PORTUGALIAE MATHEMATICA
Vol. 60 Fasc. 4 – 2003
Nova Série
UNIFORM STABILIZATION AND EXACT CONTROL
OF A MULTILAYERED PIEZOELECTRIC BODY
Boris V. Kapitonov and G. Perla Menzala
Recommended by E. Zuazua
Abstract: A transmission problem for a class of dynamic coupled system of hyperbolic equations having piecewise constant coefficients in a bounded three-dimensional
domain is considered. Assuming that in the entire boundary, dissipative mechanisms
are present and that suitable geometric conditions on the domain and the interfaces
are satisfied, we prove that the total energy associated with the model decays exponentially as t → +∞. Exact boundary controllability is then obtained through Russell’s
“controllability via stabilizability” principle.
1 – Introduction
This paper is devoted to study the uniform stabilization as t → +∞ of the
solutions of a transmission problem for a class of dynamic coupled system of
hyperbolic equations from which a distinguish example is the coupled system
of electromagneto-elasticity governed by Maxwell equations and the system of
elastic waves. Let Ω be a bounded region of R3 with smooth boundary ∂Ω = S.
Received : January 1, 2002; Revised : July 8, 2002.
AMS Subject Classification: 35Q99, 35Q60, 35L99.
Keywords: uniform stabilization; transmission problem; piezoelectric body.
412
BORIS V. KAPITONOV and G. PERLA MENZALA
We will assume that Ω is occupied by a multilayered piezoelectric body whose
motion is governed by the system (see [4] and [7]):
(1.1)

µ
¶ X
3
3
X

∂
∂
∂u



ρ
u
−
(A∗i E) = 0
A
+
tt
ij


∂x
∂x
∂x

i
j
i

i,j=1
i=1



)
(

3

X

∂
∂u



− curl H = 0
DE +
Ai


∂t
∂x

i
i=1


βHt + curl E = 0




(
)


3

X
∂u



Ai
div DE +
= 0


∂xi


i=1




div H = 0
in Ω × (0, +∞). Here x = (x1 , x2 , x3 ) ∈ Ω and t denotes the time variable.
In (1.1) we denote by
u = (u1 , u2 , u3 ) = the displacement vector
E = (E1 , E2 , E3 ) = the electric field
H = (H1 , H2 , H3 ) = the magnetic field
β(x) = the electric permeability
ρ = the density
and the 3 × 3 matrices Aij (x), Ai and D(x) will satisfy suitable assumptions
given below. In the simplest case, when we consider an isotropic medium, then,
we will have that
3
X
µ
∂
∂
Aij
∂x
∂x
i
j
i,j=1
¶
= µ∆ + (λ + µ)∇ div
where λ and µ are the Lame’s constants (µ > 0, λ + µ > 0), D will be the identity
matrix, ∇ the gradient operator, ∆ the (vector) Laplacian,
3
X
∂
i=1
and
∂xi
(A∗i E) = α curl E
3
∂u
∂ X
Ai
= −α curl ut
∂t i=1
∂xi
413
UNIFORM STABILIZATION AND EXACT CONTROL
where α is a coupling constant. Here A∗i denotes the adjoint of Ai . The coupled
system (1.1) is complemented with initial conditions

 u(x, 0) = f1 (x) ,
(1.2)
 E(x, 0) = f (x) ,
3
ut (x, 0) = f2 (x)
H(x, 0) = f4 (x)
in Ω
and boundary conditions
(1.3)

3
3
X
X

∂u


A
A∗i E ηi = − a(x) ut − b(x) u
η
−

ij
i

∂x
i,j=1
j
i=1
Z th

i
³
´



 η x (E x η) = α(x) Hx η + γ(x)
H(x, τ ) x η exp −σ(x) (t − τ ) dτ
0
on ∂Ω × (0, +∞) where “ x ” denotes the usual vector product and η = η(x)
denotes the unit outward normal to ∂Ω = S at x. The functions a(x), b(x),
α(x), γ(x) and σ(x) will satisfy suitable conditions given below. In the simplest
case they are just positive constants.
Finding uniform rates of decay of the solution of problem (1.1), (1.2) and
(1.3) as t → +∞ is of interest to understand the evolution of the model and
consequently for the phenomenon described by it. Even more interesting is the
so called transmission problem associated with model (1.1)–(1.3). Let us describe
our main result of this article: Let Ω ⊆ R3 be as above and consider a finite number of subsets of Ω, {Bk }nk=1 which are open, connected, with smooth boundary
∂Bk = Sk and such that B k ⊂ Bk+1 for 1 ≤ k ≤ n − 1. We denote by Ω0 = B1 ,
Ωk = Bk+1 \B k for k = 1, 2, ..., n−1 and Ωn = Ω\B n . Now, we consider system
(1.1) restricted to each set Ωk × (0, T ), k = 0, 1, 2, ..., n. We complement (1.1)
with the initial data (1.2) also restricted to Ωk , k = 0, 1, 2, ..., n. The boundary
conditions on S × (0, +∞) are given by (1.3). Furthermore, we will require the
following interface conditions to be satisfied
(1.4)
 (k−1)
u
= u(k)




3
3
3
3

(k)
(k−1)
X
X
X
X


(k) ∂u
(k−1) ∂u


Aij
A∗i E (k−1) ηi =
A∗i E (k) ηi
ηi −
ηi −
Aij
i,j=1
∂xj



(k−1)

= η x E (k)

ηxE



(k−1)
(k)
ηxH
i=1
i,j=1
∂xj
i=1
= ηxH
for any (x, t) ∈ Sk ×(0, +∞), k = 1, 2, ..., n. Here, η = η(x) = (η1 , η2 , η3 ) is the
(k)
unit normal vector pointing the exterior of Bk and Aij , u(k) , E (k) and H (k) are
the restrictions of Aij , u, E and H to Ωk respectively.
414
BORIS V. KAPITONOV and G. PERLA MENZALA
We will assume to be valid the following conditions
HYPOTHESIS I.
h
ij
1) Aij = Aij (x) are 3 × 3 matrices given by Aij (x) = Ckh
(x)
½
i
3×3
where
ij
Ckh
(x) = (1 − δih δik ) aikjh (x) + δik δjh aihjk (x)
1 if ` = k
and aijkh are Cartesian components of the elastic
0 if ` 6= k
tensor with the symmetric properties
with δ`k =
aijkh = ajikh = akhij .
2) Ai and D = D(x) are 3 × 3 matrices given by
Ai = [ekhi ]
and
D(x) = [dij (x)]
where ekhi and dij (x) are Cartesian components of the piezoelectric and
electric permittivity tensors respectively and satisfy the following conditions:
dkh = dhk ,
3
X
k,h=1
dkh ξk ξh ≥ d0 |ξ|2
for some d0 > 0 and any vector ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 .
3) The matrices Aij (x) satisfy the condition
3
X
i,j=1
Aij (x) vj ¦ vi ≥ c0
3
X
i=1
|vi |2
for some co > 0 and any vector vi = (vi1 , vi2 , vi3 ) ∈ R3 . Here the dot ¦
denotes the inner product in R3 .
4) We assume that aijkh (x), dij (x) and β(x) > 0 are piecewise constant
functions which lose continuity only on S1 , S2 , ..., Sn .
5) ρ and ekhi are real constants, ρ > 0.
6) The functions a = a(x), b = b(x), α(x), γ(x) and σ(x) are real-valued
and continuously differentiable functions on S = ∂Ω. Furthermore, a > 0,
b > 0, α > 0, γ ≥ 0 and σ > 0 for all x ∈ S.
415
UNIFORM STABILIZATION AND EXACT CONTROL
Observe that from the symmetry of the aijkh it follows that A∗ij = Aji . Also,
for an isotropic medium, the constants aijkh are given by
aijkh = λ δij δkh + µ (δik δjh + δih δjk )
where λ and µ are the Lame’s constants. Furthermore, assumption 3) in Hypothesis I holds for an isotropic medium with the constant co = µ > 0. In fact, in
that case, direct calculation shows that
3
X
Aij vj ¦ vi = (λ + µ)
i,j=1
µX
3
vii
i=1
¶2
3
X
+µ
(vij )2
i,j=1
≥ µ
3
X
i=1
|vi |2 .
Let {u, E, H} be the global solution of problem (1.1) satisfying the initial conditions (1.2), the boundary conditions (1.3) and the interface conditions (1.4).
We consider the (total) energy E(t) given by
E(t) =
(1.5)
n Z
X
k=0
+D
+
Ωk
(k)
(
E
(k)
Z (
S
(k)
ρ |ut |2 +
b |u
¦E
(k)
3
X
(k)
Aij
i,j=1
+β
(k)
|H
∂u(k) ∂u(k)
¦
∂xj
∂xi
(k) 2
|
)
dx
)
¯Z t h
i
³
´ ¯¯2
¯
¯
¯
| +γ¯
H(x, τ ) x η exp −σ(t−τ ) dτ ¯
dS
(n) 2
0
where β (n) = β and u(n) = u.
We (formally) calculate the derivative of E(t), use the equations together with
the boundary conditions as well as the interface conditions to obtain that
(1.6)
dE(t)
= −2
dt
Z (
S
a |ut |2 + α |H x η|2
)
¯Z t h
i
³
´ ¯¯2
¯
dS .
H(x, τ ) x η exp −σ(t − τ ) dτ ¯¯
+ σ γ ¯¯
0
Thus
dE(t)
≤ 0.
dt
Assuming suitable geometric conditions on Ω (and Sk ) as well as monotonicity
assumptions on the coefficients of the system, we are able to prove that
E(t) ≤ c exp(−wt) E(0)
for any t ≥ 0 where c and w are positive constants.
416
BORIS V. KAPITONOV and G. PERLA MENZALA
As an application of the above result, we study the following exact controllability problem: Assume that γ ≡ 0. Given a time T > 0, the initial distribution F (x) = (f1 (x), f2 (x), f3 (x), f4 (x)) and a desired terminal state G(x) =
(g1 (x), g2 (x), g3 (x), g4 (x)) where F and G belong to an appropriate function
space, to find vector-valued functions p~(x, t) and ~q(x, t) such that the solution
of (1.1), (1.2) and (1.4) with boundary conditions
(1.7)
 3
X






