1, (B^∗x)⁰(·)∈L²(0, T;U). So, equation (4.2) makes sense. The result above shows that theL²norm of||(B^∗x)⁰(·)||^U is equivalent to theL²norm of||(B^∗φ)⁰(·)||^U (notice that||(B^∗φ)⁰(·)||^U

∈L²(0, T) by Prop. 3.3).

Proof of Lemma 4.1. We prove (4.4) forx(t) satisfying (1.3, 1.4) andφ(t) solution of (2.3, 2.4).

Relation (4.1) implies that Z T

0 ||(B^∗φ)⁰(·)||²Udt≤2 (Z T

0 ||(B^∗x)⁰(·)||²Udt+ Z T

0 ||(B^∗ψ)⁰(·)||²Udt )

·

Estimate above combined with inequality (3.6) in Proposition 3.2 implies the existence of a constantC1 >0, independent of (x⁰, x¹), such that

C1

Z T 0

||(B^∗φ)⁰(·)||²Udt≤Z T 0

||(B^∗x)⁰(·)||²Udt. (4.5) On the other hand, according to Remark 4.2 and to relation (4.1) we have that

||(B^∗φ)⁰(·)||^U ∈L²(0, T).

This means that (4.2) can be rewritten as

ψ⁰⁰(t) +Aψ(t) +B(B^∗ψ)⁰(t) =−B(B^∗φ)⁰(t). (4.6) We denote now byw(t) the extension of (B^∗φ)⁰ obtained by defining w(t) = 0, t∈R\[0, T]. We still denote byψ(t) the solution of

ψ⁰⁰(t) +Aψ(t) +B(B^∗ψ)⁰=−Bw(t), t∈R,

ψ(0) =ψ⁰(0) = 0. (4.7)

We clearly haveψ(t) = 0 fort∈R\[0, T].

Taking the Laplace transform we get

λ²ψ(λ) +b Aψ(λ) +b λ BB^∗ψ(λ) =b −Bw(λ),b ∀λ=γ+iη, γ >0.

The equality above holds in (D(A¹²))⁰.

By applying ¯λψ¯b∈ D(A¹²) to the equality above, we get

λ|λ|²||ψ(λ)b ||²X+ ¯λ||A¹²ψ(λ)b ||²X+||λ B^∗ψ(λ)b ||²U =− hw(λ),λ B¯ ^∗ψ(λ)¯b i^U. Taking the real part of each term, we get

Z

Rη

||λ B^∗ψ(λ)b ||²Udη≤ 1 2 Z

Rη

||w(λ)b ||²Udη+1 2 Z

Rη

||λ B^∗ψ(λ)b ||²Udη.

Parseval identity implies

k(B^∗ψ)⁰(t)k²L²(0,T;U)≤ k(B^∗φ)⁰(t)k²L²(0,T;U). (4.8) Relation (4.1) and inequality above imply that

k(B^∗x)⁰(t)k²L²(0,T;U)≤4k(B^∗φ)⁰(t)k²L²(0,T;U). (4.9) Inequalities (4.5) and (4.9) obviously yield the conclusion (4.4).

We can now prove the first main result.

Proof of Theorem 2.2. All finite energy solutions of (1.3, 1.4) satisfy the estimate

E(x(t))≤Me^−ωtE(x(0)), ∀t≥0, (4.10)

whereM, ω >0 are constants independent of (x⁰, x¹), if and only if there exist a time T >0 and a constant C >0 (depending onT) such that

E(x(0))−E(x(T))≥CE(x(0)), ∀(x⁰, x¹)∈V ×X.

By (2.1) relation above is equivalent to the inequality Z T

0 ||(B^∗x)⁰(s)||²Uds≥CE(x(0)), ∀(x⁰, x¹)∈V ×X.

From Lemma 4.1 it follows that the system (1.3, 1.4) is exponentially stable if and only if Z T

0 ||(B^∗φ)⁰(s)||²Uds≥CE(x(0)), ∀(x⁰, x¹)∈ D(A)×V

holds true. By density it follows that (1.3, 1.4) is exponentially stable if and only if (2.5) holds true. This ends up the proof of Theorem 2.2.

Remark 4.3. By analyzing the proof above we notice that the proof of the inequality (4.9) does not require assumption (H). More precisely, the inequality (4.8) can be also obtained in the following direct manner:

Let (x⁰, x¹)∈V ×X such thatB^∗φ∈H¹(0, T;U). Then, by formally multipying (4.7) byψ⁰, it follow that the functionψ=x−φsatisfies

||(ψ(t), ψ⁰(t)||²V×X+ Z t

0

||B^∗ψ⁰(s)||²U+||B^∗x⁰(s)||²U

ds=

Z t 0

||B^∗φ⁰(s)||²Uds.

