c 2000 Society for Industrial and Applied Mathematics
SIAM J. CONTROL OPTIM.
Vol. 39, No. 4, pp. 1160–1181
STABILIZATION OF BERNOULLI–EULER BEAMS BY MEANS OF
A POINTWISE FEEDBACK FORCE∗
KAIS AMMARI† AND MARIUS TUCSNAK‡
Abstract. We study the energy decay of a Bernoulli–Euler beam which is subject to a pointwise
feedback force. We show that both uniform and nonuniform energy decay may occur. The uniform
or nonuniform decay depends on the boundary conditions. In the case of nonuniform decay in the
energy space we give explicit polynomial decay estimates valid for regular initial data. Our method
consists of deducing the decay estimates from observability inequalities for the associated undamped
problem via sharp trace regularity results.
Key words. pointwise stabilization, observability inequality, unbounded feedback, exponential
stability
AMS subject classifications. 35E15, 93D15, 93D20, 35B37, 35Q72
PII. S0363012998349315
1. Introduction. The aim of this paper is to study the pointwise feedback stabilization of a Bernoulli–Euler beam. More precisely, we consider the following initial
and boundary value problems:
(1.1)
(1.2)
(1.3)
∂2u
∂4u
∂u
(ξ, t) δξ = 0,
(x, t) +
(x, t) +
2
∂t
∂x4
∂t
u(0, t) = u(π, t) =
u(x, 0) = u0 (x),
0 < x < π, t > 0,
∂2u
∂2u
(0, t) =
(π, t) = 0,
2
∂x
∂x2
∂u
(x, 0) = u1 (x),
∂t
t > 0,
0 < x < π,
and
(1.4)
(1.5)
(1.6)
∂2u
∂4u
∂u
(x, t) +
(x, t) +
(ξ, t) δξ = 0,
2
∂t
∂x4
∂t
u(0, t) =
0 < x < π, t > 0,
∂2u
∂u
∂3u
(π, t) =
(0, t) =
(π, t) = 0,
2
∂x
∂x
∂x3
u(x, 0) = u0 (x),
∂u
(x, 0) = u1 (x),
∂t
t > 0,
0 < x < π.
Here u denotes the transverse displacement of the beam, δξ is the Dirac mass concentrated in the point ξ ∈ (0, π), and we suppose that the length of the beam is equal
to π. The boundary condition (1.2) means that both ends of the beam are simply
supported, whereas (1.5) signifies that the end x = 0 is simply supported and at x = π
there is a shear hinge end. Simple calculation shows that (1.1) is equivalent to the
∗ Received by the editors December 11, 1998; accepted for publication (in revised form) March 9,
2000; published electronically December 5, 2000.
http://www.siam.org/journals/sicon/39-4/34931.html
† Centre de Mathématiques Appliquées, UMR CNRS 7641, École Polytechnique, 91128 Palaiseau
Cedex, France ([email protected]).
‡ Institut Elie Cartan, Université de Nancy I, 54506 Vandoeuvre les Nancy Cedex, France
([email protected]).
1160
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POINTWISE STABILIZATION
equations modelling the vibrations of two Bernoulli–Euler beams with a dissipative
joint (see [4] for further discussion of the model).
Pointwise stabilization of Bernoulli–Euler beams, or, equivalently, stabilization of
serially connected beams with dissipative joints, has been widely studied in recent
literature (see [4], [5], [6], [7], [17], [23]). In [4], [5], [6], [20], and [23] the authors
give several examples showing that both uniform and nonuniform decay may occur.
Their method is based on a classical result of Huang and Prüss (see [14] and [22])
combined with elaborate eigenvalues, calculations, or with concepts in system theory.
In the cases when we have strong, but not exponential decay, as far as we know, no
estimates were given in the literature.
In the present paper we give a simple proof of the fact that for any ξ ∈ (0, π)
solutions of (1.1)–(1.3) are not uniformly stable in the energy space. For the solutions
of (1.4)–(1.6) we give a complete characterization of points ξ for which the solutions
are uniformly stable in the energy space. The main novelty of the paper consists in
the fact that, even in the cases when we have no uniform energy decay, we give explicit
decay estimates for regular initial data. These estimates depend on the diophantine
approximations properties of ξ. As far as we know, the results in previous literature
concerning beam equations with pointwise feedbacks are essentially devoted to exponential stabilization. In the case of strong, but not exponential stability, no estimates
were given. In the case of bounded feedback controls, similar estimates were given by
Russell in [24]. Russell’s method cannot be extended to unbounded feedbacks and,
namely, to the case of pointwise stabilizers.
Even for the particular case of exponential decay our method is different from
those previously used in pointwise stabilization problems. Our approach, avoiding
frequency domain methods and spectrum calculations, is based on sharp trace regularity results combined with observability inequalities valid for solutions of appropriate
conservative problems. As far as we know this is the first example in which observability estimates for the undamped problem are used to derive stability estimates in
the presence of an unbounded feedback. Due to the appropriate choice of the associated undamped problem the basic observability estimates are simply obtained by
applying Ingham’s inequality. For bounded feedbacks a similar method was used in
[13] in order to study uniform stabilization of second-order equations. Since in our
case the feedback is unbounded we have to use new arguments, namely, some sharp
trace regularity results.
The plan of the paper is as follows. In section 2 we give precise statements of
the main results. Section 3 contains some new trace regularity results needed in the
following sections. In section 4 we prove exact pointwise observability results for the
associated undamped problem. The proof of the main result is given in section 5.
2. Statement of the main results. If u is a solution of (1.1)–(1.3) or of (1.4)–
(1.6), we define the energy of u at instant t by
2 2
2 ∂ u
1 π ∂u
E(u(t)) =
(x, t) + 2 (x, t)
(2.1)
dx.
2 0
∂t
∂x
Simple formal calculations show that a sufficiently smooth solution of (1.1)–(1.3)
or of (1.4)–(1.6) satisfies the energy estimate
(2.2)
2
t
∂u
E(u(0)) − E(u(t)) =
∂t (ξ, s) ds
0
∀ t ≥ 0.
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KAIS AMMARI AND MARIUS TUCSNAK
In particular, (2.2) implies that
E(u(t)) ≤ E(u(0)) ∀t ≥ 0.
The estimate above suggests that the natural wellposedness spaces for (1.1)–(1.3)
(respectively, for (1.4)–(1.6)) are V1 × L2 (0, π) (respectively, V2 × L2 (0, π)), where
dφ
V1 = H 2 (0, π) ∩ H01 (0, π), V2 = φ ∈ H 2 (0, π)|φ(0) =
(π) = 0
dx
are Hilbert spaces for the inner product
π 2
d u1 d2 ū2
u1
u2
,
=
+
v
v̄
1 2 , i = 1, 2.
v1
v2
dx2 dx2
0
V ×L2 (0,π)
i
Denote
Y = H 2 (0, π) ∩ H 4 (0, ξ) ∩ H 4 (ξ, π) × H 2 (0, π),
(2.3)
2
D(A1 ) = (u, v) ∈ Y, u(0) = v(0) = u(π) = v(π) = ddxu2 (0) =
(2.4) 2
d u
d2 u
d3 u
d3 u
dx2 (ξ+) = dx2 (ξ−), dx3 (ξ+) − dx3 (ξ−) = −v(ξ) ,
(2.5)
D(A2 ) = (u, v) ∈ Y, u(0) = v(0) =
d2 u
dx2 (ξ+)
=
d2 u
d3 u
dx2 (ξ−), dx3 (ξ+)
−
du
dx (π)
d3 u
dx3 (ξ−)
=
dv
dx (π)
=
d2 u
dx2 (0)
d2 u
dx2 (π)
=
= 0,
d3 u
dx3 (π)
= 0,
= −v(ξ) .