i,j=1
Aij
3
X
∂u
ηi −
A∗i E ηi + b u = p~(x, t)
∂xj
i=1
η x E = ~q(x, t)
on S × (0, T ), satisfy
(1.8)
u(x, T ) = g1 (x), ut (x, T ) = g2 (x), E(x, T ) = g3 (x), H(x, T ) = g4 (x) .
Let us mention some bibliographical comments: Boundary controllability in transmission problems for the wave equation has been considered by J.-L. Lions [22]
and S. Nicaise in [24] and [25]. Uniform stabilization and exact control for the
Maxwell system in multilayered media were studied by B. Kapitonov in [10].
Boundary controllability in transmission problems for a class of second order
hyperbolic systems has been studied by J. Lagnese [18]. Stabilization and exact
boundary controllability for the system of elasticity were considered by J. Lagnese
[17], [18], F. Alabau and V. Komornik [1] and M. Horn [6] among others.
The exact controllability problem for the Maxwell system has been studied by
D. Russell [27] for a circular cylindrical region, by K. Kime [14] for a spherical
region and by J. Lagnese for a general region. In [9] and [19] the exact controllability problem has been studied by means of the Hilbert Uniqueness Method
introduced by J.-L. Lions [20], [21]. Uniform exponential decay of solutions of
Maxwell’s equation with boundary dissipation was proved by B. Kapitonov in
[10] and [11], including the uniform “simultaneous” stabilization for a pair of
Maxwell’s equations.
The results obtained in this article generalize previous work of the authors
[12], [13] where a transmission problem was considered either for Maxwell system
with boundary conditions with memory and for the system of electromagnetoelasticity.
Let us describe the sections of this paper: Solvability of (1.1)–(1.4) in the
appropriate class of functions is shown in Section 2. This is done via semigroup
theory and the main technical difficulty comes from the memory term (see (1.3))
on ∂Ω × (0, +∞). In Section 3 we prove the uniform exponential decay of the
UNIFORM STABILIZATION AND EXACT CONTROL
417
energy E(t) via the multiplier method. At this point, we needed to modified
“slightly” the usual multipliers in order to take care of the additional boundary terms which appear after integration (in space) of the fundamental identity.
We also needed to assume suitable geometric conditions on Ω and Sk as well as
(k)
some monotonicity assumptions on Aij , D(k) and β (k) . In the last section, the
controllability problem (1.1), (1.2), (1.4), (1.7)–(1.8) when γ ≡ 0 is solved.
Since ρ is a positive constant we may assume without lost of generality that
ρ ≡ 1. When studying system (1.1) the restrictions of u, E, H, β, Aij , D, to
Ωk (k = 0, 1, 2, ..., n−1) will be denoted by u(k) , E (k) , H (k) , etc. When k = n
we will write u(n) = u, E (n) = E, etc. At each point x belonging to one of the
boundaries S = ∂Ω, S1 , S2 , ..., Sn , the unit normal vector pointing the exterior
will be denote by η = η(x) and its components by ηi . We use the standard
notations, for example H m (Ω) and H r (∂Ω) will denote the Sobolev spaces of
order m and r on Ω and ∂Ω respectively. The norm of a vector v ∈ R3 will be
denote by |v|. Due to the techniques we use in this article (the multiplier method)
in order to achieve the result on the exponential decay we needed to assume that
b = b(x) is bounded above by a suitable constant (see (3.15) in Theorem 3.5).
This is, apparently, a limitation of the method.
We conclude this introduction with some comments on the boundary conditions (1.3). The second line in (1.3) combines the so-called Leontovich’s boundary
condition (when γ ≡ 0) and a dissipative term of memory type with an exponentially decaying kernel. A boundary condition in electromagnetism with memory
was introduced by M. Fabrizio and A. Morro in [5]. Later V. Berti [2] studied
the asymptotic stability of such models. When γ ≡ 0 and α(x) > 0 then, Leontovich’s boundary condition is also of dissipative type. In V. Komornik’s book
[16] (pg. 120) a nice geometrical meaning of such boundary condition is given in
case α ≡ 1: The tangential component of the magnetic field H is obtained from
the tangential component of the electrical field E by a rotation of angle 90 ◦ in
the positive direction in the tangent plane. The term −a(x)ut in the first line of
(1.3) is also a dissipative mechanism and the left hand side (of the first line of
(1.3)) could be interpreted as an stress tensor for the system at the boundary S.
2 – Well-posedness
In this section we will prove the well-posedness of problem (1.1)–(1.4) using
semigroup theory. The main (technical) difficulty arises from the memory term
appearing on the boundary condition (1.3).
418
BORIS V. KAPITONOV and G. PERLA MENZALA
Let us consider the Hilbert space X consisting on triples v = (v1 , v2 , v3 ) of
three-component vector-value functions vj (x) such that
v1 , v2 ∈ [L2 (Ωk )]3 ,
k = 0, 1, 2, ..., n ,
curl v1 , curl v2 ∈ [L2 (Ωk )]3 ,
and
k = 0, 1, 2, ..., n
v3 ∈ [L2 (S)]3 ,
v3 ¦ η = 0 on S .
We define the inner product in X as follows. If v, w ∈ X, then
(v, w)X =
n Z
X
k=0
+
Ωk
½
(k)
(k)
(k)
(k)
curl v1 ¦ curl w1 + curl v2 ¦ curl w2
(k)
D(k) v1
¦
(k)
w1
+
(k)
β (k) v2
¦
(k)
w2
¾
dx +
Z
S
γ v3 ¦ w3 dS .
The following lemma was proved in [12] (see also B.V. Kapitonov [8]):
Lemma 2.1. Assume that α(x) and γ(x) belong to C 1 (S). Then, the
mapping
u = (u1 , u2 , u3 ) 7→ u1 − η(u1 ¦ η) − α u2 x η − γ u3
from [Ce 1 (Ω)]3 = {u(k) ∈ [C 1 (Ωk )]3 , k = 0, 1, 2, ..., n} into [C 1 (S)]3 extends by
continuity to a continuous linear mapping from X into [H −1/2 (S)]3 which we also
denote by
u 7→ u1 − η(u1 ¦ η) − α u2 x η − γ u3 ≡ w(u; α, γ) .
Remark 2.2. Well known results (see for instance the book of G. Duvaut
and J.-L. Lions [3]) imply that for any u ∈ X, the expressions η x u1 and η x u2
where η = η(x) is the unit normal vector pointing the exterior of Sk , are well
defined on Sk and belong to [H −1/2 (Sk )]3 .
Lemma 2.1 and Remark 2.2 make it possible to introduce in X, the closed
subspace
V =
½
u = (u1 , u2 , u3 ) ∈ X such that
(k−1)
η x u1
(k)
(k−1)
= η x u 1 , η x u2
(k)
= η x u2
on Sk , k = 1, 2, ..., n
and u1 − η(u1 ¦ η) − α u2 x η − γ u3 = 0 on S
¾
.
UNIFORM STABILIZATION AND EXACT CONTROL
419
Let us denote by Z the (real) Hilbert space which consists of all elements
w = (w1 , w2 , w3 , w4 , w5 ) of three-component vector-valued functions wj (x) such
(k)
(k)
(k)
(k)
that w1 ∈ [H 1 (Ωk )]3 , w2 , w3 , w4 ∈ [L2 (Ωk )]3 , k = 0, 1, ..., n, w5 ∈ [L2 (S)]3 ,
(k)
(k−1)
w1 = w 1
on Sk , k = 1, 2, ..., n. The inner product in Z is given by:
If w, v ∈ Z, then
n Z
X
(w, v)Z =
k=0
+D
+
Z
Ωk
(k)
S
(
3
X
(k)
Aij
i,j=1
(k)
w3
¦
(k)
v3
(k)
(k)
∂v1
∂w1
(k)
(k)
¦
+ w 2 ¦ v2
∂xj
∂xi
+β
(k)
(k)
w4
¦
(k)
v4
)
dx
(b w1 ¦ v1 + γ w5 ¦ v5 ) dS .
1/2
The norm in the space Z will be denote by k · kZ = ( · , · )Z . In Z we define
the unbounded operator A with domain D(A) which consists of all elements
w = (w1 , w2 , w3 , w4 , w5 ) ∈ Z such that, for k = 1, 2, ..., n
3
X
Aij
3
X
Aij
(k)
i,j=1
(k)
3
X
∂w1
(k)
−
A∗i w3 ∈ [H 1 (Ωk )]3
∂xj
i=1
(k)
w2 ∈ [H 1 (Ωk )]3 , (w3 , w4 , w5 ) ∈ V, w4 x η ∈ [L2 (S)]3
i,j=1
3
X
∂w1
A∗i w3 ηi + a w2 + b w1 = 0
ηi −
∂xj
i=1
(k−1)
w2
(k)
= w2
on S
on S
and
3
X
(k−1)
Aij
i,j=1
(k−1)
∂w1
∂xj
=
3
X
ηi −
(k)
Aij
i,j=1
3
X
(k−1)
A∗i w3
ηi =
i=1
(k)
3
X
∂w1
(k)
ηi −
A∗i w3 ηi
∂xj
i=1
then A: D(A) ⊆ Z 7→ Z is defined as
Aw =
Ã
w2 ,
−β
3
X
µ
¶
on Sk , k = 1, 2, ..., n ,
µ
¶
3
3
X
X
∂
∂w1
∂ ∗
∂w2
Ai w3 , D−1 curl w4 −
Aij
−
,
Ai
∂xi
∂xj
∂xi
∂xi
i,j=1
i=1
i=1
−1
curl w3 , w4 x η − σ w5
!
whenever w = (w1 , w2 , w3 , w4 , w5 ) ∈ D(A).
420
BORIS V. KAPITONOV and G. PERLA MENZALA
Next, we consider the adjoint operator A∗ . We can verified in a similar manner
as in [12] that the domain of A∗ coincides with the following subspace
∗
D(A ) =
(
(k)
v = (v1 , v2 , v3 , v4 , v5 ) ∈ Z such that v2 ∈ [H 1 (Ωk )]3 , (v3 , v4 , v5 ) ∈ Ve ,
3
X
(k)
(k) ∂v1
Aij
i,j=1
∂xj
−
3
X
i=1
(k)
A∗i v3 ∈ [H 1 (Ωk )]3 ,
v4 x η ∈ [L2 (S)]3 ,
3
X
∂v1
A∗i v3 ηi − a v2 + b v1 = 0 on S ,
ηi −
Aij
∂x
j
i=1
i,j=1
3
X
(k−1)
= v2
3
X
(n−1)
(k−1) ∂v1
v2
(k)
Aij
on Sk , k = 1, 2, ..., n ,
∂xj
i,j=1
ηi −
on Sk , k = 1, 2, ..., n
)
3
X
(k−1)
A∗i v3
i=1
ηi =
3
X
(k)
(k) ∂v1
Aij
i,j=1
∂xj
ηi −
3
X
(k)
A∗i v3 ηi
i=1
where Ve is as in the definition of V with −α(x) instead of α(x). Given v =
(v1 , v2 , v3 , v4 , v5 ) ∈ D(A∗ ) then, we have that
∗
Ã
A v = − v2 ,
3
X
µ
∂
∂v1
Aij
∂xi
∂xj
i,j=1
¶
−
− β −1 curl v3 , v4 x η + σ v5
3
X
∂
i=1
!
∂xi
µ
A∗i v3 , D−1 curl v4 −
3
X
i=1
Ai
¶
∂v2
,
∂xi
.
We note that the operator A is closed since it coincides with the adjoint operator
of A∗ . Clearly A is densely defined. Furthermore, we have
Lemma 2.3. Assuming Hypothesis I given in the introduction (with ρ = 1),
then, the operators A and A∗ are dissipative, that is,
(2.1)
(Aw, w)Z ≤ 0
for any w ∈ D(A)
(A∗ v, v)Z ≤ 0
for any v ∈ D(A∗ ) .
and
(2.2)
421
UNIFORM STABILIZATION AND EXACT CONTROL
Proof: It is enough to prove (2.1) for a dense subset of D(A). In fact, the
set of piecewise smooth vector-valued functions w = (w1 , w2 , w3 , w4 , w5 ) ∈ D(A)
such that w1 ∈ [C 2 (Ωk )]3 , w2 , w3 , w4 ∈ [C 1 (Ωk )]3 , w5 ∈ [C(S)]3 , k = 0, 1, 2, ..., n
is dense in D(A). Let w be an element of such dense subset. Taking the inner
product of Aw with w in Z and using the divergence theorem we obtain that
(Aw, w)Z =
n Z
X
k=0
Ωk

3
 X

(k)
(k) ∂w2
Aij
i,j=1
"
∂xj
(k)
¦
∂w1
∂xi
µ
3
X
(k) ¶
∂
(k) ∂w1
+
Aij
∂xi
∂xj
i,j=1
"
(k)
+ curl w4 −
+
Z n
S
=
3 Z
X
i=1
(2.3)
3
X
Ai
i=1
−
(k)
∂w2
∂xi
#
3
X
∂
i=1
∂xi
(k)
(A∗i w3 )
(k)
#
(k)
o
"
3
 X
Sk 
(k−1)
(k−1) ∂w1
Aij
∂xj
i,j=1
−
"
(k)
(k) ∂w1
Aij
∂xj
i,j=1
3
X
(k−1)
(k−1)
¦ (w3

Z " X
3
ηi −
ηi −
3
X
(k−1)
A∗i w3
ηi
i=1
3
X
(k)
A∗i w3 ηi
i=1
(k)
#
#
(k−1)
(k)
(k)
x η) − w4 ¦ (w3 x η)
#



(− a w2 − b w1 ) ¦ w2 + w4 ¦ (w3 x η) + b w2 ¦ w1
+ γ(w4 x η − σ w5 ) ¦ w5
¾
dS .
Now, we use the fact that (w3 , w4 , w5 ) ∈ V . Therefore
w3 − η(w3 ¦ η) − α w4 x η − γ w5 = 0

¦ w2
dSk
+ w4 ¦ (w3 x η) + b w2 ¦ w1 + γ(w4 x η − σ w5 ) ¦ w5
Z ½


¦ w2
3
X
∂w1
ηi −
A∗i w3 ηi ¦ w2
+
Aij
∂xj
S  i,j=1
i=1
S
(k)
¦ w3 − curl w3 ¦ w4
b w2 ¦ w1 + γ(w4 x η − σ w5 ) ¦ w5 dS
+ w4
=
(k)
¦ w2
on S



dS
dx
422
BORIS V. KAPITONOV and G. PERLA MENZALA
which together with the fact that w5 ¦ η = 0 on S give us that
(− aw2 − bw1 ) ¦ w2 + w4 ¦ (w3 x η) + bw2 ¦ w1 + γ(w4 x η − σw5 ) ¦ w5 =
= −a|w2 |2 − bw1 ¦ w2 + w3 ¦ (η x w4 ) + bw2 ¦ w1 + γ(w4 x η) ¦ w5 − γσ|w5 |2
(2.4)
= −a|w2 |2 − γσ|w5 |2 + (w4 x η) ¦ (γw5 − w3 )
h
= −a|w2 |2 − γσ|w5 |2 + w4 x η ¦ −η(w3 ¦ η) − αw4 x η
= −a|w2 |2 − γσ|w5 x η|2 − α|w4 x η|2 .
i
Therefore, from (2.3) and (2.4) we obtain that
(Aw, w)Z = −
Z h
S
i
a|w2 |2 + γσ|w5 x η|2 + α|w4 x η|2 dS ≤ 0 .
The proof that (2.2) also holds for A∗ can be done in a similar way.
Therefore, A and A∗ are dissipative operators and clearly A is a densely
defined closed operator. We use a classical result (see [26], Corollary I.4.4, which
says “Let A be a densely defined closed linear operator. If both A and A∗ are
dissipative, then A is the infinitesimal generator of a C0 semigroup of contractions
on the Hilbert space Z ”)) to conclude that A is a generator of a C 0 semigroup
of contractions {U (t)}t≥0 on Z.
Lemma 2.4.
Let M1 be the orthogonal complement of the subspace
∗
M = {v ∈ D(A ) such that Av ∗ = 0} in Z. Assume Hypothesis I given in
the Introduction (with ρ = 1). Then, the following properties are valid:
1) U (t) takes M1 ∩ D(A) into itself.
2) Any element w = (w1 , w2 , w3 , w4 , w5 ) ∈ M1 ∩ D(A) has the following
property
div
(
(k)
D(k) w3
(k)
3
X
∂w1
+
Ai
∂xi
i=1
)
= 0,
(k)
div w4 = 0 ,
k = 0, 1, ..., n
in the sense of distributions.
3) Any element w = (w1 , w2 , w3 , w4 , w5 ) ∈ M1 ∩ D(A) satisfies the additional
interface conditions
(k−1)
Ã
β (k−1) w4
(k−1)
D(k−1) w3
3
X
!
(k−1)
∂w
+
Ai 1
∂xi
i=1
for any x ∈ Sk , k = 1, 2, ..., n.
(k)
¦ η = β (k) w4 ¦ η
¦η =
Ã
(k)
D(k) w3
3
X
(k)
∂w
+
Ai 1
∂xi
i=1
!
¦η
423
UNIFORM STABILIZATION AND EXACT CONTROL
Proof: First, we observe that the kernel of A∗ is nonempty. In fact, it
contains elements of the form v = (v1 , 0, ∇ϕ1 , ∇ϕ2 , 0) where ϕ1 and ϕ2 belong to
H 2 (Ω) ∩ H01 (Ω) and v1 is a solution of the following problem
(2.5)

