SIAM J. CONTROL OPTIM.
Vol. 42, No. 1, pp. 239–259
c 2003 Society for Industrial and Applied Mathematics
BOUNDARY FEEDBACK STABILIZATION OF SHALLOW SHELLS∗
SHUGEN CHAI† , YUXIA GUO‡ , AND PENG-FEI YAO§
Abstract. We consider the stabilization of the shallow shell by boundary feedbacks where the
model has a middle surface of any shape. First, we put the shallow shell model in a suitable semigroup
scheme. The existence, the uniqueness, and the properties of solutions to the shallow shell are then
treated by the semigroup approach and the regularity of elliptic boundary value problems. Finally,
we establish the uniform energy decay rate for the shallow shell under some checkable geometric
conditions on the middle surface.
Key words. shallow shell, regularity, boundary stabilization
AMS subject classifications. 35A, 35L, 35Q, 49A, 49B, 49E
PII. S0363012901397156
1. Introduction. We are concerned with the stabilization of the shallow shell by
boundary feedbacks. This issue has been analyzed a great deal for the wave equation
and plates; see Lagnese [8], Lagnese and Lions [9], Lasiecka and Triggiani [10], [11],
and many others. For thin shells, we know very little about this problem. A circular
cylindrical shell is considered by Chen, Coleman, and Liu [3] and a spherical shell
by Lasiecka, Triggiani, and Valente [12], and Triggiani [17]. In the above cases the
models are expressed in terms of special coordinates and all the work takes place in
those coordinates.
We study the shallow shell model where the tensor of change of curvature is given
by the Hessian of the normal displacement; see Ciarlet [4], Mason [13], Niordson [14],
or Koiter [7]. The model is written into a coordinate free form by using the global
geometry analysis in Yao [20]. This is one of the simplest thin shell models. For other
models, for example, the Koiter model where the change of the curvature tensor is
much more complicated, the control problems seem to be even more difficult; see Chai
and Yao [2].
We shall carry out the control scheme, which is given in Lagnese [8] for the boundary stabilization of thin plates, to study the boundary stabilization of the shallow shell
and we obtain the exponential stabilization under very weak geometrical conditions.
There are some difficulties we have to overcome.
One of the key problems in getting the uniform energy decay rate is obtaining the
regularity of solutions to the shallow shell. By using some ideas in Lagnese [8] and the
geometry approach, we address the resulting closed-loop system of the shallow shell
after exerting the boundary feedback controls in an appropriate semigroup scheme
so that the regularity in the time variable follows from the semigroup theory; see
∗ Received by the editors October 29, 2001; accepted for publication (in revised form) September
10, 2002; published electronically March 26, 2003. This work is supported by the NSF of China grant
60074006.
http://www.siam.org/journals/sicon/42-1/39715.html
† Department of Mathematics, Shanxi University, Taiyuan 030006, China; Institute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080,
China ([email protected]).
‡ Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China (yguo@math.
tsinghua.edu.cn).
§ Institute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of
Sciences, Beijing 100080, China; Department of Electrical and Electronic Engineering, Imperial College of Science, Technology and Medicine, Exhibition Road, London SW7 2BT, UK ([email protected]).
239
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SHUGEN CHAI, YUXIA GUO, AND PENG-FEI YAO
Pazy [15]. In addition, the regularity we need in the spatial variables is obtained by
using the elliptic boundary value theory; see Agmon, Douglis, and Nirenberg [1]. We