.
The corresponding operators A1 and A2 will be defined in section 5. If (u0 , u1 ) ∈ Y ,
we denote
||(u0 , u1 )||2Y = u0 2H 4 (0,ξ) + u0 2H 4 (ξ,π) + u1 2H 2 (0,π) .
(2.6)
We first check that (1.1)–(1.3) (respectively, (1.4)–(1.6)) are well posed in the spaces
above. Then we study the behavior of E(u(t)) when t → ∞. The wellposedness and
strong stability properties are summarized in the result below.
Proposition 2.1. The following assertions hold true.
1. Suppose that (u0 , u1 ) ∈ D(A1 ) (respectively, that (u0 , u1 ) ∈ D(A2 )). Then
the problem (1.1)–(1.3) (respectively, (1.4)–(1.6)) admits a unique solution
u
u
∈ C(0, T ; D(A1 )) respectively,
∈ C(0, T ; D(A2 )) .
∂u
∂u
∂t
0
1
∂t
2
0
1
2. If (u , u ) ∈ V1 × L (0, π) (respectively, (u , u ) ∈ V2 × L2 (0, π)), then the
problem (1.1)–(1.3) (respectively, (1.4)–(1.6)) admits a unique solution
u ∈ C(0, T ; V1 ) ∩ C 1 (0, T ; L2 (0, π)) (respectively, u ∈ C(0, T ; V2 )
∩C 1 (0, T ; L2 (0, π))),
such that u(ξ, ·) ∈ H 1 (0, T ) and
(2.7)
u(ξ, ·)
2H 1 (0,T ) ≤ C(
u0 2H 2 (0,π) + u1 2L2 (0,π) ),
where the constant C > 0 depends only on ξ and T . Moreover, u satisfies the
energy estimate (2.2).
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POINTWISE STABILIZATION
3. The estimate limt→∞ E(u(t)) = 0 holds true for any finite energy solution
of (1.1)–(1.3) (respectively, of (1.4)–(1.6)) if and only if πξ ∈ Q (respectively,
ξ
2p
π = 2q−1 ∀p, q ∈ N).
Remark 1. The result above shows, in particular, that one cannot expect strong
stabilization ∀ξ ∈ (0, π).
The main results in this paper concern the precise asymptotic behavior of the
solutions of (1.1)–(1.3) and of (1.4)–(1.6). As we will see below, the systems (1.1)–
(1.3) and (1.4)–(1.6) are generally not uniformly stable in the natural energy spaces.
However, we prove that, in some cases of strong but not exponential stability, the
energy decay is uniform for all initial data lying in more regular spaces.
Denote by Q the set of all rational numbers. Let us also denote by S the set of
all numbers ρ ∈ (0, π) such that πρ ∈ Q and if [0, a1 , . . . , an , . . .] is the expansion of
ρ
π as a continued fraction, then (an ) is bounded. Let us notice that S is obviously
uncountable and, by classical results on diophantine approximation (cf. [3, p. 120]),
its Lebesgue measure is equal to zero. Roughly speaking, the set S contains the
irrationals which are “badly” approximable by rational numbers. In particular, by
the Euler–Lagrange theorem (cf. [18, p. 57]) S contains all ξ ∈ (0, π) such that πξ is
an irrational quadratic number (i.e., satisfying a second degree equation with rational
coefficients). According to a classical result (see, for instance, [26] and the references
therein), if ξ ∈ S, then there exists a constant Cξ > 0 such that
| sin (nξ)| ≥
(2.8)
Cξ
n
∀ n ≥ 1.
Our main results can now be stated as follows.
Theorem 2.2.
1. For any ξ ∈ (0, π), the system described by (1.1)–(1.3) is not exponentially
stable in V1 × L2 (0, π).
2. ∀ξ ∈ S and ∀t ≥ 0 we have
(2.9)
E(u(t)) ≤
Cξ
||(u0 , u1 )||2Y
(t + 1)2
∀ (u0 , u1 ) ∈ D(A1 ),
where Cξ > 0 is a constant depending only on ξ.
3. ∀ > 0 there exists a set B ⊂ [(0, π) \ πQ], the Lebesgue measure of B being
equal to π, such that ∀ξ ∈ B and ∀t ≥ 0 we have
(2.10)
E(u(t)) ≤
Cξ,
(t + 1)
2
1+
||(u0 , u1 )||2Y
∀ (u0 , u1 ) ∈ D(A1 ),
where Cξ, > 0 is a constant depending only on ξ and .
Theorem 2.3.
1. The system described by (1.4)–(1.6) is exponentially stable in V2 × L2 (0, π) if
and only if πξ is a rational number with coprime factorization
p
ξ
= , where p is odd.
π
q
(2.11)
2. ∀ξ ∈ S and ∀t ≥ 0 we have
(2.12)
E(u(t)) ≤
Cξ
||(u0 , u1 )||2Y
(t + 1)2
∀ (u0 , u1 ) ∈ D(A2 ),
where Cξ > 0 is a constant depending only on ξ.
1164
KAIS AMMARI AND MARIUS TUCSNAK
3. ∀ > 0 there exists a set B ⊂ [(0, π) \ πQ], the Lebesgue measure of B being
equal to π, such that ∀ξ ∈ B and ∀t ≥ 0 we have
(2.13)
E(u(t)) ≤
Cξ,
2
(t + 1) 1+
||(u0 , u1 )||2Y
∀ (u0 , u1 ) ∈ D(A2 ),
where Cξ, > 0 is a constant depending only on ξ and .
Remark 2. In the case of a string with pointwise stabilizer, the explicit eigenvalue
calculation in [27] suggests that one cannot expect polynomial decay estimates like
(2.10) for any ξ satisfying the assumption in the third assertion of Proposition 2.1.
By analogy with the result in [27] we conjecture that ∀ > 0 there exists ξ satisfying
the assumption in the third assertion of Proposition 2.1 and the sequences (tn ) (of
real numbers), and (un ) (of finite energy solutions), with tn → ∞, such that
lim tn
n→∞
E(un (tn ))
= ∞.