Ã
!
(k)
3
3
X
X

∂
∂

(k) ∂v1

A∗i ∇ϕ1 in Ωk , k = 0, 1, ..., n
A
=

ij


∂x
∂x
∂x
i
j
i

i=1
i,j=1





(k−1)
(k)


v1
= v1 ,




(k)
(k−1)
3
3
X
X
(k) ∂v1
(k−1) ∂v1


η
=
ηi on Sk , k = 1, 2, ..., n ,
A
A

i
ij
ij


∂xj
∂xj

i,j=1
i,j=1






3
3

X
 X
∂v1


Aij
A∗i ηi ∇ϕ1 on S .
ηi + bv1 =


∂xj
i,j=1
i=1
The proof of 1) is simple. Indeed, if v ∈ Ker(A∗ ) and w ∈ M1 ∩ D(A), then
d
(U (t)w, v)Z = (AU (t)w, v)Z = (U (t)w, A∗ v)Z = 0
dt
which proves 1). Now, let us prove 2): We will prove that
Z
(2.6)
Ωk
"
D
(k)
(k)
w3
(k)
3
X
∂w1
+
Ai
∂xi
i=1
#
¦ ∇ϕ1 dx = 0
for an arbitrary ϕ1 ∈ H 2 (Ω) with support contained in Ωk . Clearly (2.6) implies
that
)
(
(k)
3
X
∂w1
(k) (k)
= 0
div D w3 +
Ai
∂xi
i=1
in the sense of distributions. Let us take any such ϕ1 and v1 a solution of problem
(2.5). We consider the element
ṽ = (v1 , 0, ∇ϕ1 , 0, 0)
which belongs to the kernel of A∗ . Then, for any
w = (w1 , w2 , w3 , w4 , w5 ) ∈ M1 ∩ D(A)
we have that
0 = (w, ṽ)Z =
(2.7)
n Z
X
k=0
+
Z
S
Ωk
(
3
X
(k)
Aij
i,j=1
b w1 ¦ v1 dS .
(k)
(k)
)
∂w1
∂v
(k)
¦ 1 + D(k) w3 ¦ ∇ϕ1 dx
∂xj
∂xi
424
BORIS V. KAPITONOV and G. PERLA MENZALA
However, using the divergence theorem and (2.5) we deduce that
n Z
X
k=0
3
X
Ωk i,j=1
=
(k)
(k) ∂w1
Aij
n Z
X
k=0
+
∂xj
(2.8)
=
Z
Sk
3
X
Ωk
Aji
!
Sk
(
i=1
3
X
Ωk i=1
(k)
(k) ∂v1
Aji
−
∂xi
i,j=1
3
X
ηj ¦
(k)
w1
)
dSk
A∗i ∇ϕ1 ¦ w1 dx
(k−1)
(k−1) ∂v1
Aij
∂xj
i,j=1
3
X
A∗i ηi ∇ϕ1
(k)
(k) ∂v1
Aij
ηi −
∂xj
i,j=1
3
X
)
ηi
)
(k)
¦ w1
dSk
− bv1 ¦ w1 dS
(k)
Ai
ηj ¦
(k−1)
w1
(k)
∂xi
i=1
(k)
¦ w1 dx
∂v1
ηj ¦ w1 dS
∂xi
3
X
∂
Z (X
3
S
Z
−
3 Z
X
k=1
=
(k)
(k−1)
(k−1) ∂v1
Aji
∂xi
i,j=1
S i,j=1
k=0
+
Ã
∂
(k) ∂v1
−
Aji
∂xi
Ωk i,j=1 ∂xj
( 3
n Z
X
X
n Z
X
+
∂v1
dx =
∂xi
3
X
k=1
+
(k)
¦
∂w1
¦ ∇ϕ1 dx −
∂xj
Z
S
b w1 ¦ v1 dS .
Substitution of (2.8) into (2.7) completes the proof of (2.6). It can be shown in
(k)
a similar way that div w4 = 0 by taking in this case ṽ = (0, 0, 0, ∇ϕ2 , 0) where
ϕ2 is an arbitrary element of H 2 (Ω) with support in Ωk .
Finally, let us prove 3): Since ṽ = (0, 0, 0, ∇ϕ2 , 0) belongs to the kernel of A∗
for an arbitrary ϕ2 ∈ H 2 (Ω) ∩ H01 (Ω) it follows that for w ∈ M1 ∩ D(A) we have
that
0 = (w, ṽ)Z =
n Z
X
k=0
=
Z
β
+
Ωk
(0)
S1
Z
(k)
Sn
β (k) w4 ¦ ∇ϕ2 dx
(0)
w4
¦ η ϕ2 dS1 −
(n−1)
β (n−1) w4
Z
(1)
S1
β (1) w4 ¦ η ϕ2 dS1 + · · ·
¦ η ϕ2 dSn −
Z
(n)
Sn
β (n) w4 ¦ η ϕ2 dSn .
Now, we choose ϕ2 such that ϕ2 ≡ 0 on S1 , ..., Sk−1 , Sk+1 . Then
Z
Sk
n
(k−1)
β (k−1) w4
(k)
o
¦ η − β (k) w4 ¦ η ϕ2 dSk = 0
425
UNIFORM STABILIZATION AND EXACT CONTROL
which implies that
(k−1)
β (k−1) w4
(k)
¦ η = β (k) w4 ¦ η
on Sk , k = 1, 2, ..., n .
Now, elements of the form ṽ = (v1 , 0, ∇ϕ1 , 0, 0) belong to the kernel of A∗ for
an arbitrary ϕ1 ∈ H02 (Ω) with v1 being a solution of (2.5). Thus, for any w ∈
M1 ∩ D(A) we have that
n Z
X
0 = (w, ṽ)Z =
(2.9)
k=0
+
Z
S
(
Ωk
3
X
(k)
)
(k)
∂w1
∂v
(k)
¦ 1 + D(k) w3 ¦ ∇ϕ1 dx
∂xj
∂xi
(k)
Aij
i,j=1
b w1 ¦ v1 dS .
Using the divergence theorem and (2.5) we deduce that
n Z
X
k=0
3
X
(k)
Ωk i,j=1
=
Aij
n Z
X
k=0
+
k=1
+
(2.10)
=
k=0
=
n Z
X
k=0
Ã
3
X
Sk
3
X
S i,j=1
n Z
X
(k)
(k)
∂
(k) ∂v1
−
Aji
∂xi
Ωk i,j=1 ∂xj
n Z
X
Z
(k)
∂w1
∂v
¦ 1 dx =
∂xj
∂xi
−
Ωk
Ωk
(
!
(k−1)
(k−1) ∂v1
Aji
∂xi
i,j=1
3
X
Aji
ηj ¦
(k−1)
w1
−
(k)
(k) ∂v1
Aji
∂xi
i,j=1
3
X
ηj ¦
(k)
w1
)
dSk
∂v1
ηj ¦ w1 dS
∂xi
3
X
∂
i=1
(k)
¦ w1 dx
∂xi
A∗i ∇ϕ1
¦
(k)
w1 dx
+
3
X
∂
(k)
w1 ¦ ∇ϕ1 dx −
Ai
∂xi
i=1
Z
Z
S
S
−bv1 ¦ w1 dS
b v1 ¦ w1 dS .
Substitution of (2.10) into (2.9) implies that
0 = (w, ṽ)Z =
(2.11)
=
n Z
X
k=0
n
X
k=1
−
Z
Ωk
Sk
(
(k)
D(k) w3
"
n Z
X
k=1
D
Sk
(k−1)
"
D
3
X
(k−1)
w3
(k)
(k)
∂w1
Ai
+
∂xi
i=1
(k)
w3
3
X
)
¦ ∇ϕ1 dx
(k−1)
∂w1
+
Ai
∂xi
i=1
3
X
(k)
∂w1
+
Ai
∂xi
i=1
#
#
¦ η ϕ1 dSk
¦ η ϕ1 dSk .
426
BORIS V. KAPITONOV and G. PERLA MENZALA
Now, we choose ϕ1 such that ϕ1 ≡ 0 on S1 , ..., Sk−1 , Sk+1 , ..., Sn and obtain from
(2.11) that
"
D
(k−1)
(k−1)
w3
(k−1)
3
X
∂w1
Ai
+
∂xi
i=1
#
"
¦η = D
(k)
(k)
w3
3
X
(k)
∂w1
Ai
+
∂xi
i=1
#
¦η
on Sk , k = 1, 2, ..., n, which completes the proof of Lemma 2.4.
Theorem 2.5. Let M1 be the orthogonal complement of the subspace
{w ∈ D(A∗ ) such that A∗ w = 0} in Z. Assume Hypothesis I given in the
Introduction (with ρ = 1) and let f = (f1 , f2 , f3 , f4 , 0) ∈ M1 ∩ D(A) then, there
exists a unique solution {u, E, H} of problem (1.1)–(1.4) such that
(2.12)
"
β (k−1) H (k−1) ¦ η = β (k) H (k) ¦ η
D
(k−1)
E
(k−1)
#
3
X
"
for any x ∈ Sk , k = 1, 2, ..., n and t ≥ 0. Furthermore
µ
u, ut , E, H,
Z th
0
#
3
X
∂u(k−1)
∂u(k)
+
Ai
¦ η = D(k) E (k) +
¦η
Ai
∂xi
∂xi
i=1
i=1
i
³
´
H(x, τ ) x η exp −σ(x) (t − τ ) dτ
¶
∈ M1 ∩ D(A)
for any t ≥ 0 and (1.6) is valid for any t ≥ 0 where E(t) is given by (1.5).
Proof: Let w = (w1 , w2 , w3 , w4 , w5 ) = U (t)f ∈ M1 ∩D(A) then (w1 , w3 , w4 ) =
(u, E, H). The relation
d
d
w =
U (t)f = Aw
dt
dt
give us that
(2.13)





















∂w1
= w2
∂t
µ
¶ X
3
3
X
∂w1
∂w2
∂
∂
=
A∗i w3
Aij
−
∂t
∂x
∂x
∂x
i
j
i
i,j=1
i=1
Ã
3
X
∂w3
∂w2
−1
=
D
curl
w
−
Ai
4


∂t
∂xi


i=1



∂w4



= β −1 curl w3


∂t





∂w5



= w4 x η − σw5 .
∂t
!
UNIFORM STABILIZATION AND EXACT CONTROL
427
From the last equation in (2.13) we obtain the identity
´
∂³
exp(σ(x) t) w5
∂t
(w4 x η) exp(σ(x) t) =
which implies that
(2.14)
w5 (x, t) =
Z th
0
i
³
´
w4 (x, τ ) x η exp −σ(t − τ ) dτ
because w5 (x, 0) = 0. Since w ∈ M1 ∩ D(A) then (w3 , w4 , w5 ) ∈ V (see the definition of V after Remark 2.2). Consequently
(2.15)
w3 − η(w3 ¦ η) − α w4 x η − γ w5 = 0
on S .
Substitution of (2.14) into (2.15) and writting w3 = E, w4 = H implies that
η x (E x η) − α Hx η − γ
Z th
0
i
³
´
H(x, τ ) x η exp −σ(t − τ ) dτ = 0
on S because η x (E x η) = E − η(E ¦ η), (|η| = 1). The first boundary condition
in (1.3) is also satisfy because w ∈ M1 ∩ D(A) and the interface conditions (1.4)
for the same reason.
Lemma 2.4 implies the validity of (2.12) as well as the last equation in (1.1).
Finally, let us prove (1.6) for a dense subset of M1 ∩ D(A), namely, the set
of piecewise smooth vector-valued functions w = (w1 , w2 , w3 , w4 , 0) belonging to
M1∩D(A) such that w1 ∈ [C 2 (Ωk )]3 , wj ∈ [C 1 (Ωk )]3 , j = 2, 3, 4 and k = 0, 1, 2, ..., n.
Let us take the inner product of 2 ut , 2E and 2H by the first, second and third
equation of (1.1) respectively. We obtain the identity
(2.16)


µ
3
X
∂
∂u
0 = 2 ut ¦ utt −
Aij

∂xi
∂xj
i,j=1
¶
+
3
X
∂
A∗i E



i=1

 Ã
!
3

∂
n
o
X
∂u
− curl H + 2H ¦ βHt + curl E .
+ 2E ¦
DE +
Ai

 ∂t
∂xi
∂xi
i=1
Using the identity div(U ×V ) = V ¦curl U − U ¦curl V valid for any pair of vectors
U and V in R3 , we obtain from (2.16) that
0 =
(2.17)
o
∂ n 2
|ut | + DE ¦ E + β|H|2 − 2 div(H×E)
∂t
( 3
)
3
X ∂
X
∂ut
+ 2 ut ¦
A∗i E + 2E ¦
Ai
∂xi
∂xi
i=1
i=1
−
3
X
∂
i=1
∂xi
(
3
X
∂u
Aij
2 ut ¦
∂xj
j=1
)
+2
3
X
i,j=1
Aij
∂u ∂ut
¦
.
∂xj ∂xi
428
BORIS V. KAPITONOV and G. PERLA MENZALA
Since A∗ij = Aji and
3
X
3
X
∂ut
∂
Ai
2
¦E =
{2 ut ¦ A∗i E}
∂x
∂x
i
i
i=1
i=1
2 ut ¦
3
X
∂
i=1
∂xi
A∗i E + 2
3
X
Ai
i=1
3
3
X
X
∂ut
∂
∂ut
¦E =
{2 ut ¦ A∗i E} − 2
¦ A∗i E .
∂xi
∂x
∂x
i
i
i=1
i=1
Then, we can rewritte (2.17) as follows


3


X
∂u ∂u
∂
|ut |2 +
Aij
¦
+ DE ¦ E + β|H|2
0 =

∂t 
∂xj ∂xi
i,j=1

à 3
3
X
X
∂ 
∂u
(2.18)
− 2 div(H x E) −
∂xi 
i=1
Aij
2 ut ¦
j=1
∂xj
− A∗i E
!