mention that, for the static problem, the existence, uniqueness, and regularity (i.e.,
the ellipticity) have been thoroughly treated by Ciarlet [4].
Another problem is that we have to develop some trace estimates for solutions
of the closed-loop system which permit certain boundary traces to be expressed in
terms of other traces modulo lower-order interior terms. We use Horn [6] to obtain
the trace estimates on the tangential component of solutions of the shallow shell
and Lasiecka and Triggiani [11] on the bending component. Those results allow us
to have the stabilization under very weak geometrical conditions. We mention that
trace estimates for the wave equation were developed in Lasiecka and Triggiani [10]
to eliminate the geometric constraints.
1.1. Some notation. We introduce some notations in preparation for the shallow shell.
Denote the usual inner product in R3 by ·, ·, i.e., the dot product. Let M be
a surface in R3 . For simplicity, M is assumed to be smooth. Surface M produces
a natural Riemannian manifold of dimension 2 with the induced metric in R3 . We
denote this induced metric on surface M by g or by ·, ·, as is convenient. For each
x ∈ M , Mx is the tangential space of M at x. It is assumed that surface M is
orientable with the unit normal field N on M . Denote the set of all vector fields on
M by X (M ). Denote the set of all k-order tensor fields and the set of all k-forms on
M by T k (M ) and Λk (M ), respectively, where k is a nonnegative integer. Then
Λk (M ) ⊂ T k (M ).
In particular, Λ0 (M ) = T 0 (M ) = C ∞ (M ) is the set of all C ∞ functions on M and
T 1 (M ) = T (M ) = Λ(M ) = X (M ),
where Λ(M ) = X (M ) is in the following isomorphism: for X ∈ X (M ) given, the
equation
U (Y ) = Y, X
∀ Y ∈ X (M )
determines a unique U ∈ Λ(M ).
It is well known that, for each x ∈ M , k-order tensor space Txk on Mx is an inner
product space defined as follows. Let e1 , e2 be an orthonormal basis of Mx . For any
α, β ∈ Txk , x ∈ M , the inner product is given by
(1.1)
α, βT k =
x
2
α(ei1 , . . . , eik )β(ei1 , . . . , eik )
at x.
i1 ,...,ik =1
In particular, for k = 1 definition (1.1) becomes
g(α, β) = α, βTx = α, β
∀ α, β ∈ Mx ,
that is, the induced inner product of Mx in R3 .
Let Ω be a bounded region of surface M with a regular boundary Γ or without
boundary (when Γ is empty). From (1.1), T k (Ω) are then inner product spaces in the
following sense:
(1.2)
T1 , T2 T k dx
∀ T1 , T2 ∈ T k (Ω),
(T1 , T2 )T k (Ω) =
Ω
x
BOUNDARY FEEDBACK STABILIZATION OF SHALLOW SHELLS
241
where dx is the volume element of surface M in its Riemaniann metric g.
The completions of T k (Ω) in inner products (1.2) are denoted by L2 (Ω, T k ). In
particular, L2 (Ω, Λ) = L2 (Ω, T ). L2 (Ω) is the completion of C ∞ (Ω) in the following
inner product:
(f, h)L2 (Ω) =
f (x)h(x) dx
∀ f, h ∈ C ∞ (Ω).
Ω
Let D be the Levi–Civita connection on M in the induced metric g of surface
M . For U ∈ X (M ), DU is the covariant differential of U which is a 2-order covariant
tensor field in the following sense:
DU (X, Y ) = DY U (X) = DY U, X
(1.3)
∗
∀ X, Y ∈ Mx , x ∈ M.
2
We also define D U ∈ T (M ) by
D∗ U (X, Y ) = DU (Y, X)
(1.4)
∗
∀ X, Y ∈ Mx , x ∈ M,
2
that is, D U ∈ T (M ) is the transpose of DU . For any T ∈ T 2 (M ), the trace of T
at x ∈ M is defined by
trT =
2
T (ei , ei ),
i=1
where e1 , e2 is an orthonormal basis of Mx . It is obvious that trT ∈ C ∞ (M ) if
T ∈ T 2 (M ).
For T ∈ T k (M ) and X ∈ X (M ), we define lX T ∈ T k−1 (M ) by
lX T (X1 , . . . , Xk−1 ) = T (X, X1 , . . . , Xk−1 )
∀ X1 , . . . , Xk−1 ∈ X (M ).
The Sobolev space H k (Ω) is the completion of C ∞ (Ω) with respect to the norm
f 2H k (Ω) =
(1.5)
k
i=1
Di f 2L2 (Ω,T i ) + f 2L2 (Ω) ,
f ∈ C ∞ (Ω),
i
where D f is the ith covariant differential of f in the induced metric g of M which is