This implies that (4.9) holds.

This means that the result mentioned in Remark 2.3 can be also established by a direct method.

Before giving the proof of Theorem 2.4, we need a technical lemma. This lemma extends a result in Jaffard et al.[11]. For a proof we refere to Ammariet al.[1].

Lemma 4.4. Let(E^k)be a sequence of positive real numbers satisfying

Ek+1≤ Ek−CEk+1^2+α, ∀k≥0, (4.11)

whereC >0andα >−1are constants. Then there exists a positive constantM such that E^k≤ M

(k+ 1)^1+α1 , ∀k≥0. (4.12)

Proof of Theorem 2.4. By density (2.8) implies that for all (x⁰, x¹)∈V ×X we have

Z T 0

||(B^∗φ)⁰(t)||²Udt≥C||(x⁰, x¹)||²V×XG

||(x⁰, x¹)||²X1×X2

||(x⁰, x¹)||²V×X

·

By applying Lemma 4.1 we obtain that the solution x(t) of (1.3, 1.4) satisfies the following inequality Z T

0 ||(B^∗x)⁰(t)||²Udt≥C||(x⁰, x¹)||²V×XG

||(x⁰, x¹)||²X₁×X₂

||(x⁰, x¹)||²V×X

, ∀(x⁰, x¹)∈V ×X.

Relation above and (2.1) imply the existence of a constantK >0 such that

||(x(T), x⁰(T))||²V×X≤ ||(x⁰, x¹)||²V×X−K||(x⁰, x¹)||²V×XG

||(x⁰, x¹)||²X1×X2

||(x⁰, x¹)||²V×X

, ∀(x⁰, x¹)∈ D(A^d).

(4.13) By using (2.7) (see again [24]), we obtain for fixedθ∈(0,1)

||(x⁰, x¹)||²X1×X2

||(x⁰, x¹)||²V×X

≥ ||(x⁰, x¹)||X^2−2θ1^θ×X2

||(x⁰, x¹)||V²⁻×^θ2θX

, ∀(x⁰, x¹)∈ D(A^d). (4.14)

By using (4.14) combined with the fact that the functiont→ ||(x(t), x⁰(t))||²V×X is nonincreasing, the functionG is increasing and relation (4.22) we obtain the existence of a constantK1>0 such that

||(x(T), x⁰(T))||²V×X ≤ ||(x⁰, x¹)||²V×X−K1||(x⁰, x¹)||²V×XG



||(x(T), x⁰(T))||V^2−2θ×^θX

||(x⁰, x¹)||Y²⁻₁^θ×^2θY₂



· (4.15)

Estimate (4.15) remains valid in successive intervals [kT,(k+ 1)T], so, we have

||(x((k+ 1)T), x⁰((k+ 1)T))||²V×X ≤ ||(x(kT), x⁰(kT))||²V×X

−K1||(x(kT), x⁰(kT))||²V×XG



||(x((k+ 1)T), x⁰((k+ 1)T))||V^2−2θ×^θX

||(x(kT), x⁰(kT))||Y^2−2θ1^θ×Y2



·

SinceA^dgenerates a semigroup of contractions inD(A^d) and the graph norm onD(A^d) is equivalent to||.||^Y1×Y₂, relations above imply the existence of a constantK2>0 such that

||(x((k+ 1)T), x⁰((k+ 1)T))||²V×X ≤ ||(x(kT), x⁰(kT))||²V×X

−K2||(x(kT), x⁰(kT))||²V×XG



||(x((k+ 1)T), x⁰((k+ 1)T))||V^2−2θ×^θX

||(x⁰, x¹)||_D^2−2θ(^θAd)



,

∀(x⁰, x¹)∈ D(A^d). (4.16)

If we adopt now the notation

E^k=G



||(x(kT), x⁰(kT))||V^2−2θ×^θX

||(x⁰, x¹)||_D^2−2θ(^θAd)



, (4.17)

the inequalities (4.23) implies

||(x((k+ 1)T), x⁰((k+ 1)T))||²V×X

||(x(kT), x⁰(kT))||²V×X

E^k

E^k+1 E^k+1≤ E^k−K2E^kE^k+1. (4.18) Since, the function t → ||(x(t), x⁰(t))||²V×X is nonincreasing and the function G is increasing, relation (4.18) implies

||(x((k+ 1)T), x⁰((k+ 1)T))||²V×X

||(x(kT), x⁰(kT))||²V×X

E^k

E^k+1E^k+1≤ E^k−K2Ek+1² . (4.19) According to (4.24), relation (4.19) gives,

1

2

6

4

||(x(kT),x0(kT))||2−2θ θ V×X

||(x0,x1)||2−2θ D(Aθd)

3

7

5

1−θθ G ^{||(x(kT),x0(kT))||}

2−2θθ V×X

||(x⁰,x¹)||_D(Ad^2−2θθ ₎

!