.
Integration of identity (2.18) over Ωk and summation in k from zero up to n give
us
n
∂ X
∂t k=0
(2.19)
Z


Ωk 
(k)
|ut |2 +
3
X
(k)
Aij
i,j=1


∂u(k) ∂u(k)
¦
+ D(k) E (k) ¦ E (k) + β|H (k) |2 dx =

∂xj
∂xi
n Z
X
=
k=1 Sk
where
³
´
(k)
F (k) = 2 H (k) x E (k) ¦ η − 2 ut
and
(2.20)
¦

3
X

³
(k)
Aij
i,j=1
´
F (k−1) − F (k) dS +
Z
F dS
S

3

X
∂u(k)
ηi −
A∗i E (k) ηi

∂xj
i=1


3
X
3

X
∂u
Aij
A∗i E ηi .
ηi −
F = 2 (H x E) + 2 ut ¦


∂xj
i,j=1
i=1
Due to the interface conditions (1.4), the integrals over Sk on the right hand side
of (2.20) are equal to zero for k = 1, 2, ..., n. Now, we use the boundary conditions
(1.3) to get the identities
2 (H x E) ¦ η = 2 E ¦ (η x H)
n
o
= 2 η(E · η) + η x (E x η) ¦ {η x H} ,
(2.21)
= 2 {η x (E x η)} ¦ {η x H}
½
= 2 {η x H} ¦ α(H x η) + γ
Z
t
0
|η| = 1 ,
³
´
(H x η) exp −σ(t−τ ) dτ
¾
=
429
UNIFORM STABILIZATION AND EXACT CONTROL
∂
= − 2 α |H x η| −
∂t
2
)
( ¯Z
³
´ ¯¯2
¯ t
γ ¯¯ (H x η) exp −σ(t−τ ) dτ ¯¯
0
¯Z t
³
´ ¯¯2
¯
¯
− 2 γ σ ¯ (H x η) exp −σ(t−τ ) dτ ¯¯
0
and
(
3
X
3
X
∂u
2 ut ¦
Aij
ηi −
A∗i E ηi
∂x
j
i,j=1
i=1
(2.22)
)
= 2 ut ¦ {−aut − bu}
= −2 a |ut |2 −
∂
(b |u|2 ) .
∂t
Using (2.21) and (2.22) together with (2.19) where F is given by (2.20), we obtain
that

n
∂ X
∂t k=0
Z (
= −
(k)
|ut |2 +
Ωk
Z (
3
X
(k)
Aij
i,j=1
)
¯Z
¯
2 α |H x η|2 + 2 a |ut |2 − 2 γ σ ¯¯
S


∂u(k) ∂u(k)
¦
+ D(k) E (k) ¦ E (k) + β (k) |H (k) |2 dx =

∂xj ∂xi
t
0
³
¯2
¯
´
(H x η) exp −σ(t−τ ) dτ ¯¯
à ¯Z
!)
³
´ ¯¯2
¯ t
∂
2
¯
¯
γ ¯ (H x η) exp −σ(t−τ ) dτ ¯ + b |u|
+
dS
∂t
0
which implies (1.6). This concludes the proof of Theorem 2.5.
Corollary 2.6. Under the assumptions of Theorem 2.5, let f= (f1 ,f2 ,f3 ,f4 ,0)
∈ Z, then U (t)f is the weak solution of the problem
dw
= Aw ,
dt
³
(m)
Proof: Let f (m) = f1
(m)
, f2
(m)
, f3
w(0) = f .
(m)
, f4
´
, 0 ∈ D(A) such that f (m) → f
in Z as m → ∞. Then, U (t)f (m) satisfies the following identity
(2.23)
Z
0
T
(µ
dψ
U (t)f (m) ,
dt
¶
µ
+ U (t)f
Z
(m)
∗
,A ψ
¶ )
Z
µ
dt = − f
(m)
, ψ(0)
¶
Z
for any ψ ∈ L2 (0, T ; D(A∗ )) such that ψt ∈ L2 (0, T ; Z) and ψ(T ) = 0. Passing
to the limit in (2.23) as n → +∞, we obtain
(2.24)
Z
T
0
(µ
dψ
U (t)f,
dt
which proves Corollary 2.6.
¶
Z
µ
∗
+ U (t)f, A ψ
¶ )
Z
µ
dt = − f, ψ(0)
¶
Z
430
BORIS V. KAPITONOV and G. PERLA MENZALA
Remark 2.7. We note that U (t) takes M1 into itself. Indeed, if g ∈ Ker(A∗ )
and take ψ(t) = (T − t)g, then from (2.17) it follows that
Z
T
(U (t)f, g)Z = T (f, g)Z
0
which implies that (U (t)f, g)Z = (f, g)Z , ∀ t ≥ 0 whenever f ∈ M1 .
3 – Stabilization
In this section we will prove the main result of this article, that is, the exponential stabilization of the solution of problem (1.1)–(1.4). The proof is based
on the theory of multipliers and it is motivated by the invariance of system (1.1)
(with constant coefficients) relative to the one-parameter group of dilations in
all variables. The multipliers have to be conveniently modified in such a way
that the extra boundary terms appearing in the identities can be estimated by
appropriate bounds. Let ϕ = ϕ(x) be an auxiliary (scalar) smooth function on
Ω which we will choose later. Let us fix t0 > 0 and consider the multiplier
(3.1)
where ∇ =
µ
L1 u = (t + t0 )ut + (∇ϕ ¦ ∇)u + u
¶
∂
∂
∂
,
,
,
∂x1 ∂x2 ∂x3
∇ϕ ¦ ∇ =
∂ϕ ∂
∂ϕ ∂
∂ϕ ∂
+
+
∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3
and u = u(x, t) = (u1 , u2 , u3 ).
We also consider the multipliers
(3.2)
and
(3.3)
L2 = L2 (E, H) = (t + t0 )E + β ∇ϕ x H
"
3
X
∂u
Ai
L3 = L3 (H, E, u) = (t + t0 )H − ∇ϕ x DE +
∂xi
i=1
We take the inner product (in R3 ) of L1 u, L2 and L3 with
3
X
µ
∂
∂u
Aij
utt −
∂xi
∂xj
i,j=1
and
Ã
3
X
∂
∂u
DE +
Ai
∂t
∂x
i
i=1
βHt + curl E ,
!
¶
+
3
X
∂
i=1
∂xi
− curl H
(A∗i E) ,
#
.
431
UNIFORM STABILIZATION AND EXACT CONTROL
½
3
P
¾
∂u
by E ¦ ∇ϕ and div H
∂xi
i=1
by βH ¦ ∇ϕ. Since {u, E, H} is a solution of (1.1) then, adding the identities we
obtain that
respectively. Finally, we multiply div DE +
3
X
∂Ii
∂F
− divx G −
−J = 0
∂t
∂xi
i=1
(3.4)
where


F = (t + t0 )
(3.5)
Ai

3
X
|ut |2 +
Aij
i,j=1
∂u ∂u
¦
+ DE ¦ E + β |H|2
∂xj ∂xi
+ 2 ut ¦ (∇ϕ ¦ ∇)u + 2 ut ¦ u + 2 β(∇ϕ x H) ¦ DE
+ 2 β(∇ x H) ¦
¯
Ã
!
∂u 
Ai
,
∂xi 
i=1
3
X
G = 2 (t + t0 ) H x E + ∇ϕ (DE ¦ E) + (∇ϕ) β |H|2 − 2 DE (E ¦ ∇ϕ)
(3.6)
Ã
3
X
∂u
Ai
− 2 βH(H ¦ ∇ϕ) + 2 E x ∇ϕ x
∂xi
i=1
h
i
"
!
,
3
X
∂u
Ii = 2 (t + t0 ) ut + (∇ϕ ¦ ∇) u + u ¦
− A∗i E
Aij
∂x
j
j=1
(3.7)
(
3
X
∂ϕ
∂u ∂u
2
Apq
¦
+
|ut | −
∂xi
∂x
q ∂xp
p,q=1
)
#
and
J = (∆ϕ − 1)
3
X
Aij
i,j=1
X
∂u ∂u
∂2ϕ
∂u ∂u
¦
−2
Aij
¦
∂xj ∂xi
∂xi ∂xp
∂xj ∂xp
i,j,p=1
(3.8)
+2
3
X
3
X
∂2ϕ
dij Ej Ek − (∆ϕ − 1) DE ¦ E
∂xi ∂xk
i,j,k=1
+ (3 − ∆ϕ) |ut |2 + 2
∂2ϕ
β Hi Hj − (∆ϕ − 1) β|H|2
∂x
∂x
i
j
i,j=1

3
X
∂2ϕ
∂u
+ 2E ¦
Ak
+

∂xi ∂xk
∂xi
i,k=1
− (∆ϕ − 1)
3
X
k=1

∂u 
Ak
.
∂xk 
Ã
3
X
!
∂u
Ak
¦ ∇ ∇ϕ
∂xk
k=1
432
BORIS V. KAPITONOV and G. PERLA MENZALA
1
|x − x0 |2 for some fixed x0 ∈ Ω, then J ≡ 0.
2
In this case (3.4) will be a conservation law. However, due to the expressions
of G and Ii we can see that we will need (after integration in Ωk of identity
∂ϕ
(3.4)) a definite sign for
· We will choose ϕ(x) as a “little” perturbation of
∂η
1
|x − x0 |2 for some x0 ∈ Ω. Let f = (f1 , f2 , f3 , f4 , 0) ∈ M1 ∩ D(A) and {u, E, H}
2
be the corresponding solution of problem (1.1)–(1.4) obtained in Theorem 2.5.
Integration over Ωk ×(0, t) of the identity (3.4) and summation over k implies
that
Observe that if we consider ϕ(x) =
(t+t0 )
n Z
X
(
k=0 Ωk
n Z
X
3
(k) ∂u(k)
X
(k) ∂u
(k)
Aij
|ut |2 +
¦
+D(k) E (k)¦E (k) +β (k) |H (k) |2
∂x
∂x
j
i
i,j=1
+2
k=0
+β
(k)
(3.9)
=
Ωk
(
(k)
ut
(∇ϕ x H
Z TZ
n
X
0
k=1
(k)
(k)
¦ (∇ϕ ¦ ∇)u(k) + ut
)¦
Ã
3
X
∂u(k)
Ai
∂xi
i=1
!)
(Vk−1 − Vk ) dSk dt +
Sk
)
¯
¯t=T
¯
dx ¯ +
¯
t=0
¦ u(k) + β (k) (∇ϕ x H (k) ) ¦ D(k) E (k)
¯t=T
¯
¯
dx ¯
=
¯
t=0
Z TZ
0
S
Vn dS dt +
n Z TZ
X
0
k=0
Jk (x, t) dx dt
Ωk
where
n
Vk = 2 (t +
(k)
t0 )ut
+ (∇ϕ ¦ ∇)u
(
(k)
+u
(k)
o
¦
(
)
3
X
(k) ∂u
Aij
i,j=1
(k)
∂xj
ηi −
3
X
A∗i E (k) ηi
i=1
+
3
(k) ∂u(k)
X
∂ϕ
(k)
(k) ∂u
|ut |2 −
Aij
¦
∂η
∂xj
∂xi
i,j=1
+
∂ϕ (k) (k) 2
∂ϕ (k) (k)
D E ¦ E (k) +
β |H | − 2 (D (k) E (k) ¦ η) (E (k) ¦ ∇ϕ)
∂η
∂η
(3.10)
− 2β
(k)
(H
(k)
¦ η) (H
(k)
(
)
+ 2 (t + t0 ) η ¦ (H (k) x E (k) )
Ã
3
X
∂u(k)
¦ ∇ϕ) + 2 ∇ϕ x
Ai
∂xi
i=1
!)
n
¦ η x E (k)
o
and Jk (x, t) is the restriction of J(x, t) (given in (3.8)) to the subset Ωk .
∂ϕ
denotes the normal derivative of ϕ at x ∈ Sk .
Here
∂η
The proof of the main result will follow as long as we can get appropriate
estimates for all terms on the right hand side of identity (3.9). The following
three Lemmas will take care of such estimates. Since their proofs are quite long
433
UNIFORM STABILIZATION AND EXACT CONTROL
and technical we prefer to give the precise statement postponing their proofs to
the end of the section. The first Lemma tell us that the differences Vk−1 − Vk
will have a “good” sign if we choose ϕ conveniently together with a monotonicity
(k)
condition on {Aij }, {D (k) } and {β (k) }.
Lemma 3.1. Let f = (f1 , f2 , f3 , f4 , 0) ∈ M1 ∩ D(A) and {u, E, H} be the
corresponding solution of problem (1.1)–(1.4) obtained in Theorem 2.5. Then,
the identity
Vk−1 − Vk