an i-order tensor field on Ω, and · L2 (Ω,T i ) and · L2 (Ω) are the induced norms in
inner products (1.1)–(1.2), respectively. For details on Sobolev spaces on Riemannian
manifolds, we refer to Hebey [5] or Taylor [16].
Another important Sobolev space for us is H k (Ω, Λ), defined by
H k (Ω, Λ) = { U | U ∈ L2 (Ω, Λ), Di U ∈ L2 (Ω, T i+1 ), 1 ≤ i ≤ k }
with inner product
(U, V )H k (Ω,Λ) =
k
(Di U, Di V )L2 (Ω,T i+1 )
∀ U, V ∈ H k (Ω, Λ);
i=0
for example, see Wu [18]. In particular, H 0 (Ω, Λ) = L2 (Ω, Λ).
For Γ̂ ⊂ Γ, set
(1.6)
(1.7)
HΓ̂1 (Ω, Λ) = { W | W ∈ H 1 (Ω, Λ), W |Γ̂ = 0 },
HΓ̂2 (Ω)
∂w w | w ∈ H (Ω), w Γ̂ =
=0 .
∂n Γ̂
=
2
In particular, H01 (Ω, Λ) = HΓ1 (Ω, Λ) and H02 (Ω) = HΓ2 (Ω).
242
SHUGEN CHAI, YUXIA GUO, AND PENG-FEI YAO
1.2. Model. We assume that the middle surface of the shell is a bounded region
Ω of surface M in R3 before the deformation takes place. The shell, a body in R3 , is
defined by
S = { p | p = x + zN (x), x ∈ Ω, −h/2 < z < h/2 },
where h is the thickness of the shell, small.
Denote by η(x) the displacement vector of point x of the middle surface. We
decompose the displacement vector η into a sum
(1.8)
η(x) = W (x) + w(x)N (x),
x ∈ Ω, W (x) ∈ Mx ,
i.e., W and w are components of η on the tangent plane and on the normal of the undeformed middle surface Ω, respectively. The linearized strain tensor and the change
of curvature tensor of the middle surface Ω are given by
Υ(η) =
(1.9)
1
(DW + D∗ W ) + wΠ
2
and
ρ(η) = −D2 w
(1.10)
in a coordinate free form, respectively, where Π is the second fundamental form of
surface M and D2 w the Hessian of w, which are justified for a shallow shell. For (1.9)
and (1.10), we refer to Ciarlet [4], Niordson [14], Mason [13], or to Koiter [7].
Remark 1.1. If we express the two tensors (1.9) and (1.10) by a coordinate, they
look complicated. Let the middle surface of the shell be given by a coordinate
ϕ = (ϕ1 (x1 , x2 ), ϕ2 (x1 , x2 ), ϕ3 (x1 , x2 )),
Set
aα =
∂ϕ1 ∂ϕ2 ∂ϕ3
,
,
∂xα ∂xα ∂xα
(x1 , x2 ) ∈ R2 .
,
W = w1 a1 + w2 a2 .
Then the tensors (1.9) and (1.10) become
Υαβ =
1
(wα|β + wβ|α ) − bαβ w,
2
ραβ = −w|αβ ,
where 1 ≤ α, β ≤ 2, bαβ = −∂aα N · ∂aβ is the second fundamental form, and
wα|β = ∂aβ wα − Γλαβ wλ ,
w|αβ = ∂aβ ∂aα w − Γλαβ w.
The shell strain energy associated with a displacement field η of the middle surface
Ω can be written as
Eh
(1.11)
B(η, η) dx,
B1 (η, η) =
1 − µ2 Ω
where
(1.12)
B(η, η) = a(Υ(η), Υ(η)) + γa(ρ(η), ρ(η)),
γ = h2 /12,
BOUNDARY FEEDBACK STABILIZATION OF SHALLOW SHELLS
(1.13)
a(T1 , T1 ) = (1 − µ)T1 , T1 Tx2 + µ(trT1 )2 ,
243
T1 ∈ T 2 (Ω),
for x ∈ Ω, where E, µ, respectively, denote Young’s modulus and Poisson’s coefficient
of the material.
Thus, with expression (1.11) we are able to associate the following symmetric
bilinear form, directly defined on the middle surface Ω:
B(η, ζ) =
(1.14)
B(η, ζ) dx,
Ω
where η is given in (1.8) and
ζ = U + uN,
U (x) ∈ Mx ,
x ∈ Ω.
Denote by H and by k the mean curvature and the Gauss curvature of surface
M , respectively. From Yao [20], we have the following Green’s formula for the shallow
shell.
Formula I. Let the bilinear form B(·, ·) be given in (1.14). For all sufficiently
smooth η = (W, w) and ς = (U, u), we have
B(η, ς) = (Aη, ς)L2 (Ω,Λ)×L2 (Ω) + ∂(Aη, ς)dΓ,
(1.15)
Γ
where
(1.16)
∂(Aη, ς) = v1 (η)U, n + v2 (η)U, τ + v3 (η)
∂u
+ v4 (η)u,
∂n
n, τ are the normal and the tangential along curve Γ, respectively,
−∆µ W − (1 − µ)kW − F(w)
(1.17)
,
Aη =
γ[∆2 w − (1 − µ)δ(kdw)] + (H 2 − 2(1 − µ)k)w + G(W )
∆µ is of the Hodge-Laplacian type, applied to the 1-form (or equivalently vector
fields), defined by
1−µ
(1.18)
δd + dδ ,
∆µ = −
2
d the exterior differential, δ the formal adjoint of d, ∆ the Laplacian on manifold M ,
F(w) = (1 − µ)ldw Π + µHdw + wdH,
(1.19)
G(W ) = (1 − µ)DW, ΠTx2 − µHδW,
and
(1.20)