1

2

6

4

||(x((k+1)T),x0((k+1)T))||2−2θ θ V×X

||(x0,x1)||2−2θ D(θAd)

3

7

5

1−θθ G ^||(x((k+1)T),x⁰((k+1)T))||V^2−2θ×X^θ

||(x⁰,x¹)||_D(A^2−2θθ

d)

!E^k+1

≤ E^k−K2Ek+1² . (4.20)

Relation (4.20) combined with that the functionx→ ¹

x1−θ^θ G(x) is increasing in (0,1), gives

E^k+1≤ E^k−K2Ek+1² , ∀k≥0. (4.21)

By applying Lemma 4.4 and using relation (4.24) we obtain the existence of a constantM >0 such that

||(x(kT), x⁰(kT))||²V×X≤

G⁻¹ M

k+ 1 _1−θ^θ

||(x⁰, x¹)||²_D(Ad), ∀k≥0, which obviously implies (2.9).

Proof of second assertion of Theorem 2.4. By density (2.10) implies that for all (x⁰, x¹)∈V ×X we have Z T

0 ||(B^∗φ)⁰(t)||²Udt≥C||(x⁰, x¹)||²X₁×X₂.

Then, Lemma 4.1 combined with (2.7) and (2.1) imply the existence of a constantK >0 such that

||(x(T), x⁰(T))||²V×X ≤ ||(x⁰, x¹)||²V×X−K||(x⁰, x¹)||X^2θ₁×X₂

||(x⁰, x¹)||V^2−2θ×^θX

,

∀(x⁰, x¹)∈ D(A^d). (4.22)

Following the same steps as in the proof of the first assertion of Theorem 2.4 we obtain the existence of a constantC >0 such that for all k≥0 we have

||(x((k+ 1)T), x⁰((k+ 1)T))||²V×X ≤ ||(x(kT), x⁰(kT))||²V×X

−C||(x((k+ 1)T), x⁰((k+ 1)T))||V^2θ×X

||(x⁰, x¹)||_D²⁻(^θA^2θd)

, ∀(x⁰, x¹)∈ D(A^d). (4.23)

If we adopt the notation

H^k= ||(x(kT), x⁰(kT))||²V×X

||(x⁰, x¹)||²_D(Ad)

, (4.24)

relation (4.23) gives

H^k+1≤ H^k−CHk+1^1θ , ∀k≥0. (4.25)

By applying Lemma 4.4 and using relation (4.25) we obtain the existence of a constantM >0 such that

||(x(kT), x⁰(kT))||²V×X ≤M||(x⁰, x¹)||²_D(Ad)

(k+ 1)^1−θθ , ∀k≥0,

which obviously implies (2.10).

5. Some applications

Now, we give some applications of Theorem 2.2 and Theorem 2.4. Some of them are new and some were obtained by different methods in previous literature.

5.1. First example: Stabilization of the string We consider the following initial and boundary problem:

(I)











∂²u

∂t² −∂²u

∂x²+∂u

∂t(ξ, t)δξ = 0,(x, t)∈(0,1)×(0,+∞), u(0, t) =u(1, t) = 0, t∈(0,+∞),

u(x,0) =u⁰(x), ∂u

∂t(x,0) =u¹(x), x∈(0,1),

whereξ∈(0,1) and δξ is the Dirac mass concentrated in the pointξ∈(0,1).

In this case, we have:

X =L²(0,1), U =R, V =H₀¹(0,1), and

A=− d²

dx²,D(A) =H²(0,1)∩H₀¹(0,1), Bk=k δξ,∀k∈R. (5.1) Then,A^d is given by

A^d u

v

=



 v d²u

dx² −v(ξ)δξ



,

∀(u, v)∈ D(Ad) =

(u, v)∈[H₀¹(0,1)∩H²(0, ξ)∩H²(ξ,1)]×H₀¹(0,1), du

dx(ξ⁺)−du

dx(ξ⁻) =v(ξ)

· Denote byQ the set of all rational numbers. Let us also denote by S the set of all numbers ρ ∈(0,1) such that ρ6∈Q and if [0, a1, . . . , an, . . .] is the expansion ofρ as a continued fraction, then (an) is bounded. Let us notice thatS is is obviously uncountable and, by classical results on diophantine approximation (cf. [7], p.

120), its Lebesgue measure is equal to zero. In particular, by Euler–Lagrange theorem (see Lang [15], p. 57)S contains allξ∈(0,1) such thatξis an irrational quadratic number (i.e. satisfying a second degree equation with rational coefficients). According to a classical result (see for instance Tucsnak [25] and the references therein) ifξ∈ S then there exists a constantCξ >0 such that

|sin (nπξ)| ≥Cξ

n, ∀n≥1. (5.2)

Stability results for (I) are then an immediate consequence of Theorem 2.2 and Theorem 2.4. We have the following result:

Theorem 5.1. 1. For anyξ∈(0,1)the system described by (I) is not exponentially stable in V ×L²(0,1).