2
n
(un (0), ∂u
∂t (0))
Y
3. Some regularity results. Consider the initial and boundary value problems
(3.1)
(3.2)
∂2v
∂4v
(x,
t)
+
(x, t) = k(t)δξ ,
∂t2
∂x4
v(x, 0) = 0,
∂v
(x, 0) = 0,
∂t
0 < x < π, t > 0,
0 < x < π,
and either
(3.3)
v(0, t) = v(π, t) =
∂2v
∂2v
(0, t) =
(π, t) = 0,
2
∂x
∂x2
t > 0,
or
(3.4)
v(0, t) =
∂2v
∂3v
∂v
(π, t) =
(0,
t)
=
(π, t) = 0,
∂x
∂x2
∂x3
t > 0.
The equations above are models for the vibrations of an undamped Bernoulli–Euler
beam, in the presence of a pointwise force. The main result of this section gives
regularity properties of the solutions of (3.1)–(3.3) and of (3.1),(3.2), and (3.4). These
regularity results are sharp (according to Remark 4 below).
Proposition 3.1. Suppose that k ∈ L2 (0, T ). Then the problem (3.1)–(3.3)
(respectively, (3.1),(3.2),(3.4)) admits a unique solution having the regularity
(3.5)
v ∈ C(0, T ; V1 ) ∩ C 1 (0, T ; L2 (0, π)) (respectively,
(3.6)
v ∈ C(0, T ; V2 ) ∩ C 1 (0, T ; L2 (0, π))).
Moreover, v(ξ, ·) ∈ H 1 (0, T ) and there exists a constant C > 0, depending only on T ,
such that
(3.7)
v(ξ, ·)
H 1 (0,T ) ≤ C
k
L2 (0,T )
∀ k ∈ L2 (0, T ).
Remark 3. We notice that the interior regularity (3.5) (respectively, (3.6)) does
not follow from the Sobolev regularity of the right-hand side of (3.1) or from the results
1165
POINTWISE STABILIZATION
in [25]. Moreover, estimate (3.7) is not a consequence of the interior regularity (3.5)
(respectively, (3.6)).
In order to prove Proposition 3.1 we first study the case of free vibrations of an
undamped beam, i.e., we consider the initial and boundary value problem
∂2φ
∂4φ
(x,
t)
+
(x, t) = 0,
∂t2
∂x4
(3.8)
(3.9)
φ(0, t) = φ(π, t) =
φ(x, 0) = u0 (x),
(3.10)
0 < x < π, t > 0,
∂2φ
∂2φ
(0,
t)
=
(π, t) = 0,
∂x2
∂x2
∂φ
(x, 0) = u1 (x),
∂t
t > 0,
0 < x < π,
and the problem formed by (3.8), (3.10), and the boundary conditions
(3.11)
φ(0, t) =
∂2φ
∂3φ
∂φ
(π, t) =
(0,
t)
=
(π, t) = 0,
∂x
∂x2
∂x3
t > 0.
The following result, besides showing that the problems above are well posed in the
natural energy spaces, gives a sharp inequality on the trace of φ at the point ξ.
Lemma 3.2. Suppose that (u0 , u1 ) ∈ V1 × L2 (0, π) (respectively, (u0 , u1 ) ∈ V2 ×
2
L (0, π)). Then the initial and boundary value problem (3.8)–(3.10) (respectively,
(3.8),(3.10), and (3.11)) admits a unique solution
φ ∈ C(0, T ; V1 ) ∩ C 1 (0, T ; L2 (0, π)),
(3.12)
respectively,
φ ∈ C(0, T ; V2 ) ∩ C 1 (0, T ; ; L2 (0, π)),
(3.13)
satisfying
φ(ξ, ·) ∈ H 1 (0, T ).
Moreover, there exists a constant C > 0, depending only on T , such that
φ(ξ, ·)
2H 1 (0,T ) ≤ C(
u0 2H 2 (0,π) + u1 2L2 (0,π) ).
(3.14)
Proof. We first notice that problem (3.8)–(3.10) can be written as
φ
φ
∂
∂φ = A0 ∂φ ,
∂t
∂t
∂t
where
D(A0 ) =
(3.15)
u
∈ H 4 (0, π) ∩ H01 (0, π) × H 2 (0, π) ∩ H01 (0, π)
v
2
d u
d2 u
dx2 (0) = dx2 (π) = 0 ,
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KAIS AMMARI AND MARIUS TUCSNAK
and
v u
d4 u .
A0 : D(A0 ) → V1 × L (0, π), A0
=
v
− 4
dx
2
(3.16)
One can easily check that A0 is skew-adjoint. So, by Stone’s theorem, it generates
a semigroup of isometries in V1 × L2 (0, π). This implies that (3.8)–(3.10) admits a
unique solution φ satisfying (3.12).
In order to prove (3.14) we put
u0 (x) =
(3.17)
∞
an sin (nx), u1 (x) =
n=1
with
∞
(3.18)
n=1
∞
n2 bn sin (nx),
n=1
n4 (a2n + b2n ) < ∞. In this case the solution of (3.8)–(3.10) is given by
φ(x, t) =
[an cos (n2 t) sin (nx) + bn sin (n2 t) sin (nx)],
n≥1
which implies that
(3.19)
φ(ξ, t) =
∞
an cos (n2 t) sin (nξ) + bn sin (n2 t) sin (nξ) .
n=1
If we consider the right-hand side of (3.19) as a Fourier series in t (see Theorem 4.1
from [10] for details) we obtain the existence of a constant C depending on T such
that
φ(ξ, ·)
2H 1 (0,T ) ≤ C
(3.20)
∞
n4 (a2n + b2n ),
n=1
which obviously implies (3.14).
The problem (3.8), (3.10), (3.11) can be treated in a completely similar manner.
It suffices to replace formulas (3.17) and (3.18) by the relations
u0 (x) =
∞
an sin
n=0
φ(x, t) =
∞
(2n + 1)2
2n + 1
2n + 1
x , u1 (x) =
bn sin
x ,
2
4
2
n=0
∞ 2n + 1
(2n + 1)2
t sin
x
an cos
4
2
n=0
(3.21)
+ bn sin
2n + 1
(2n + 1)2
t sin
x .
4
2
The relations above clearly imply (3.14).
In order to prove Proposition 3.1 we need the following technical result.
Lemma 3.3. Let γ > 0, ξ ∈ (0, π) be two fixed real numbers and
Cγ = {w ∈ C | Re(w)Im(w) = − γ2 }. Then the functions
sin(w ξ) sin[w (ξ − π)] sh(w ξ) sh[w (ξ − π)]
i
(3.22)
−
+
f1 (w) =
2w
sin(wπ)
sh(wπ)
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POINTWISE STABILIZATION
and
(3.23)
i
f2 (w) =
2w
sin(w ξ) cos[w (ξ − π)] sh(w ξ) ch[w (ξ − π)]
−
+
cos(wπ)
ch(wπ)
are bounded on Cγ , uniformly with respect to ξ ∈ [0, π].
Proof. Let us suppose that f1 is not bounded on Cγ . In this case there exists a
sequence (wn ) ⊂ Cγ such that
(3.24)
lim |f1 (wn )| = +∞.
n→+∞
As f1 is analytical in the open set D = {w ∈ C | Re(w)Im(w) < 0} and Cγ ⊂ D,
relation (3.24) clearly implies that |wn | → +∞. Due to the definition of Cγ this can
happen in two situations :
(3.25)
|Re(wn )| → +∞, |Im(wn )| =
γ
→ 0,
2 |Re(wn )|
|Im(wn )| → +∞, |Re(wn )| =
γ
→ 0.