3
(k−1) ∂u(k−1)
∂ϕ  X
(k−1)
(k) ∂u
= −
(Aij
− Aij )
¦
∂η 
∂xj
∂xi
i,j=1
+
3
X
(k)
Aij
i,j=1
(3.11)
µ
∂u(k) ∂u(k−1)
−
∂xj
∂xj
¶ µ
¦
∂u(k) ∂u(k−1)
−
∂xi
∂xi
¶
+ (D(k) − D(k−1) ) E (k) ¦ E (k)
+ D(k−1) (E (k) − E (k−1) ) ¦ (E (k) − E (k−1) )
)
(

β (k)
+ (β (k) − β (k−1) ) |H (k) x η|2 + (k−1) |H (k) ¦ η|2

β
holds for k = 1, 2, ..., n.
Let us choose a convenient function ϕ(x): Let Φ(x) be the solution of the
Neumann problem


∆Φ = 1
in Ω


measure(Ω)
∂Φ


=

∂η
area(S)
on ∂Ω
which admits a solution Φ ∈ C 2 (Ω) ∩ C 1 (Ω). Let δ > 0 and x0 ∈ Ω (to be chosen
later) and define
(3.12)
ϕ(x) = δ Φ(x) +
1
|x − x0 |2 .
2
Thus, the normal derivative of ϕ is given by
∂ϕ
∂Φ(x)
= δ
+ (x − x0 ) ¦ η .
∂η
∂η
Now, we concentrate our discussion in estimating the term
n Z TZ
X
k=0
in (3.9), where Jk is given by (3.8).
0
Jk (x, t) dx dt
Ωk
434
BORIS V. KAPITONOV and G. PERLA MENZALA
Lemma 3.2. Under the assumptions of Lemma 3.1 and Hypothesis I (with
ρ ≡ 1) and choosing ϕ(x) as in (3.12), then, the following estimate
≤ δ c5
n Z TZ
X
k=0
0

3
X
Ωk 
i,j=1
n Z TZ
X
k=0
(k)
Aij
0
∂u(k)
∂xj
Jk (x, t) dx dt ≤
Ωk
¦
∂u(k)
∂xi
+ D(k) E (k) ¦ E (k) + β (k) |H (k) |2



dx dt
holds for any δ > 0 and some positive constant c5 which depends only on Φ and
the norms of the matrices Aij , Ai and D.
We will impose some geometric assumptions on Ω and Sk :
HYPOTHESIS II. There exists a positive constant δ1 ≥ 0 such that
a) δ1 c5 < 1 ,
b) δ1
∂Φ
+ (x − x0 ) ¦ η ≥ 0 for some point x0 ∈ Ω and all x ∈ Sk ,
∂η
c) δ1
measure(Ω)
+ (x − x0 ) ¦ η > 0 for all x ∈ S,
area(S)
where c5 is given as in the conclusion of Lemma 3.2 and η = η(x) denotes the
unit outward normal to Sk (or to S in c)).
Remark 3.3. We note that the above assumptions on Hypothesis II hold
with δ1 = 0 for star-shaped surfaces S1 , S2 , S3 , ..., Sn and strictly star-shaped
surface S with respect to x0 , i.e.
(x − x0 ) ¦ η > 0
for all x ∈ S .
If all surfaces S1 , S2 , ..., Sn are strictly star-shaped with respect to a point x0 ∈ Ω,
then conditions a) and b) hold with δ1 > 0 for a class of domains Ω which includes
star-shaped regions.
Lemma 3.4. Under the assumptions of Lemma 3.1, Hypothesis I and II
(with ρ = 1) then, the following estimate
Z TZ
0
S
Vn dS dt ≤
435
UNIFORM STABILIZATION AND EXACT CONTROL
≤ −(t + t0 )
−
Z
S
S
−
Z TZ ½
−
Z TZ n
−
0
t=0
0
S
S
S
S
S
¾
∂ϕ
2 (t + t0 ) a −
− c7 |ut |2 dS dt
∂η
Z TZ n
0
0
t=0
−
0
2
¯t=T
Z TZ
¯
2
¯
−
(1 − c6 b) b |u|2 dS dt
a |u| dS ¯
Z TZ ½
0
¯t=T
¯2 )
¯Z t
¯
¯
¯
¯
¯
dS ¯¯
b|u| + γ ¯ [H x η] exp(−σ(t − τ )) dτ ¯
Z (
∂ϕ
− δ0 |∇ϕ|
∂η
¾ X
3
Aij
i,j=1
∂u ∂u
¦
dS dt
∂xj ∂xi
o
2 (t + t0 ) α − (β + c8 α2 ) (3 + δ0−1 ) |∇ϕ| − c9 |H x η|2 dS dt
2 (t + t0 ) σ − 1 − γ c10 (3 + δ0−1 ) |∇ϕ| − c11
¯Z t
³
´ ¯¯2
¯
¯
· γ ¯ [H x η] exp −σ(t−τ ) dτ ¯¯ dS dt
o
0
holds, for some positive constants cj , 6 ≤ j ≤ 11 (which will be defined in the
proof of Lemma 3.4).
Finally, we will requere the following monotonicity assumptions:
(k)
HYPOTHESIS III. We assume the monotonicity conditions on {Aij },
{D(k) } and {β (k) }:
1)
3 ³
P
i,j=1
³
(k−1)
Aij
(k)
− Aij
´
´
vj ¦ vi ≥ 0 for any vi ∈ R3 and all 1 ≤ k ≤ n.
2) D(k) −D(k−1) v ¦ v ≥ 0 for any v ∈ R3 and all 1 ≤ k ≤ n and k = 1, 2, ..., n.
3) β (k) ≥ β (k−1) for all 1 ≤ k ≤ n.
Let us consider the following quantities: Let δ0 > 0 be such that
∂ϕ
≥ δ0 |∇ϕ|
∂η
(3.13)
which is possible because
(3.14)
for any x ∈ S
∂ϕ
> 0 on S and S is compact. Let
∂η
n
o
λ0 = max |x − x0 | + δ1 |∇Φ| ,
x∈Ω
where x0 ∈ Ω and δ1 are as in Hypothesis II.
436
BORIS V. KAPITONOV and G. PERLA MENZALA
With the help of the above Lemmas now we can prove the main result of this
paper.
Theorem 3.5. Let us assume Hypothesis I, II and III and
(3.15)
c0 δ 0
2 λ0
b(x) ≤
where the constants δ1 and c5 appeared in Hypothesis II, δ0 in (3.13), c0 in
Hypothesis I and λ0 in (3.14). Let f = (f1 , f2 , f3 , f4 , 0) belong to M1 ∩ D(A) and
{u, E, H} be the unique solution of problem (1.1)–(1.4) obtained in Theorem 2.5.
Then, there exist positive constants c and w such that
E(t) ≤ c exp(−wt) E(0)
for any t ≥ 0 where E(t) is given by (1.5).
Proof: We will use identity (3.9). First, we observe that we need to get a
bound for the term
I = 2
n Z
X
k=0
(3.16)


Ωk 
(k)
(k)
ut ¦ (∇ϕ ¦ ∇)u(k) + ut
Ã
3
X
∂u(k)
+ β (k) (∇ϕ x H (k) ) ¦
Ai
∂xi
i=1
¦ u(k) + β (k) (∇ϕ x H (k) ) ¦ D(k) E (k)
¯
¯t=T
¯
.
dx ¯¯

¯
!

t=0
Each term on the integrand of (3.16) can be bound in the same way as in the
proof
Z (which we will give later of Lemma 3.4). Except that will appear the term
n
P
|u(k) |2 dx. However, since u(k) ∈ [H 1 (Ωk )]3 for k = 0, 1, ..., n then, we know
k=0 Ωk
that the following inequality
c12
n
X
k=0
ku(k) k2[L2 (Ωk )]3 ≤
n Z
X
k=0
(k)
Aij
Ωk
∂u(k) ∂u(k)
¦
dx +
∂xj
∂xi
Z
S
b |u|2 dS
holds for some positive constant c12 . Here u(n) = u. Thus, the term I in (3.16)
can be estimated by
(3.17)
|I| ≤ c13 E(T ) ≤ c13 E(0)
for some positive constant c13 . Observe that all terms on the right hand side of
the conclusion of Lemma 3.3 can assume to be with a fixed sign provided we take
UNIFORM STABILIZATION AND EXACT CONTROL
t0 > 0 large enough. In fact, 1 − c6 b ≥ 0 by assumption (3.15),
¾
½
437
∂ϕ
− δ0 |∇ϕ| ≥ 0
∂η
∂ϕ
− c7 as well as the last two
∂η
coefficients on the inequality in Lemma 3.3 will be positive for all t ≥ 0 as long as
we choose t0 = T0 large enough. Now, we use Lemmas 3.1, 3.2 and 3.3 together
with (3.17) to conclude from identity (3.9) that
on S by (3.13). The coefficient
(3.18)
2 (t + t0 ) a −
(T + T0 ) E(T ) ≤ c13 E(0) + δ1 c5
Z
T
0
E(t) dt
for any T > 0. Recall that δ1 c5 < 1. Let us denote by g(T ) the right hand side of
g 0 (T )
δ 1 c5
(T + T0 )p
(3.18). Clearly
g(0) where
≤
which implies that g(T ) ≤
g(T )
T + T0
T0p
p = δ1 c5 < 1. Returning to (3.18) we obtain that
(3.19)
E(T ) ≤
c14
E(0)
(T + T0 )1−p
where c14 = c13 T0−p . Now, we can choose T > 0 large enough in (3.19) so that
c14 /(T + T0 )1−p is strictly less than one. The semigroup property then implies
the conclusion of Theorem 3.5.
Corollary 3.6. Under the assumptions of Theorem 3.5, let f = (f1 ,f2 ,f3 ,f4 , 0)
∈ M1 , then
a) The same conclusion as in Theorem 3.5 holds.
b) If γ ≡ 0, then, the semigroup {U (t)}t≥0 associated with problem
(1.1)–(1.4) takes the closed subspace M1 into itself and kU (t)kL(Z,Z) < 1
¸
¶
·µ
c13 1/1−p
−1 .
for any t > T0
T0
Proof: a) follows from a density argument and Theorem 3.5. Item b) is a
consequence of (3.19), again by a density argument.
Now, we will prove the technical Lemmas 3.1, 3.2 and 3.4.
Proof of Lemma 3.1: The idea is to use the interface conditions (1.4).
e H (k−1) = H,
In order to simplify notations let us denote by E (k−1) = E, E (k) = E,
(k−1)
(k)
e D (k−1) = D, D (k) = D,
e A
H (k) = H,
= Pij , Aij = Peij , β (k−1) = β, β (k) = β̃,
ij
(k−1)
(k)
u
= u and u = ũ. Using the interface conditions (1.4) and (3.10) we find
438
BORIS V. KAPITONOV and G. PERLA MENZALA
that
∂ϕ e 2
∂ϕ
β|H|2 −
β̃|H| − 2 β(H ¦ η) (H ¦ ∇ϕ)
∂η
∂η
e ¦ η) (H
e ¦ ∇ϕ) + ∂ϕ DE ¦ E − ∂ϕ (D
eE
e ¦ E)
e
+ 2 β̃(H
∂η
∂η
Vk−1 − Vk = L +
−2
(3.20)
+2
(Ã
(Ã
3
X
∂u
Ai
DE +
∂xi
i=1
3
X
∂ ũ
eE
e+
Ai
D
∂xi
i=1
(
3
X
∂ϕ
∂u
+2
E¦
Ai
∂η
∂xi
i=1
!
!
)
¦ η {E ¦ ∇ϕ}
e ¦ ∇ϕ}
¦ η {E
3
X
∂u
A∗i E ηi
ηi −
Pij
L = 2 (∇ϕ ¦ ∇)u ¦
∂x
j
i=1
i,j=1
n
o
3
X
+
∂ϕ
∂η
3
X
i,j=1
Peij
∂ ũ ∂ ũ
¦
∂xj ∂xi
)
(
)
)
(
3
n
o
∂u ∂u
∂ϕ X
−
Pij
¦
− 2 |∇ϕ ¦ ∇)ũ ¦
∂η i,j=1
∂xj ∂xi
(
)
3
X
∂ϕ e
∂ ũ
E¦
−2
Ai
∂η
∂xi
i=1
where
(
)
3
X
i,j=1
3
X
∂ ũ
e ηi
A∗i E
ηi −
Peij
∂xj
i=1
)
.
Using (2.12) we obtain the identities
(3.21)
(3.22)
e x η|2 +
β|H x η|2 + β|H ¦ η|2 = β|H
n
β̃ 2 e
|H ¦ η|2
β
o
e ¦ η) η(H ¦ η) + η x (H x η) ¦ ∇ϕ
β(H ¦ η)(H ¦ ∇ϕ) = β̃(H
because H = η(H ¦ η) + η x (H x η) since |η| = 1. Observe also that (3.21) it is
equal to β|H|2 because |H|2 = |H x η|2 + |H ¦ η|2 . Furthermore (3.22) can be
written as
½
e ¦ η) η
β̃(H
¾
β̃ e
e x η) ¦ ∇ϕ =
H ¦ η + η x (H
β
½
¾
e ¦ η) η β̃ H
e¦η+H
e − η(H
e ¦ η) ¦ ∇ϕ
= β̃(H
β
e ¦ η) (H
e ¦ ∇ϕ) +
= β̃(H
β̃
e ¦ η|2 .
(β̃ − β) (∇ϕ ¦ η) |H
β
439
UNIFORM STABILIZATION AND EXACT CONTROL
From the above discussion, we can write the identity
∂ϕ
∂ϕ e 2
e ¦ η) (H
e ¦ ∇ϕ) =
β̃|H| − 2 β(H ¦ η) (H ¦ ∇ϕ) + 2 β̃(H
β|H|2 −
∂η
∂η
∂ϕ
=
∂η
(3.23)
(
2
e ¦ η|2
e x η|2 + β̃ |H
β|H
β
−
o
∂ϕ n e
e ¦ η|2
β̃|H x η|2 + β̃|H
∂η
∂ϕ e
β̃
e ¦ η) (H
e ¦ ∇ϕ)
(β̃ − β)
|H ¦ η|2 + 2 β̃(H
β
∂η
e ¦ η) (H
e ¦ ∇ϕ) − 2
− 2 β̃(H
∂ϕ
= −
∂η
)
(
e ¦ η|2
e x η|2 + β̃ (β̃ − β)|H
(β̃ − β)|H
β
)
.
Using the interface conditions
(3.24)