2. For allξ∈ S and for allt≥0we have

(u(t),∂u

∂t(t))

²_V

×L²(0,1)

≤ Cξ

t+ 1||(u⁰, u¹)||²_D(Ad),

∀(u⁰, u¹)∈ D(A^d), (5.3)

whereCξ>0is a constant depending only onξ.

3. If >0then, for almost all ξ∈(0,1) and for allt≥0we have

(u(t),∂u

∂t(t))

²

V×L²(0,1)

≤ Cξ,

(t+ 1)¹⁺¹ ||(u⁰, u¹)||²_D(Ad),

∀(u⁰, u¹)∈ D(A^d), (5.4) whereCξ,>0is a constant depending only on ξ and.

Remark 5.2. The first assertion of the theorem above was proved by a different method in Bambergeret al.[3].

In this case the problem (2.3, 2.4) becomes

∂²φ

∂t² −∂²φ

∂x² = 0,(0,1)×(0,+∞), (5.5)

φ(0, t) =φ(1, t) = 0,(0,+∞), (5.6)

φ(x,0) =u⁰(x), ∂φ

∂t(x,0) =u¹(x),(0,1). (5.7)

Lemma 5.3. The operatorsA andB defined by(5.1)satisfy assumption (H).

Proof. Letk∈R. It can be easily cheked thatv= (λ²+A)⁻¹Bk satisfies:

λ²v(x)− d²v

dx²(x) = 0, x∈(0, ξ)∪(ξ,1), Reλ >0, (5.8)

v(0) =v(1) = 0, (5.9)

[v]ξ= 0, dv

dx

ξ

=k, (5.10)

where we denote by [g] the jump of the functiong at the pointξ.

The solutions of (5.8, 5.9) have the form v(x) =

Ash(λx), x∈(0, ξ), Bsh[λ(x−1)], x∈(ξ,1), whereA, B are constants.

Consequently, the solutions of (5.8–5.10) have the following form

v(x) =







 1 λ

sh[λ(ξ−1)] sh(λx)

sh(λ) k, x∈(0, ξ), 1

λ

sh(λξ) sh[λ(x−1)]

sh(λ) k, x∈(ξ,1).

Then, the functionH(λ) =λ B^∗(λ²+A)⁻¹B associated to problem (I) is given by the following expression H(λ) = sh(λξ) sh[λ(ξ−1)]

sh(λ) , Reλ >0.

We easily check that

sup

λ∈C_β|H(λ)| ≤ ch(βξ) ch[β(ξ−1)]

sh(β) ·

Thus (H) is satisfied.

The observability inequality concerning the trace at the pointx=ξ of the solutions of (5.5–5.7) is given in the proposition below.

Proposition 5.4. LetT >0be fixed. Then the following assertions hold true.

1. For allξ∈ S the solutionφ of (5.5–5.7) satisfies Z T

0

∂φ

∂t(ξ, t) 2

dt≥Cξ

ku⁰k²L²(0,1)+ku¹k²H⁻¹(0,1)

,

∀(u⁰, u¹)∈V ×L²(0,1), (5.11)

whereCξ>0is a constant depending only onξ.

2. For all >0and for almost allξ∈(0,1) the solutionφof (5.5–5.7) satisfies Z T

0

∂φ

∂t(ξ, t) 2

dt≥Cξ,

ku⁰k²H⁻(0,1)+ku¹k²H⁻¹⁻(0,1)

,

∀(u⁰, u¹)∈V ×L²(0,1), (5.12)

whereCξ,>0is a constant depending only on ξ and.

3. The result in assertion 1 is sharp in the sense that, for allξ ∈(0,1), there exists a sequence (u⁰_m, u¹_m)

⊂ V ×L²(0,1) such that the corresponding sequence of solutions (φm) of (5.5–5.7) with initial data (u⁰_m, u¹_m)satisfies

mlim→∞

RT 0

h∂φ

∂t(ξ, t) i2

dt ku⁰k²H(0,1)+ku¹k²H⁻¹⁺(0,1)

= 0. (5.13)

Proof. If we put

u⁰(x) = X∞ n=1

ansin(nπx), u¹(x) = X∞ n=1

nbnsin(nπx) (5.14)

with

(nan),(nbn)⊂l²(R), then we clearly have

∂φ

∂t(ξ, t) =X

n≥1

−nπansin(nπt) sin(nπξ) +nπbncos(nπt) sin(nπξ)

. (5.15)