2 |Im(wn )|
or
(3.26)
Suppose that (3.25) holds true. In this case a simple calculation shows that
sh(wn ξ) sh[wn (ξ − π)] 1
= ,
lim (3.27)
2
n→+∞
sh(wn π)
and
lim sup | sin(wn ξ) sin[wn (ξ − π)]| ≤ 1.
(3.28)
n→+∞
Relations (3.24), (3.27), and (3.28) imply that
lim |wn sin(wn π)| = 0.
(3.29)
n→+∞
Since lim |wn | = +∞, relation (3.29) yields
n→+∞
lim | sin(wn π)| = 0.
n→+∞
It is easily checked that the relation above implies the existence of a subsequence of
(wn ), denoted also by (wn ), and of the sequences (αn ) ⊂ N, (βn ) ⊂ [0, 1[, satisfying
(3.30)
|Re(wn )| = αn + βn ,
lim βn = 0.
n→+∞
We obviously have
(3.31)
γ
wn sin βn π − iπ 2Re(wn ) γ |wn sin(wn π)| = πRe(w
)
−
iπ
.
β
n
γ
n
Re(wn ) βn π − iπ 2Re(w
2
n)
On the other hand, (3.25) and (3.30) imply
sin β π − iπ γ
n
wn 2Re(wn ) = 1.
lim (3.32)
=
1,
lim
γ
n→+∞ n→∞ Re(wn ) βn π − iπ 2Re(w
n)
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KAIS AMMARI AND MARIUS TUCSNAK
Moreover, we obviously have
γ πγ
(3.33)
βn πRe(wn ) − iπ ≥
2
2
∀ n ≥ 1.
Relations (3.31)–(3.33) contradict (3.29). It follows that (3.24) and (3.25) cannot
both be true. By a similar method we can show that (3.24) and (3.26) cannot both
hold true. This means that assumption (3.24) is false, i.e., that f1 is bounded on Cγ .
The proof that f2 is also bounded on Cγ can be done in a completely similar manner.
The bounds are uniform with respect to ξ since supw∈Cγ |fi (w)|, i = 1, 2 depends
continuously on ξ ∈ [0, π].
We can now give the proof of the main result of this section.
Proof of Proposition 3.1. We use the method of transposition. Let D(A0 ) be the
space defined in (3.15), and denote by D[(A0 )] the dual space of D(A0 ) with respect
to the pivot space V1 × L2 (0, π). It is well known that A0 can be extended to a
skew-adjoint operator (denoted also by A0 ),
A0 : V1 × L2 (0, π) → [D(A0 )] ,
such that A0 generates a group of isometries in [D(A0 )] , denoted by S(t).
Moreover, we define the operator
0
B0 : R → [D(A0 )] , B0 r =
∀r ∈ R.
(3.34)
rδξ
With the notation above the problem (3.1)–(3.3) can be written as a Cauchy
problem in [D(A0 )] under the form
∂
v(t)
v(t)
(3.35)
= A0 ∂v
+ B0 k(t) ∀ t > 0,
∂t ∂v
∂t (t)
∂t (t)
(3.36)
v(0) =
∂v
(0) = 0.
∂t
After a simple calculation we get that the operator B0∗ : D(A0 ) → R is given by
u
u
∗
= v(ξ)
∀
∈ D(A0 ).
B0
v
v
This implies that
(3.37)
B0∗ S ∗ (t)
u0
u1
=
∂φ
(ξ, t)
∂t
∀
u0
u1
∈ D(A0 ),
with φ satisfying (3.8)–(3.10). From (3.14) and (3.37) we deduce that there exists a
constant C > 0 such that
0 2
0 2
0
T
∗ ∗
u u u
∈ D(A0 ).
(3.38)
∀
1
B0 S (t) u1 dt ≤ C u1 u
2
0
V ×L (0,π)
1
According to Theorem 3.1 in [2, p. 173], inequality (3.38) implies that (3.35), (3.36)
admit a unique solution
v
∈ C(0, T ; V1 × L2 (0, π)),
∂v
∂t
1169
POINTWISE STABILIZATION
which obviously implies the conclusion (3.5). The proof that the interior regularity
property (3.6) holds true for all solutions of (3.1), (3.2), and (3.4) can be obtained in
a completely similar manner, so we skip it here.
We still have to prove the trace regularity property (3.7).
As (3.1) is time reversible, after extending k by zero for t ∈ R \ [0, T ], we can
solve (3.1)–(3.3) for t ∈ R. In this way we obtain a function, denoted also by v, such
that
(3.39)
v ∈ C(0, T ; V1 ) ∩ C 1 (0, T ; L2 (0, π)), v(x, t) = 0,
∀ t ≤ 0,
and v satisfies (3.1)–(3.3) ∀(x, t) ∈ [0, π] × R.
Let v(x, λ), where λ = γ + iη, γ > 0, and η ∈ R, be the Laplace (with respect to
time) transform of v. Since v satisfies (3.39), estimate (3.7) is equivalent to the fact
that the function t → e−γt v(ξ, t) belongs to H 1 (R) and that there exists a constant
M1 > 0 such that
e−γ· v(ξ, ·)
2H 1 (−∞,∞) ≤ M1 k
2L2 (−∞,∞) .
Equivalently, by the Parseval identity (see, for instance, [9, p. 212]), it suffices to prove
that the function
η → (γ + iη)
v (ξ, γ + iη)
belongs to L2 (Rη ) for some γ > 0, and that there exists a constant M2 > 0 such that
∞
(3.40)
≤ M2
|k(γ + iη)|2 dη.
(γ + iη)
v (ξ, γ + iη)
2 2
L (
Rη )
−∞
It can be easily checked that v satisfies
∂ 4 v
(x, λ) = 0, x ∈ (0, ξ) ∪ (ξ, 1), Reλ > 0,
∂x4
(3.41)
λ2 v(x, λ) +
(3.42)
v(0, λ) = v(π, λ) =
∂ 2 v
∂ 2 v
(0,
λ)
=
(π, λ) = 0, Reλ > 0,
∂x2
∂x2
(3.43)
∂
v
[
v ]ξ =
∂x
(3.44)
∂ 3 v
∂x3
ξ
ξ
∂ 2 v
=
∂x2
ξ
= 0,
=
k(λ), Reλ > 0,
where we denote by [f ]ξ the jump of the function f at the point ξ. As the equations
above are linear, we deduce that, for every λ ∈ C, Reλ > 0, we can find H1 (λ) ∈ C,
such that
(3.45)
k(λ) ∀Re λ > 0.