u = ũ



3
3
3
3
X
X
X
X
∂u
∂ ũ
∗
e ηi
e

Pij
ηi −
ηi −
Ai E ηi =
A∗i E
Pij


∂x
∂x
j
j
i,j=1
i=1
i=1
i,j=1
∂
∂
(u − ũ) = ηi
(u − ũ) on Sk because u − ũ = 0 for
∂xi
∂η
x ∈ Sk , we deduce the following identities
on Sk and the fact that
2 (∇ϕ ¦ ∇) u ¦
Ã
3
X
∂u
A∗i E ηi
Pij
ηi −
∂x
j
i=1
i,j=1
3
X
!
− 2 (∇ϕ ¦ ∇) ũ ¦
= (∇ϕ ¦ ∇) (u − ũ) ¦
(3.25)
Ã
∂ϕ ∂
=
(u − ũ) ¦
∂η ∂η
3
X
i,j=1
3
X
∂ ũ
e ηi
Peij
ηi −
A∗i E
∂xj
3
X
Ã
3
X
3
X
i,j=1
Ã
!
3
X
∂ ũ
e ηi
Peij
A∗i E
ηi −
∂xj
i=1
3
X
∂u
A∗i E ηi
Pij
ηi −
∂x
j
i,j=1
i=1
∂ϕ ∂
+
(u − ũ) ¦
∂η ∂η
i=1
3
X
∂u
A∗i E ηi
ηi −
Pij
∂x
j
i=1
i,j=1
+ (∇ϕ ¦ ∇) (u − ũ) ¦
Ã
Ã
−
!
3
X
∂ ũ
e ηi
A∗i E
ηi −
Peij
∂x
j
i=1
i,j=1
3
X
!
!
=
!
=
440
BORIS V. KAPITONOV and G. PERLA MENZALA
(
3
X
∂
∂
∂u
A∗i E ¦
¦
(u − ũ) −
(u − ũ)
Pij
∂xj ∂xi
∂xi
i=1
i,j=1
∂ϕ
=
∂η
3
X
∂ϕ
+
∂η
(
3
X
i,j=1
)
3
X
∂
∂ ũ
e ¦ ∂ (u − ũ)
¦
(u − ũ) −
Peij
A∗i E
∂xj
∂xi
∂xi
i=1
)
.
Substitution of identity (3.25) into the expression of L (given after (3.20)) give
us that
∂ϕ
L = −
∂η
(3.26)
−
−
(
3
X
3
X
∂u ∂ ũ
∂ ũ ∂u
Pij
¦
−
¦
Peij
∂xj ∂xi
∂xj ∂xi
i,j=1
i,j=1
)
3
3
∂ϕ X
∂u
∂ ũ
∂ϕ X
¦E +
¦E
Ai
Ai
∂η i=1
∂xi
∂η i=1
∂xi
3
3
∂ϕ X
∂ϕ X
∂u e
∂ ũ e
¦E +
¦E .
Ai
Ai
∂η i=1
∂xi
∂η i=1
∂xi
The following identities will be useful:
3
X
Pij
i,j=1
3
X
∂ ũ ∂u
∂u ∂ ũ
Peij
¦
−
¦
=
∂xj ∂xi
∂xj ∂xi
i,j=1
3
X
=
i,j=1
(3.27)
3
X
=
i,j=1
=
3
X
i,j=1
+
¶
µ
3
X
∂u
∂u
∂u ∂u
∂ ũ
e
e
¦
(Pij − Pij )
(Pij − Pij )
¦
−
−
∂xj
∂xi
∂xj
i,j=1
3
X
∂u ∂u
∂u
(Pij − Peij )
(Pij − Peij )
¦
−
¦
∂xj
(Pij − Peij )
3
X
i,j=1
Peij
∂xi
∂xj
i,j=1
3
X
∂u ∂u
∂u
Pij
¦
−
¦
∂xj ∂xi
∂x
j
i,j=1
∂u
¦
∂xj
µ
∂u
∂ ũ
−
∂xi ∂xi
¶
µ
∂xj
µ
∂xi
∂ ũ
∂u
−
∂xi ∂xi
∂ ũ
∂u
−
∂xi ∂xi
¶
¶
.
Substitution of (3.27) into (3.26) give us that

(3.28)
µ
¶
3
3
X
∂u
∂ ũ
∂ϕ  X
∂u
∂u ∂u
¦
−
¦
−
Pij
L = −
(Pij − Peij )
∂η i,j=1
∂xj ∂xi
∂xj
∂xi ∂xi
i,j=1
+
3
X
i,j=1
+
3
X
i=1
∂u
Peij
¦
Ai
∂xj
µ
∂u
∂ ũ
−
∂xi ∂xi
∂u e
¦E −
∂xi
3
X
i=1
Ai
¶
+
3
X
Ai
i=1


∂ ũ e
¦E .

∂xi
3
X
∂ ũ
∂u
Ai
¦E −
¦E
∂xi
∂xi
i=1
441
UNIFORM STABILIZATION AND EXACT CONTROL
Again, we use the interface conditions (3.24) on Sk to obtain
3
X
i,j=1
Pij
∂u
¦
∂xj
µ
¶
∂ ũ
∂u
−
∂xi ∂xi
3 µ
X
∂ ũ
=
(3.29)
(
3
X
A∗i E ¦
i=1
∂u
−
∂η
∂η
i,j=1
=
−
3
X
i,j=1
¶
µ
∂ ũ
∂u
−
∂xi ∂xi
¶
=
µ
3
X
∂ ũ ∂u
∂u
ηi −
−
¦ Pij
∂xj
∂η
∂η
i=1
3
X
∂ ũ
e ηi
ηi −
A∗i E
Peij
∂xj
i=1
) ½
¦
¾
∂ ũ ∂u
−
∂η
∂η
¶
¦ A∗i E ηi
.
Substitution of (3.29) into (3.30) give us that

3
∂u ∂u
∂ϕ  X
(Pij − Peij )
L = −
¦
∂η i,j=1
∂xj ∂xi
+
"
+
3
X
# µ
3
X
∂ ũ
e ηi ¦
e
A∗i E
ηi −
Pij
∂x
j
i=1
i,j=1
3
X
i,j=1
Peij
∂u
¦
∂xj
µ
∂u
∂ ũ
−
∂xi ∂xi
¶
+
3 µ
X
∂u
i=1

(3.30)
∂ ũ ∂u
−
∂η
∂η
∂xi
−
¶
∂ ũ
∂xi
¶
3
3
X
∂ϕ  X
∂u ∂u
∂ ũ
= −
(Pij − Peij )
¦
+
¦
Peij
∂η i,j=1
∂xj ∂xi
∂xj
i,j=1
+
3
X
i=1
+
e
A∗i E
3 µ
X
∂u
∂xi
i=1

¦
µ
−
∂ ũ
∂u
−
∂xi ∂xi
∂ ũ
∂xi
¶
¶
e
¦ A∗i E
+


3
X
i,j=1
+
i,j=1
−2
Peij
3
X
i=1
µ
Ai
∂ ũ
∂u
−
∂xj
∂xj
¶ µ
∂ ũ e
¦E + 2
∂xi
∂xj

3
∂ϕ  X
∂u ∂u
¦
= −
(Pij − Peij )
∂η i,j=1
∂xj ∂xi
3
X
∂u
Peij
¦
¦
3
X
i=1
∂ ũ
∂u
−
∂xi ∂xi
Ai
¶


∂u e
¦E .

∂xi
µ
e
¦ A∗i E
µ



∂ ũ
∂u
−
∂xi ∂xi
∂ ũ
∂u
−
∂xi ∂xi
¶
¶
442
BORIS V. KAPITONOV and G. PERLA MENZALA
Now, we return to (3.20) and use (3.23) with (3.30) to obtain that
Vk−1 − Vk

3
∂u ∂u
∂ϕ  X
(Pij − Peij )
= −
¦
∂η i,j=1
∂xj ∂xi
µ
¶ µ
¶
3
X
∂ ũ
∂u
∂u
∂ ũ
e
Pij
+
−
−
¦
i,j=1
3
X
−2
i=1
∂xj
Ai
∂xj
∂xi
∂xi
3
X
∂u e
∂ ũ e
¦E + 2
¦E
Ai
∂xi
∂xi
i=1
β̃
e ¦ η|2
(β̃ − β) |H
β

" 3
#
" 3
#

X
X
∂u
∂ ũ
e¦
eE
e¦E
e
Ai
Ai
− 2E ¦
+ 2E
− DE ¦ E + D

∂xi
∂xi
i=1
i=1
e x η|2 +
+ (β̃ − β) |H
(3.31)
+2
−2
(Ã
(Ã
eE
e+
D
3
X
∂ ũ
Ai
∂xi
i=1
3
X
∂u
Ai
DE +
∂xi
i=1
!
!
¦η
¦η
)
)
e ¦ ∇ϕ}
{E
{E ¦ ∇ϕ} .
Let us write in a more convenient form some of the terms in (3.31):
K ≡
∂ϕ e e e
∂ϕ
DE ¦ E −
DE ¦ E
∂η
" 3
" 3 ∂η
#
#
X
∂u
∂u
∂ϕ
∂ϕ e X
E¦
Ai
E¦
Ai
+2
−2
∂η
∂xi
∂η
∂xi
i=1
i=1
+2
(3.32)
−2
= 2
(Ã
(Ã

 ∂ϕ
 ∂η
3
X
∂ ũ
eE
e+
Ai
D
∂x
i
i=1
3
X
∂u
DE +
Ai
∂x
i
i=1
DE ¦ E −
"
!
!
¦η
¦η
)
)
e ¦ ∇ϕ}
{E
{E ¦ ∇ϕ}
∂ϕ e e e
eE
e ¦ η) (E
e ¦ ∇ϕ) − (DE ¦ η) (E ¦ ∇ϕ)
D E ¦ E + (D
∂η
#
"
3
3
X
∂ϕ
∂u
∂ϕ e X
∂u
+
E¦
Ai
−
E¦
Ai
∂η
∂xi
∂η
∂xi
i=1
i=1
"
#
"
#
#

3

X
∂ ũ
∂ ũ
e ¦ ∇ϕ} −
¦ η {E
Ai
¦ η {E ¦ ∇ϕ}
+
Ai

∂xi
∂xi
i=1
i=1
−
3
X
∂ϕ e e e
∂ϕ
DE ¦ E +
DE ¦ E .
∂η
∂η
443
UNIFORM STABILIZATION AND EXACT CONTROL
Next we use the identity
(a x b) ¦ (c x d) = (a ¦ c) (b ¦ d) − (b ¦ c) (a ¦ d)
valid for any vectors a, b, c, d ∈ R3 to obtain the following identities
n
2 (∇ϕ x DE) ¦ (η x E) = 2 (∇ϕ ¦ η) (DE ¦ E) − (DE ¦ η) (E ¦ ∇ϕ)
= 2
Ã
e E)
e ¦ (η x E)
e = 2
2 (∇ϕ x D
!
3
X
∂ϕ
DE ¦ E − 2 (DE ¦ η) (E ¦ ∇ϕ) ,
∂η
o
∂ϕ e e e
eE
e ¦ η) (E ¦ ∇ϕ) ,
D E ¦ E − 2 (D
∂η
Ã
!
Ã
!
Ã
!
Ã
!
3
3
X
X
∂u
∂ϕ
∂u
∂u
2 ∇ϕ x Ai
¦ (η x E) = 2
Ai
−2
Ai
¦ η(E ¦ ∇ϕ)
E¦
∂xi
∂η
∂xi
∂xi
i=1
i=1
i=1
and
Ã
!
3
X
3
3
X
X
∂u
∂ ũ
∂ ũ
e¦
e = 2 ∂ϕ E
e ¦ ∇ϕ) .
Ai
2 ∇ϕ x Ai
¦ (η x E)
−2
Ai
¦ η(E
∂x
∂η
∂x
∂x
i
i
i
i=1
i=1
i=1
Substitution of the above identities in (3.32) give us that
e E)
e ¦ (η x E)
e
K = 2 (∇ϕ x DE) ¦ (η x E) − 2 (∇ϕ x D
Ã
!
Ã
!
3
X
∂u
∂ ũ
e
+ 2 ∇ϕ x Ai
¦ (η x E) − 2 ∇ϕ x Ai
¦ (η x E)
∂x
∂x
i
i
i=1
i=1
−
∂ϕ e e e
∂ϕ
DE ¦ E +
DE ¦ E
∂η
∂η
+2
(3.33)
3
X
3
3
∂ ũ
∂ϕ e X
∂u
∂ϕ e X
Ai
−2
Ai
E¦
E¦
∂η
∂xi
∂η
∂xi
i=1
i=1
Ã
(
3
X
∂u
= 2 (η x E) ¦ ∇ϕ x DE +
Ai
∂xi
i=1
)!
)!
Ã
(
3
X
∂ ũ
e
e
e
− 2 (η x E) ¦ ∇ϕ x DE +
Ai
i=1
−
∂ϕ
∂ϕ e e e
DE ¦ E
DE ¦ E +
∂η
∂η
+2
∂xi
3
3
∂ϕ e X
∂ϕ e X
∂ ũ e
∂u
E¦
−2
E¦
Ai E
Ai
.
∂η
∂η
∂η
∂x
i
i=1
i=1
444
BORIS V. KAPITONOV and G. PERLA MENZALA
e together with (2.12) we can simIf we use the interface conditons η x E = η x E
plify some terms of K:
Ã
(
3
X
∂u
2 (η x E) ¦ ∇ϕ x DE + Ai
∂xi
i=1
)!
Ã
(
)!
3
X
∂ ũ
e
e
e
− 2 (η x E) ¦ ∇ϕ x DE + Ai
=
∂xi
i=1
)!
Ã
(
3
X
∂u
e
= 2 (η x E) ¦ ∇ϕ x DE + Ai
(3.34)
Ã
(
eE
e+
e ¦ ∇ϕ x D
− 2 (η x E)
(
∂xi
i=1
3
X
∂ϕ e
∂u
E ¦ DE + Ai
= 2
∂η
∂xi
i=1
(
3
X
∂ ũ
Ai
∂xi
i=1
)
(
)
3
X
∂u
e
− 2 (E ¦ ∇ϕ) DE + Ai
¦η
3
X
∂ ũ
∂ϕ e
eE
e+
Ai
−2
E¦ D
∂η
∂xi
i=1
(
3
X
∂u
∂ϕ e
E ¦ DE + Ai
= 2
∂η
∂xi
i=1
)
)!
)
(
∂xi
i=1
eE
e+
e ¦ ∇ϕ) D
+ 2 (E
(
3
X
∂ ũ
Ai
∂xi
i=1
3
X
∂ ũ
∂ϕ e
eE
e+
Ai
E¦ D
−2
∂η
∂xi
i=1
Substitution of identity (3.34) into (3.33) give us that
(3.35)
½
∂ϕ
e ¦ (E − E)
e + (D
e − D)E
e¦E
e
D(E − E)
K = −
∂η
¾
.
Using (3.35) in (3.31) we finally deduce that

3
∂ϕ  X
∂u ∂u
¦
Vk−1 − Vk = −
(Pij − Peij )

∂η i,j=1
∂xj ∂xi
+
3
X
i,j=1
Peij
µ
∂ ũ
∂u
−
∂xj
∂xj
"
¶ µ
¦
∂ ũ
∂u
−
∂xi ∂xi
e ¦ η|2
e x η|2 + β̃ |H
+ (β̃ − β) |H
β
#
¶
e ¦ (E − E)
e + (D
e − D)E
e¦E
e
+ D(E − E)
which completes the proof of Lemma 3.1.



)
.
)
¦η
UNIFORM STABILIZATION AND EXACT CONTROL
445
Proof of Lemma 3.2: Straightforward calculations using (3.8) and ϕ(x)
chosen as in (3.12) lead us to the identity
Jk = δ
3
X
(k)
Aij
i,j=1
3
(k) ∂u(k)
X
∂u(k) ∂u(k)
∂2Φ
(k) ∂u
¦
− 2δ
Aij
¦
∂xj
∂xi
∂xp ∂xi
∂xj
∂xp
i,j,p=1
3
X
∂2Φ
∂2Φ
(k) (k)
(k) (k)
dij Ej Ep(k) + 2 δ
β (k) Hi Hj
∂xi ∂xp
∂xi ∂xj
i,j,p=1
i,j=1
(k)
(3.36)
3
X
− δ|ut |2 + 2 δ
− δ D (k) E (k) ¦ E (k) − δ β (k) |H (k) |2

3
X
Ã
!

3
3
X
X
∂u(k)
∂u(k)
∂u(k)
∂2Φ
Aj
Ap
+
¦ ∇ ∇Φ − Aj
.
+ 2 δ E (k) ¦

∂xi ∂xp
∂xi
∂xj
∂xj 
j=1
j=1
i,p=1
Let us estimate the terms on the right hand side of (3.36): We claim that for any
ε > 0 we have that
− 2δ
3
X
(k) ∂u(k)
∂2Φ
(k) ∂u
Aij
¦
≤
∂xp ∂xi
∂xj
∂xp
i,j,p=1
≤ δε
(3.37)
3
X
(k) ∂u
Aij
∂xj
i,j=1
+ δε
−1
(k)
3
X
(k)
Aij
i,j=1
In fact, let vi =
−2
3
X
(k) ∂u
Aij
Ã
3
X
∂u(k)
∂xi
3
X
∂ 2 Φ ∂u(k)
∂xp ∂xj ∂xp
p=1
! Ã
¦
3
X
∂ 2 Φ ∂u(k)
∂xp ∂xi ∂xp
p=1
!
∂ 2 Φ ∂u(k)
and ε > 0 then, we can write
∂xp ∂xi ∂xp
p=1
(k)
∂xj
i,j=1
¦
¦ vi = −
3
X
(k)
Aij
i,j=1
3
X
+ε
Ã
√ ∂u(k)
1
ε
+ √ vj
∂xj
ε
(k) ∂u
Aij
∂xj
i,j=1
≤ ε
3
X
i,j=1
(k)
(k) ∂u
Aij
(k)
∂xj
! Ã
√ ∂u(k)
1
ε
+ √ vi
¦
∂xi
ε
3
∂u(k)
1 X
(k)
¦
+
A vj ¦ v i
∂xi
ε i,j=1 ij
3
X
∂u(k)
(k)
−1
Aij vj ¦ vi
¦
+ε
∂xi
i,j=1
(k)
because Aij satisfies assumption 3) of Hypothesis I. This proves (3.37). Let
c2 = max kAij (x)k ,
x∈Ω
i,j=1,2,3
¯
¯
¯ ∂2Φ ¯
¯
¯
c3 = max ¯
¯
¯
∂xi ∂xj ¯
x∈Ω
i,j=1,2,3
!
.
446
BORIS V. KAPITONOV and G. PERLA MENZALA
where k k denotes the norm of the matrix. With this notations, we have that

¯
¯
3 ¯
X
(k) ¯
∂u
¯
¯
|vi | ≤ c3
¯
¯
¯ ∂xj ¯

j=1
and
(3.38)
¯
¯
!2
à 3
¯ 3
¯
3
X
X
¯ X (k)
¯
(k)
¯
kAij (x)k |vj | |vi | ≤ c2
|vj |
Aij vj ¦ vi ¯¯ ≤
¯
¯
¯i,j=1
i,j=1
j=1
¯
¯)2
¯
( 3 ¯
3 ¯
(k) ¯2
X ¯¯ ∂u(k) ¯¯
X
¯ ∂u ¯
2
2
≤ 9 c 2 c3
¯
¯
¯ ≤ 27 c2 c3
¯ .
¯ ∂xi ¯
¯ ∂xi ¯
i=1
i=1
From (3.38) we deduce that
(3.39)
¯
¯
¯X
¯
3
(k) ∂u(k)
X
¯ 3
¯
(k)
(k) ∂u
¯
¯ ≤ 27 c2 c2 c−1
¦
¦
v
A
v
A
j
i
3 0
ij
ij
¯
¯
∂xj
∂xi
¯i,j=1
¯
i,j=1
where c0 is the positive constant in Hypothesis I (item 3)). Using (3.39) into
(3.37) we get that
− 2δ
(3.40)
3
X
(k) ∂u(k)
∂2Φ
(k) ∂u
Aij
¦
≤
∂xp ∂xi
∂xj
∂xp
i,j,p=1
n
−1
≤ δ ε + 27 c2 c23 c−1
0 ε
3
o X
(k)
Aij
i,j=1
∂u(k) ∂u(k)
¦
.
∂xj
∂xi
Let us bound the last term on the right hand side of (3.36). Let
c4 = max kAj k
j=1,2,3
then
(3.41)
¯
¯
¯
3 ¯
3 ¯¯
2
X
X
∂u(k) ¯¯
∂u ¯¯
¯ ∂ Φ
¯
A
¯ ≤ 3 c 4 c3
¯
¯ ∂x ¯
¯ ∂xi ∂xp p ∂xi ¯
i
i,p=1
and
(3.42)
i=1
¯
¯
¯
3 ¯
3 ¯
X
X
¯ ∂u ¯
∂u(k) ¯¯
¯
¯
¯
¯ ≤ c4
¯A j
¯ ∂x ¯ .
¯
∂xj ¯
i
j=1
i=1
447
UNIFORM STABILIZATION AND EXACT CONTROL
Also
¯
¯
(µ
¶
¶2
3
3 ¯µ
¯
X
X
∂2Φ
∂u(k)
∂2Φ
∂2Φ
¯
¯
a1j
¦ ∇ ∇Φ¯ =
+ a3j
+ a2j
¯ Aj
¯
¯
∂xj
∂x1 ∂x2
∂x1 ∂x3
∂x21
j=1
j=1
¶2
µ
2
2
2
+
+
where (a1j , a2j , a3j ) = Aj
Thus
a1j
µ
a1j
∂ Φ
∂ Φ
∂ Φ
+ a2j
+ a3j
2
∂x1 ∂x2
∂x2 ∂x3
∂x2
∂2Φ
∂2Φ
∂2Φ
+ a2j
+ a3j
∂x1 ∂x3
∂x2 ∂x3
∂x23
¶2 )1/2
∂u(k)
·
∂xj
¯
¯
¯
¯
¶
3 ¯
3 ¯µ
¯
(k) ¯
X
X
∂u(k)
¯
¯ ∂u ¯
¯
¦ ∇ ∇Φ¯ ≤ 3 c3 c4
¯ .
¯
¯ Aj
¯
¯ ∂xj ¯
¯
∂xj
(3.43)
j=1
j=1
From (3.41)–(3.43) we obtain the estimate
2 δ E (k) ¦
(3.44)

3
X

∂2Φ
∂xi ∂xp
i,p=1
Ap
∂u(k)
∂xi
+
Ã
3
X
Aj
j=1
≤ 2 δ |E (k) | {6 c3 c4 + c4 }
≤ δ {6 c3 c4 + c4 } ε1 |E
∂u(k)
∂xj
!
¦ ∇ ∇Φ −
¯
¯
(k) ¯
¯ ∂u ¯
¯
¯
¯ ∂xj ¯
3 ¯
X
j=1
(k) 2
| +
δ ε−1
1 {6 c3 c4
3
X
Aj
j=1

∂u(k) 
∂xj 
≤
¯!
¯
3 ¯
(k) ¯ 2
X
¯ ∂u ¯
+ c4 }
¯
¯
¯ ∂xj ¯
Ã
j=1
for any ε1 > 0. Since Aij satisfies assumption 3) in Hypothesis I, we get the bound
Ã
¯!
¯
¯
¯
3 ¯
3 ¯
3
(k) ∂u(k)
(k) ¯2
(k) ¯ 2
X
X
X
(k) ∂u
¯ ∂u ¯
¯ ∂u ¯
≤ 3
Aij
¦
.
¯
¯ ≤ 3 c−1
¯
¯
0
¯ ∂xj ¯
¯ ∂xj ¯
∂xj
∂xi
j=1
j=1
j=1
Thus, the left hand side of (3.44) can be bound by
(3.45)
δ (6 c3 c4 + c4 ) ε1 |E
(k) 2
| +
−1
3 δ ε−1
1 c0 (6 c3
+ 1) c4
3
X
i,j=1
(k)
Aij
∂u(k) ∂u(k)
¦
∂xj
∂xi
for any ε1 > 0. Finally,
(3.46)
2δ
3
X
3
X
∂2Φ
∂2Φ
(k) (k)
(k) (k)
dij Ej Ep(k) + 2 δ
β (k) Hi Hj ≤
∂x
∂x
∂x
∂x
i
p
i
j
i,j,p=1
i,j=1
(k)
≤ 6 δ c4 kD(k) k d−1
¦ E (k) ) + 6 δ c4 β (k) |H (k) |2
0 (DE
448
BORIS V. KAPITONOV and G. PERLA MENZALA
where kD (k) k denotes the norm of the matrix D (k) and d0 > 0 is as in Hypothesis
I (item 2)).
Using (3.40), (3.44), (3.45) and (3.46) we deduce the following estimate for
Jk given by (3.29):
n
−1
−1
Jk ≤ δ 1 + ε + 27 c2 c23 c−1
+ 3 (6 c3 + 1) c4 ε−1
0 ε
1 c0
(3.47)
n
+ δ 6 c4 kD
(k)
k d−1
0
+ (6 c3 +
1) c4 ε1 d−1
0
(k)
o
3
o X
i,j=1
− 1 (D
+ δ β (k) (6 c2 − 1) |H (k) |2 − δ |ut |2
(k)
Aij
(k)
∂u(k) ∂u(k)
¦
∂xj
∂xi
E (k) ¦ E (k) )
for any ε > 0, ε1 > 0, where we use Hypothesis I, (item 2)). Let us choose
1/2 and ε = (3 d c−1 )1/2 in (3.47) to obtain the desired estimate
ε = 3 c3 (c2 c−1
1
0 0
0 )
of Lemma 3.2 with
½
1/2
c5 = max 1 + 12 c3 (c2 c−1
+
0 )
6 c4 max kD
k
(k)
k+
√
√
3 c4 (6 c3 + 1) (c0 d0 )−1/2 ,
3 c4 (6 c3 + 1) (c0 d0 )
−1/2
, 6 c4
¾
.
Proof of Lemma 3.4: From now on we will choose Z
δ =Z δ1 in the definition
T
of ϕ(x) in (3.12). Now, let us get a bound for the term
Vn dS dt in (3.9).
Using the boundary conditions (1.3) we can rewrite Vn as
Vn
0
S