λ v(ξ, λ) = H1 (λ)
In order to compute H1 (λ) we notice that the solutions of (3.41) have the form
x ∈ (0, ξ),
Aeiwx + Be−iwx + Cewx + De−wx ,
v(x, λ) =
A1 eiw(x−π) + B1 e−iw(x−π) + C1 ew(x−π) + D1 e−w(x−π) , x ∈ (ξ, π),
1170
KAIS AMMARI AND MARIUS TUCSNAK
where A, B, C, D, A1 , B1 , C1 , and D1 are constants and w is the unique complex number satisfying the conditions
π λ = iw2 , w = reiθ , with r > 0 and θ ∈ − , 0 .
(3.46)
2
Using (3.42), we obtain
2iA sin(wx) + 2Csh(wx),
v(x, λ) =
2iA1 sin[w(x − π)] + 2C1 sh[w(x − π)],
x ∈ (0, ξ),
x ∈ (ξ, π).
Consequently, the solutions of (3.41)–(3.43) have the following form:

sin(wπ)sh[w(ξ − π)]


2iA sin(wx) − 2iA
sh(wx), x ∈ (0, ξ),



sh(wπ) sin[w(ξ − π)]



sin(wξ)
sin[w(x − π)]
(3.47) v(x, λ) = 2iA

sin[w(ξ
− π)]




sin(wπ)sh(wξ)


sh[w(x − π)],
x ∈ (ξ, π).
 −2iA
sh(wπ) sin[w(ξ − π)]
Then, using (3.44) and (3.45), we obtain
(3.48)
H1 (λ) = f1 (w),
where f1 is defined by (3.22). By (3.46), the relation Re λ = γ > 0 implies that
w ∈ Cγ , with Cγ defined in Lemma 3.3 . We can now apply Lemma 3.3 to obtain the
existence of a constant M2 > 0 such that (3.40) holds true. This ends the proof of
the fact that (3.7) holds for all solutions of (3.1)–(3.3).
If v is the solution of (3.1), (3.2), and (3.4), similar calculations (see also [23])
imply that
k(λ) ∀Re λ > 0,
λ v(ξ, λ) = H2 (λ)
where H2 (iw2 ) = f2 (w) and f2 is defined in (3.23). Again applying Lemma 3.3 and
the method above, we can easily conclude that (3.7) holds for all solutions of (3.1),
(3.2), and (3.4).
Remark 4. It can be easily checked that ∀ε > 0, λε Hi (λ), i = 1, 2, is not bounded
on Cγ . This means that estimate (3.7) is no longer valid if we replace the H 1 norm
by the H 1+ε norm in the left-hand side of (3.7). This means that (3.7) is a sharp
estimate.
Remark 5. In [23] it is shown that the system (3.1), (3.2), (3.4) can be written
as
ż + Az = Bk
in an appropriate Hilbert space, with the input k and the output y = B ∗ k. The
results we proved in this section say that this system is well posed, in the sense used
in [23]. According to classical results (see again [23] and the references therein), this
fact is equivalent to the boundedness of H2 on some half plane Re λ ≥ γ > 0. This
boundedness was proved in the appendix of [23]. Since we didn’t use H2 for the proof
of the interior regularity, we a priori needed only the boundedness of H2 on the line
Re λ = γ > 0. Due to this fact, our approach can be easily adapted for other systems
such as strings or Kirchhoff beams.
POINTWISE STABILIZATION
1171
4. Some observability inequalities. In this section we gather, for easy reference, some observability inequalities concerning the trace at the point x = ξ of the
solutions of (3.8)–(3.10) and of (3.8), (3.9), (3.11). The results in this section are similar to those obtained in [26] in a slightly different situation. Our first result concerns
problem (3.8)–(3.10), and it can be stated as follows.
Proposition 4.1. Let T > 0 be fixed and S ⊂ [0, π] be the set introduced in
section 2. Then we have the following.
1. ∀ξ ∈ S the solution φ of (3.8)–(3.10) satisfies
2
T
∂φ
1 2
(ξ, t) dt ≥ Cξ u0 2 1
+
u
−1
H (0,π)
H (0,π)
∂t
0
(4.1)
∀(u0 , u1 ) ∈ V1 × L2 (0, π),
where Cξ > 0 is a constant depending only on ξ.
2. ∀ > 0 and for almost all ξ ∈ (0, π) the solution φ of (3.8)–(3.10) satisfies
2
T
∂φ
1 2
(ξ, t) dt ≥ Cξ, u0 2 1−
+
u
−1−
(0,π)
(0,π)
H
H
∂t
0
(4.2)
∀(u0 , u1 ) ∈ V1 × L2 (0, π),
where Cξ, > 0 is a constant depending only on ξ and .
3. The result in assertion 1 is sharp in the sense that, ∀ξ ∈ (0, π), there exists
a sequence (u0m , u1m ) ⊂ V1 × L2 (0, π) such that the corresponding sequence of
solutions (φm ) of (3.8), (3.9) with initial data (u0m , u1m ) satisfies ∀ > 0
2
T ∂φm
(ξ,
t)
dt
∂t
0
lim
(4.3)
= 0.
2
2
0
1
m→∞ um 1+
(0,π) + um H −1+ (0,π)
H
Proof. Notice first that, thanks to Lemma 3.2, the left-hand side of (4.1) is well
defined and
(4.4)
∞
2
∂φ
(ξ, t) =
−n an sin (n2 t) sin (nξ) + n2 bn cos (n2 t) sin (nξ)
∂t
n=1
in L2 (0, T ), provided that u0 , u1 are given by (3.17). Moreover, from (4.4) and
the Ball–Slemrod generalization of Ingham’s inequality (cf. [1], [11]) we obtain that,
∀T > 0, there exists a constant CT > 0 such that
2
T
∞
∂φ
4 2
(ξ, t) dt ≥ CT
n an sin2 (nξ) + n4 b2n sin2 (nξ) .
(4.5)
∂t
0
n=1
Suppose now that ξ belongs to the set S defined in section 2. Then relations (4.5)
and (2.8) imply the existence of a constant KT,ξ > 0 such that
2
T
∞
∂φ
2 2
(ξ, t) dt ≥ KT,ξ
n an + n2 b2n
∀ ξ ∈ S,
∂t
0
n=1
which is exactly (4.1).
1172
KAIS AMMARI AND MARIUS TUCSNAK
In order to prove (4.2) we use a result in [3, p. 120] (see also Proposition 2.4 in
[26]) to get that ∀ > 0 there exists a set B ⊂ (0, π) having the Lebesgue measure
equal to π and a constant C > 0, such that for any ρ ∈ B
(4.6)
| sin (nρ)| ≥
C
n1+
∀ n ≥ 1.
Let us notice that by Roth’s theorem B contains all numbers in (0, π) having the
property that πξ is an algebraic irrational (see, for instance, [3, p. 104]). Inequalities
(4.5) and (4.6) obviously imply (4.2).
We still have to show the existence of a sequence satisfying (4.3). By using
continuous fractions (see again [26] and the references therein for details) we can
construct a sequence (qm ) ⊂ N such that qm → ∞ and
| sin (qm ξ)| ≤
(4.7)
π
qm
∀ m ≥ 1.
Using (4.4) and (4.7), a simple calculation shows that the sequence (φ0m , φ1m ) =
(sin (qm πx), 0) satisfies (4.3).
The observability results for (3.8), (3.9), and (3.11) are given in the proposition
below.
Proposition 4.2. Let T > 0 be fixed and S be the set introduced in section 2.
Then the following assertions hold true.
1. The existence of a constant Cξ > 0, such that the solutions φ of (3.8), (3.10),
and (3.11) satisfy
2
T
∂φ
1 2
(ξ, t) dt ≥ Cξ u0 2 2
+
u
2
H (0,π)
L (0,π)
∂t
0
(4.8)
∀(u0 , u1 ) ∈ V1 × L2 (0, π),
is equivalent to the fact that ξ satisfies (2.11).
2. ∀ξ ∈ S the solution φ of (3.8), (3.10), and (3.11) satisfies (4.1).
3. ∀ > 0 and for almost all ξ ∈ (0, π) the solution φ of (3.8), (3.10), and (3.11)
satisfies (4.2).
Proof. From (3.21) and the Ball–Slemrod generalization of Ingham’s inequality
we obtain the existence of a constant CT > 0 such that the solution φ of (3.8), (3.10),
and (3.11) satisfies
2
2
T
(2n + 1)4
∂φ
2n + 1 2
2 (ξ, t) dt ≥ CT
(4.9)
+
b
)
sin
(a
ξ
n
n ∂t
.
16
2
0
n≥0
If ξ satisfies (2.11), then, by Lemma 2.9 in [23], there exists a constant kξ > 0 such
that
sin (2n + 1)ξ ≥ kξ
(4.10)
∀ n ≥ 0.
2
Inequalities (4.9) and (4.10) imply that (4.8) holds true ∀ξ satisfying (2.11).
On the other hand, if ξ does not satisfy (2.11), we can again apply Lemma 2.9
from [23] to get the existence of a sequence (pm ) ⊂ N, limm→∞ pm = ∞ such that
(2pm + 1)ξ
(4.11)
= 0.
lim sin
m→∞
2
POINTWISE STABILIZATION
1173
If we denote by φm the solution of (3.8), (3.11) with initial data
(2pm + 1)x ∂φm
,
(x, 0) = 0
∀ x ∈ (0, π),
φm (x, 0) = sin
2
∂t
a simple calculation using (4.11) implies that
2
T ∂φm
(ξ,
t)
dt
∂t
0
= 0,
lim
∂φm
2
m→∞ φ (0)
2
m
H 2 (0,π) + ∂t (0)
L2 (0,π)
so (4.8) is false for any ξ not satisfying (2.11). Assertions 2 and 3 of the proposition can be proved by simply adapting the proof of Proposition 4.1, so we skip the
details.
5. Proof of the main results.
Proof of Proposition 2.1. The existence and uniqueness of finite energy solutions
of (1.1)–(1.3) (respectively, the problem (1.4)–(1.6)) can be obtained by standard
semigroup methods. However, for the sake of completeness we sketch the proof here.
Consider the unbounded linear operator
u
v
4
A1 : D(A1 ) → V1 × L2 (0, π), A1
=
,
v
− ddxu4 − v(ξ)δξ
where the derivatives with respect to x are calculated in D (0, π), and D(A1 ) is defined
in (2.4). If (u, v) ∈ D(A1 ), we denote by h1 (respectively, by h2 ) the function in
L2 (0, ξ) (respectively, in L2 (ξ, π)) defined by
h1 (x) =
d4 u
, calculated in D (0, ξ),
dx4
h2 (x) =
d4 u
, calculated in D (ξ, π).
dx4
4
Moreover, we define { ddxu4 } ∈ L2 (0, π) by
4 d u
h1 (x)
=
h2 (x)
dx4
A simple calculation shows that
v u
A1
= − d4 u
(5.1)
v
dx4
if
if
x ∈ (0, ξ),
x ∈ (ξ, π).
∀
u
∈ D(A1 ).
v
We remark that D(A1 ) ⊂ Y and that the graph norm in D(A1 ) is equivalent to · Y .
A simple calculation gives
u
u
u
2
A1
,
= −|v(ξ)|
∀
∈ D(A1 ),
v
v
v
V ×L2 (0,π)
1
so A1 is a dissipative operator. Moreover, it can be easily checked that A1 is onto, so,
according to Theorems 4.3 and 4.6 from [21, p. 14–15], we obtain that A1 generates
a continuous semigroup of linear contractions acting on V1 × L2 (0, π).
1174
KAIS AMMARI AND MARIUS TUCSNAK
This implies the existence and uniqueness of solutions u of (1.1)–(1.3) satisfying
u
∈ C(0, T ; D(A1 )), if (u0 , u1 ) ∈ D(A1 ),
∂u
∂t
and
u ∈ C(0, T ; V1 ) ∩ C 1 (0, T ; L2 (0, π)), if (u0 , u1 ) ∈ V1 × L2 (0, π).
In order to prove estimate (2.2) and the trace regularity property (2.7), it suffices to
remark that, through simple integration by parts, they hold true for regular solutions
u
) ∈ C(0, T ; D(A1 )). We can then use the density of D(A1 ) in V1 × L2 (0, π).
(i.e., ( ∂u
∂t
The similar properties for problem (1.4)–(1.6) can be proved by simply replacing A1
by the operator
u
v
4
A2 : D(A2 ) → V2 × L2 (0, π), A2
=
,
v
− ddxu4 − v(ξ)δξ
where D(A2 ) is defined in (2.5).
The strong stability estimates at the end of Proposition 2.1 can be obtained by
a simple application of LaSalle’s invariance principle. However, for the sake of completeness we give here the proof. Let S(t) be the semigroup of contractions generated
by the operator A1 , already introduced. In order to prove the strong stability of the
solutions of (1.1)–(1.3), it clearly suffices to show that
0
0
u
u
=0
∀
in D(A1 ).
lim S(t)
t→∞
u1
u1
We will show that this holds true provided that
ξ
∈ Q.
π
Since the imbedding D(A1 ) ⊂ V1 × L2 (0, π) is compact, the set
0
0
u
u
= ∪t≥0 S(t)
orb
u1
u1
(5.2)
0
is precompact in V1 × L2 (0, π) for any ( uu1 ) in D(A1 ). In this case the ω-limit set of
0
( uu1 ) defined by
0 0
u
u
2
ω
=
U
∈
V
→ U,
×
L
(0,
π),
∃(t
),
t
→
∞,
S(t
)
1
n
n
n
u1
u1
0
n→∞
is nonvoid for any ( uu1 ) in D(A1 ). On the other hand, by LaSalle’s invariance principle
(we refer to [8], [12, p. 18] for more details),
0
0
u
φ
∈ω
,
if
φ1
u1
then
0 φ 1 φ V
2
1 ×L (0,π)
0 u S(t
lim )
n
tn →+∞
u1 V ×L2 (0,π)
1
0 u
S(t + tn )
= lim u1 V ×L2 (0,π)
n→+∞ 1
0 φ
=
.