"
¯2 #
¯Z T h

i
³
´
¯
¯
∂
H(x, τ ) x η exp −σ(x) (t − τ ) dτ ¯¯
= −
(t + t0 ) b|u|2 + γ ¯¯

∂t 
0
½
¾
n
o
−
∂
∂ϕ
a|u|2 − b|u|2 − 2 (t + t0 ) a −
|ut |2
∂t
∂η


3

X
∂ϕ
∂u ∂u 
−
Aij
¦
− 2 (∇ϕ ¦ ∇) u ¦ (aut + bu)
∂η i,j=1
∂xj ∂xi 
(3.48)
n
o ¯¯Z t
³
´ ¯¯2
2
¯
− 2 (t+t0 ) α |H x η| − 2 (t+t0 ) σ − 1 γ ¯ [H x η] exp −σ(1−τ ) dτ ¯¯
0
∂ϕ
∂ϕ
DE ¦ E +
β|H|2 − 2 (DE ¦ η) (E ¦ ∇ϕ)
+
∂η
∂η
Ã
3
X
∂u
Ai
− 2 β(H ¦ η) (H ¦ ∇ϕ) + 2 (η x E) ¦ ∇ϕ x
∂xi
i=1
!
.
449
UNIFORM STABILIZATION AND EXACT CONTROL
Let cj , j = 7, 8, ..., 11, be the following constants
−1
c7 = 8 λ0 a21 c−1
0 δ0 ,
−1
c6 = 2 λ0 c−1
0 δ0 ,
−1
c9 = 16 α12 c24 c−1
0 δ0 ,
c8 = c10 = 2 d1 ,
−1
c11 = 16 λ0 γ1 c24 c−1
0 δ0
where a1 = max a(x), d1 = max kD(x)k, α1 = max α(x) and γ1 = max γ(x).
x∈S
x∈Ω
x∈S
x∈S
The constant c4 was defined in the proof of Lemma 3.2 (see below (3.40)) and
c0 appeared in Hypothesis I (item 3)).
We will get a bound for some of the terms on the right hand side of (3.41).
We use the identity
H = η x (H x η) + η H ¦ η
to rewrite the expression
−2 β(H ¦ η) (H ¦ ∇ϕ) = −2 β(∇ϕ ¦ η) |H ¦ η|2 − 2 β(H ¦ η) (H x η) ¦ (∇ϕ x η) .
Now we can obtain a bound for the term
∂ϕ
β|H|2 − 2 β(H ¦ η) (H ¦ ∇ϕ) =
∂η
∂ϕ
∂ϕ
β|H x η|2 −
β|H ¦ η|2 − 2 β(H ¦ η) (H x η) ¦ (∇ϕ x η)
=
∂η
∂η
(3.49)
∂ϕ
∂ϕ
≤
β|H x η|2 −
β|H ¦ η|2 + β δ0 |∇ϕ| |H ¦ η|2 + β δ0−1 |∇ϕ| |H x η|2
∂η
∂η
≤
³
´
∇ϕ ¦ η + δ0−1 |∇ϕ| β|H x η|2 .
Next, we use the identity
E = η x (E x η) + η E ¦ η
in order to obtain that
(3.50)
³
´
∂ϕ
∂ϕ
∂ϕ
(DE ¦ E) =
(Dη ¦ η) |E ¦ η|2 + 2
(E ¦ η) Dη ¦ {η x (E x η)}
∂η
∂η
∂η
∂ϕ
D{η x (E x η)} ¦ {η x (E x η)}
+
∂η
and
−2 (DE ¦ η)(E ¦ ∇ϕ) = − 2 (Dη ¦ η) (E ¦ η) (E x η) ¦ (∇ϕ x η)
³
´
∂ϕ
∂ϕ
(Dη ¦ η) |E ¦ η|2 − 2
(E ¦ η) Dη ¦ {η x (E x η)}
−2
∂η
∂η
(3.51)
− 2 D η ¦ {η x (E x η)} (E x η) ¦ (∇ϕ x η) .
450
BORIS V. KAPITONOV and G. PERLA MENZALA
Using (3.43), (3.50) and (3.51) we get a representation and therefore an inequality
as follows
∂ϕ
(DE ¦ E) − 2 (DE ¦ η) (E ¦ ∇ϕ) =
∂η
∂ϕ
(Dη ¦ η) |E ¦ η|2 − 2 (Dη ¦ η) (E ¦ η) (E x η) ¦ (∇ϕ x η)
= −
∂η
− 2 (Dη x η) ¦ (E x η) (E x η) ¦ (∇ϕ x η)
´ ³
´
∂ϕ ³
+
D η x (E x η) ¦ η x (E x η)
∂η
(3.52)
−1
≤ δ0 |∇ϕ| (Dη ¦ η) |E x η|2 − 2 (Dη x η) ¦ (E x η) (E x η) ¦ (∇ϕ x η)
´ ³
´
∂ϕ ³
+
D η x (E x η) ¦ η x (E x η)
∂η
≤
n
o
(∇ϕ ¦ η) + δ0−1 |∇ϕ| + 2 |∇ϕ| d1 |E x η|2 .
From the boundary conditions (1.3) it follows that
¯Z
¯
¯ Th
i
³
´ ¯2
¯
¯
|E x η|2 ≤ 2 α2 |H x η|2 + 2 γ 2 ¯
H(x, τ ) x η exp −σ(x) (t − τ ) dτ ¯
¯ 0
¯
which together with (3.52) and (3.49) give us the estimate
∂ϕ
∂ϕ
(DE ¦ E) +
β|H|2 − 2 (DE ¦ η) (E ¦ ∇ϕ) − 2 β(H ¦ η) (H ¦ ∇ϕ) ≤
∂η
∂η
≤
h
i
∇ϕ ¦ η + δ0−1 |∇ϕ| + 2 |∇ϕ| [2 d1 α2 + β] |H x η|2
h
+ ∇ϕ ¦ η + δ0−1 |∇ϕ| + 2 |∇ϕ|
(3.53)
i
¯Z t h
i
³
´ ¯¯2
¯
· 2 d1 γ ¯
H(x, τ ) x η exp −σ(x) (t − τ ) dτ ¯¯
≤ (3 +
2¯
0
δ0−1 ) |∇ϕ| (2 d1 α2
2
+ 2 d1 γ (3 +
+ β) |H x η|2
¯Z t h
¯
δ0−1 ) |∇ϕ| ¯¯
0
¯2
¯
H(x, τ ) x η exp −σ(t − τ ) dτ ¯¯ .
i
³
´
Let ε1 , ε2 , ε3 positive real numbers. With the notations given above we have
the following estimates
(3.54)
2
2 −1
2
−2 (∇ϕ ¦ ∇)u ¦ (aut + bu) ≤ λ0 a1 ε−1
2 |ut | + λ0 b ε3 |u|
−1
+ (ε2 a1 c−1
0 + ε3 c0 ) |∇ϕ|
3
X
i,j=1
Aij
∂u ∂u
¦
∂xj ∂xi
451
UNIFORM STABILIZATION AND EXACT CONTROL
where we used Cauchy–Schwarz inequality and Hypothesis I (item 3)). By the
same reasons and the boundary conditions (1.3) we deduce that
Ã
3
X
∂u
Ai
2 (η x E) ¦ ∇ϕ x
∂x
i
i=1
!
≤
−1
2
≤ |∇ϕ| c4 ε−1
1 |η x E| + 3 |∇ϕ| c4 ε1 c0
(3.55)
≤ 2 λ0 c4 ε−1
1
(
3
X
Aij
i,j=1
∂u ∂u
¦
∂xj ∂xi
)
¯Z t h
i
³
´ ¯¯2
¯
2
2
¯
¯
H(x, τ ) x η exp −σ(t−τ ) dτ ¯
α1 |H x η| + γ1 γ ¯
0
+ 3 |∇ϕ| c4 ε1 c−1
0
3
X
Aij
i,j=1
∂u ∂u
¦
.
∂xj ∂xi
1
1
1
c0 δ0 c−1
c0 δ0 a−1
c0 δ0 in (3.54) and
4 , ε2 =
1 and ε3 =
8
8
2
(3.55). Thus, the summation of the left hand sides is less than or equal to
Now, we choose ε1 =
−1
2
2 −1 −1
2
(8 λ0 a21 c−1
0 δ0 ) |ut | + 2 λ0 b c0 δ0 |u| + δ0 |∇ϕ|
(
3
X
i,j=1
Aij
∂u ∂u
¦
+
∂xj ∂xi
)
¯Z t h
i
³
´ ¯¯2
¯
2 −1 −1
2
2
¯
¯
+ (16 λ0 c4 c0 δ0 ) α1 |H x η| + γ1 γ ¯
H(x, τ ) x η exp −σ(t−τ ) dτ ¯
.
0
Hence, from (3.48), (3.53) and the above discussion, we obtain the estimate
(
∂
Vn ≤ −
∂t
)
¯Z t h
i
³
´ ¯¯2
¯
(t + t0 ) b|u| + γ ¯¯
H(x, τ ) x η exp −σ(t−τ ) dτ ¯¯
2
0
∂
−1
2
{a|u|}2 − {1 − 2 λ0 b c−1
0 δ0 } b|u|
∂t
½
¾
∂ϕ
2 −1 −1
− 2 (t + t0 )a −
− 8 λ 0 a 1 c0 δ 0
|ut |2
∂η
−
(3.56)
−
−
−
½
∂ϕ
− δ0 |∇ϕ|
∂η
½
¾ X
3
i,j=1
Aij
∂u ∂u
¦
∂xj ∂xi
¾
2 (t + t0 )α − (3 + δ0−1 ) (2 d1 α2 + β) |∇ϕ| − 16 λ0 c24 δ0−1 α12 |H x η|2
½
2 (t + t0 ) − 1 − 2 d1 γ1 (3 +
δ0−1 ) |∇ϕ|
´ ¯¯2
¯Z t
³
¯
· γ ¯¯ [H x η] exp −σ(t−τ ) dτ ¯¯ .
0
−
Integration of (3.56) in S×[0, T ] proves Lemma 3.4.
−1
16 λ0 c24 c−1
0 δ0 γ 1
¾
452
BORIS V. KAPITONOV and G. PERLA MENZALA
4 – Exact controllability
In this section, we use the result of Theorem 3.5 to prove exact boundary
controllability to an arbitrary state of solutions of (1.1), (1.2), (1.4) and (1.7)
when γ ≡ 0.
Theorem 4.1. Under the assumptions of Theorem 3.5 and γ ≡ 0, there exists
Te > 0 such that for any T > Te, given any initial data f ∈ M1 and any terminal
state g ∈ M1 , there exists a boundary control {~
p(x, t), ~q(x, t)} belonging to
[L2 (S×(0, T ))]6 driving the system (1.1), (1.2), (1.4), (1.7) to the terminal state
g(x) at time T :
u(x, T ) = g1 (x) ,
ut (x, T ) = g2 (x) ,
E(x, T ) = g3 (x)
and
H(x, T ) = g4 (x) .
Moreover
n
k~
pk2W + k~qk2W ≤ c kf k2Z + kgk2Z
(4.1)
for positive constant c where W = [L2 (S×(0, T ))]3 .
Proof: Let Te = T0
in M1 :
"µ
c13
T0
¶1/1−p
o
#
−1 > 0. We consider the following equation
v − U ∗ (T ) U (T ) v = f − U ∗ (T ) g
(4.2)
where {U (t)}t≥0 is the semigroup associated with problem (1.1)–(1.4). The operator F (T ) = U ∗ (T ) U (T ) takes M1 into itself and kF (T )k < 1 for any T > Te
by Corollary 3.6. Thus we can solve (4.2) for any f, g ∈ M1 and
n
kvkZ ≤ c kf kZ + kgkZ
o
.
Consequently, if we choose v = (I − F (T ))−1 (f − U ∗ (T )g) then we will have that
(u1 , u2 , u3 , u4 ) = U (t)v − U ∗ (T − t) (U (T )v − g)
≡ (ṽ1 , ṽ2 , ṽ3 , ṽ4 ) − (w1 , w2 , w3 , w4 )
is a weak solution of (1.1), (1.2), (1.4) and (1.7) with
p~(x, t) = −a ṽ2 − a w2 ,
~q(x, t) = α η x {(ṽ4 + w4 ) x η} .
We observe that
(u1 , u2 , u3 , u4 )|t=T = g(x)
therefore by the energy identity we obtain (4.1).
UNIFORM STABILIZATION AND EXACT CONTROL
453
ACKNOWLEDGEMENT – We would like to express our sincere thanks to the Referee
of this journal for several suggestions which improved the final version of our manuscript.
The first author (B.K.) was supported by a Grant of FAPERJ (Brasil) in the project
E-26/151.523 and he is a Visiting Researcher at the National Laboratory of Scientific
Computation (LNCC/MCT, Brasil). The second author (G.P.M.) was partially supported by a Grant of CNPq and PRONEX (MCT, Brasil). Part of this work was done
while he was visiting the Centro de Modelamiento Matematico (CMM) of the Universidad de Chile, as part of the Chair “Marcel Dassault”. He express his gratitude for the
kind hospitality of the members of the CMM and especially to Prof. Carlos Conca.
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Boris V. Kapitonov,
Sobolev Institute of Mathematics,
Siberian Branch of Russian Academy of Sciences – RUSSIA
E-mail: borisvk@lncc.br
and
G. Perla Menzala,
National Laboratory of Scientific Computation LNCC/MCT,
Rua Getulio Vargas 333, Quitandinha, Petropolis, 25651-070, RJ – BRASIL
and
Federal University of Rio de Janeiro, Institute of Mathematics,
P.O. Box 68530, 21945-970, Rio de Janeiro, RJ – BRASIL
E-mail: perla@lncc.br