S(t) φ1 V ×L2 (0,π)
=
1
1175
POINTWISE STABILIZATION
Thus,
φ(., t) ∂φ
(·, t) ∂t
V1 ×L2 (0,π)
0 φ =
φ1 V
2
1 ×L (0,π)
for any t ≥ 0.
The relation above and (2.2) imply that φ satisfies the system (1.1)–(1.2) together
with
∂φ
(ξ, t) = 0 ∀ t ∈ (0, T ).
∂t
(5.3)
In particular, this implies that φ is the solution of (3.8)–(3.9) with φ(x, 0) = φ0 (x),
∂φ
1
∂t (x, 0) = φ (x). If we put
φ0 (x) =
cn sin(nx), φ1 (x) =
n≥1
n2 dn sin(nx),
n≥1
with (cn ), (dn ) ⊂ l2 (R), we have
∂φ
(ξ, t) =
n2 − cn sin(n2 t) + dn cos(n2 t) sin(nξ).
∂t
n≥1
The relation above, (5.2), and Ingham’s inequality imply that cn = dn = 0 ∀n ∈ N
so φ0 ≡ φ1 ≡ 0. We can now conclude that condition (5.2) is sufficient for the strong
stability of the solutions of (1.1)–(1.3).
If we suppose that ξ doesn’t satisfy (5.2), i.e., that πξ = pq , with p, q ∈ Z, one can
easily check that the solution u of (1.1)–(1.3) with initial data u0 = sin (qx), u1 = 0
satisfies E(u(t)) = E(u(0)) ∀t ≥ 0, so (5.2) is also necessary for the strong stability
of the solutions of (1.1)–(1.3).
In a completely similar manner we can tackle the strong stability for
(1.4)–(1.6).
Let u ∈ C(0, T ; V1 ) ∩ C 1 (0, T ; L2 (0, π)) be the solution of (1.1)–(1.3). Then u can
be written as
(5.4)
u = φ + ψ,
where φ is the solution of (3.8)–(3.10) and ψ satisfies
(5.5)
(5.6)
(5.7)
∂ 2 ψ ∂ 4 ψ ∂u
(ξ, t) δξ = 0
+
+
∂t2
∂x4
∂t
ψ(0, t) = ψ(π, t) =
in (0, π) × (0, T ),
∂2ψ
∂2ψ
(0, t) =
(π, t) = 0, t ∈ (0, T ),
2
∂x
∂x2
ψ(x, 0) =
∂ψ
(x, 0) = 0, x ∈ (0, π).
∂t
1176
KAIS AMMARI AND MARIUS TUCSNAK
In the same way the solution of (1.4)–(1.6) can be decomposed as in (5.4), where φ is
the solution of (3.8), (3.10), (3.11), and ψ satisfies (5.5), (5.7) together with
ψ(0, t) =
(5.8)
∂3ψ
∂2ψ
∂ψ
(π, t) =
(0, t) =
(π, t) = 0, t ∈ (0, T ).
2
∂x
∂x
∂x3
The main ingredient of the proofs of Theorems 2.2 and 2.3 is the following result.
Lemma 5.1. Suppose that (u0 , u1 ) ∈ V1 × L2 (0, π) (respectively, (u0 , u1 ) ∈ V2 ×
2
L (0, π)). Then the solutions u of (1.1)–(1.3) (respectively, of (1.4)–(1.6)) and the
solution φ of (3.8)–(3.10) (respectively, of (3.8), (3.10), (3.11)) satisfy
2
2
2
T
T
T
∂φ
∂u
∂φ
C1
(5.9)
∂t (ξ, t) dt ≤
∂t (ξ, t) dt ≤ 4
∂t (ξ, t) dt,
0
0
0
where C1 > 0 is a constant independent of (u0 , u1 ).
2
Remark 6. By Proposition 2.1, ∂u
∂t (ξ, ·) ∈ L (0, T ). So, (5.5) makes sense. The
∂u
result above shows that the L2 norm of ∂t (ξ, ·) is equivalent to the L2 norm of ∂φ
∂t (ξ, ·).
∂φ
2
(Notice that ∂t (ξ, ·) ∈ L (0, T ) by Lemma 3.2.)
Proof of Lemma 5.1. We prove (5.9) only for u satisfying (1.1)–(1.3) and the φ
solution of (3.8)–(3.10). As for u satisfying (1.4)–(1.6) and the φ solution of (3.8),
(3.10), (3.11), the proof is a completely similar one.
Relation (5.4) implies that
2
2
2 !
T
T
T
∂φ
∂u
∂ψ
(ξ, t) dt ≤ 2
(ξ, t) dt +
∂t
∂t
∂t (ξ, t) dt .
0
0
0
The estimate above combined with inequality (3.7) in Proposition 3.1 implies the
existence of a constant C1 > 0, independent of (u0 , u1 ), such that
2
2
T
T
∂φ
∂u
C1
(5.10)
∂t (ξ, t) dt ≤
∂t (ξ, t) dt.
0
0
On the other hand, according to Remark 6 and to relation (5.4), we have that
∂φ
2
∂t (ξ, ·) ∈ L (0, T ). This means that (5.5) can be rewritten as
(5.11)
∂4ψ
∂ψ
∂φ
∂2ψ
(ξ, t)δξ = − (ξ, t) δξ
(x, t) +
(x, t) +
2
∂t
∂x4
∂t
∂t
in (0, π) × (0, T ).
If we formally multiply (5.11) by ∂∂tψ̄ (this can be done rigorously by considering a
regularizing sequence), we obtain
2
T
T ∂φ
∂ψ
∂
ψ̄
(ξ, t)
(ξ, t)dt ,
∂t (ξ, t) dt ≤ ∂t
∂t
0
0
which obviously yields
2
2
∂ψ
∂φ
≤ (ξ, t)
.
∂t (ξ, t) 2
2
∂t
L (0,T )
L (0,T )
Relation (5.4) and the inequality above imply that
2
2
∂u
∂φ
(5.12)
≤ 4 (ξ, t)
.
∂t (ξ, t) 2
2
∂t
L (0,T )
L (0,T )
Inequalities (5.10) and (5.12) obviously yield the conclusion (5.9).
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1177
Before giving the proof of the main results we need one more technical lemma.
This lemma extends a result in [16].
Lemma 5.2. Let (Ek ) be a sequence of positive real numbers satisfying
2+α
Ek+1 ≤ Ek − CEk+1
(5.13)
∀k ≥ 0,
where C > 0 and α > −1 are constants. Then there exists a positive constant M
(depending on α and C) such that
Ek ≤
(5.14)
M
1
(k + 1) (1+α)
∀k ≥ 0.
Proof. Consider the sequence
Fk =
M
1
(k + 1) 1+α
,
where M > 0 is to be determined. After a simple calculation we obtain that
1
1
1
lim (Fk − Fk+1 )k(k + 2) 1+α =
,
M k→∞
1+α
(5.15)
so there exists k0 > 0 such that
Fk − Fk+1 ≤
2M
1
(1 + α)k(k + 2) 1+α
∀k ≥ k0 .
The relation above implies that
(5.16)
Fk − Fk+1 ≤
4
F 2+α
(1 + α)M 1+α k+1
∀k ≥ k1 = max {k0 , 2}.
If we suppose now that
(5.17)
4
M
< C and
≥ Ek1 ,
1
(1 + α)M 1+α
(k1 + 1) 1+α
from (5.16) we get
(5.18)
2+α
Fk − Fk+1 ≤ CFk+1
∀k ≥ k1 .
It obviously suffices to show that
(5.19)
Ek ≤ Fk
∀k ≥ k1 .
We shall do that by induction over k.
For k = k1 , (5.19) follows directly from (5.17). If we suppose that (5.19) holds
true for k ≤ m, by combining (5.13) and (5.18) we obtain
2+α
2+α
≤ Fm+1 + CFm+1
,
Em+1 + CEm+1
which obviously implies that Em+1 ≤ Fm+1 .
1178
KAIS AMMARI AND MARIUS TUCSNAK
We can now prove the main results.
Proof of Theorem 2.2. 1. Suppose that there exists ξ ∈ (0, π) such that solutions
of (1.1)–(1.3) satisfy the estimate
(5.20)
E(u(t)) ≤ M e−ωt E(u(0))
∀ t ≥ 0,
where M, ω > 0 are constants depending only on ξ. Relation (5.20) implies the
existence of a time T > 0 and of a constant C > 0 (depending on T ) such that
∀ (u0 , u1 ) ∈ V1 × L2 (0, π).
E(u(0)) − E(u(T )) ≥ CE(u(0))
The relation above combined with (2.2) yields
2
T
∂u
(ξ, s) ds ≥ CE(u(0))
∀ (u0 , u1 ) ∈ V1 × L2 (0, π),
∂t
0
which, by Lemma 5.1, implies that the solution φ of (3.8)–(3.10) satisfies
2
T
∂φ
(ξ, s) ds ≥ C E(u(0))
∀ (u0 , u1 ) ∈ V1 × L2 (0, π).
∂t
4
0
The inequality above clearly contradicts assertion 3 in Proposition 4.1. So assumption
(5.20) is false. We end in this way the proof of the first assertion of Theorem 2.2.
We pass now to the proof of the second assertion of this theorem. Let ξ ∈ S. By
Proposition 4.1 and Lemma 5.1, the solution u of (1.1)–(1.3) satisfies the inequality
2
T
∂u
1 2
(ξ, t) dt ≥ K1 u0 2 1
∀(u0 , u1 ) ∈ V1 × L2 (0, π),
H (0,π) + u H −1 (0,π)
∂t
0
where K1 > 0 is a constant. The relation above and (2.2) imply that
{u(T ), u (T )}
2V1 ×L2 (0,π) ≤ {u0 , u1 }
2V1 ×L2 (0,π)
(5.21)
−K1 {u0 , u1 }
2H 1 (0,π)×H −1 (0,π)
∀ (u0 , u1 ) ∈ D(A1 ).
By using a simple interpolation inequality (cf. [19, p. 49]), the fact that the function t → {u(t), u (t)}
2V1 ×L2 (0,π) is nonincreasing, and relation (5.21), we obtain the
existence of a constant K2 > 0 such that
{u(T ), u (T )}
2V1 ×L2 (0,π) ≤ {u0 , u1 }
2V1 ×L2 (0,π)
3
(5.22)
−K2
{u(T ), u (T )}
V1 ×L2 (0,π)
{u0 , u1 }
Y
.
We follow now the method used in [24]. Estimate (5.22) remains valid in successive
intervals [kT, (k + 1)T ]. So, ∀k ≥ 0, we have
{u((k + 1)T ), u ((k + 1)T )}
2V1 ×L2 (0,π)
≤ {u(kT ), u (kT )}
2V1 ×L2 (0,π) − K2
{u((k + 1)T ), u ((k + 1)T )}
3V1 ×L2 (0,π)
{u(kT ), u (kT )}
Y
.
1179
POINTWISE STABILIZATION
Since A1 generates a semigroup of contractions in D(A1 ) and the graph norm on
D(A1 ) is equivalent to · Y , the relation above implies the existence of a constant
K3 > 0 such that
{u((k + 1)T ), u ((k + 1)T )}
2V1 ×L2 (0,π) ≤ {u(kT ), u (kT )}
2V1 ×L2 (0,π)
(5.23) −K3
{u((k + 1)T ), u ((k + 1)T )}
3V1 ×L2 (0,π)
∀ (u0 , u1 ) ∈ D(A1 ).
{u0 , u1 }
Y
If we adopt now the notation
Ek =
(5.24)
{u(kT ), u (kT )}
2V1 ×L2 (0,π)
{u0 , u1 }
2Y
,
relation (5.23) gives
(5.25)
3
2
Ek+1 ≤ Ek − K3 Ek+1
∀k ≥ 0.
By applying Lemma 5.2 for α = − 21 and using relation (5.25), we obtain the existence
of a constant M > 0 such that
{u(kT ), u (kT )}
2V1 ×L2 (0,π) ≤
M {u0 , u1 }
2Y
(k + 1)2
∀k ≥ 0.
The conclusion (2.9) follows now by simply using the fact that the function
t → {u(t), u (t)}
2V1 ×L2 (0,π)
is nonincreasing.
Let us now suppose that > 0 and that ξ belongs to the set B , introduced in
section 4. From (2.2), (4.2), and Lemma 5.1, it follows that
{u(T ), u (T )}
2V1 ×L2 (0,π) ≤ {u0 , u1 }
2V1 ×L2 (0,π)
−C
{u0 , u1 }
2H 1− (0,π)×H −1− (0,π) .
Using now the same method as above and the interpolation theorem from [19, p. 81],
we obtain that the sequence Ek , defined by (5.24), satisfies
3+
2
Ek+1 ≤ Ek − KEk+1
The relation above and Lemma 5.2 (with α =
Ek ≤
M
2
(k + 1) 1+
∀k ≥ 1.
−1
2 )
give
∀k ≥ 1,
which obviously implies (2.10).
Proof of Theorem 2.3. As above, we use the fact that all finite energy solutions
of (1.4)–(1.6) are exponentially stable in V2 × L2 (0, π) if and only if there exist the
positive constants T and KT such that
(5.26)
E(u(0)) − E(u(T )) ≥ KT E(u(0))
∀ (u0 , u1 ) ∈ V2 × L2 (0, π).
1180
KAIS AMMARI AND MARIUS TUCSNAK
Using now (2.2), (5.26), and Lemma 5.1, we obtain the existence of a constant Cξ > 0
such that all solutions φ of (3.8), (3.9), and (3.11) satisfy (4.8). By Proposition 4.2,
inequality (4.8) holds true if and only if ξ satisfies (2.11). Consequently, we obtain
that the finite energy solutions of (1.4)–(1.6) are exponentially stable in V2 × L2 (0, π)
if and only if ξ satisfies (2.11).
The proof of estimates (2.12), (2.13) can be done by using obvious adaptations
of the proof of estimates (2.9), (2.10), so it is omitted.
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