v1 (η)






 v2 (η)
v3 (η)






 v4 (η)
=
(1 − µ)Υ(η)(n, n) + µ(wH − δW ),
=
(1 − µ)Υ(η)(n, τ ),
= γ[∆w − (1 − µ)D2 w(τ, τ )],
∂
∂∆w
∂w
2
+ (1 − µ)
(D w(τ, n)) + k(x)
.
= −γ
∂n
∂τ
∂n
244
SHUGEN CHAI, YUXIA GUO, AND PENG-FEI YAO
By the “principle of virtual work” and Formula I, we obtain the following displacement equations for a shallow shell (see Yao [20]) after changing t to t/λ with
λ2 E/(1 − µ2 ) = 1.
Formula II. We assume that there are no external loads on the shell and the
shell is clamped along a portion Γ0 of Γ and free on Γ1 , where Γ0 ∪ Γ1 = Γ and
Γ0 ∩Γ1 = ∅. Then the displacement vector η = (W, w) satisfies the following boundary
value problem:

Wtt − [∆µ W + (1 − µ)kW + F(w)] = 0,



 w − γ∆w + γ ∆2 w − (1 − µ)δ(kdw)
tt
tt
(1.21)
in Q∞ ,
2

− 2(1 − µ)k)w + G(W ) = 0,
+(H



η(0) = η 0 , ηt (0) = η 1 ,
(1.22)
(1.23)

 W = 0,
 w = ∂w = 0,
∂n
on Σ0∞ ,
v1 (η) = v2 (η) = v3 (η) = 0 and v4 (η) + γ
∂wtt
=0
∂n
Q∞ = Ω × (0, ∞),
Σ1∞ = Γ1 × (0, ∞).
on Σ1∞ ,
where
(1.24)
Σ0∞ = Γ0 × (0, ∞),
Remark 1.2. If the shell is flat, a plate, the equations in (1.21) are uncoupled.
The equation on the component w is the same as in Lagnese [8]—a Kirchhoff plate
(see Yao [20]).
1.3. Uniform stabilization. We write (1.21) as
(1.25)
ηtt − γ(0, ∆wtt ) + Aη = 0
and define the total energy of shell by
1
E(t) = [Wt 2L2 (Ω,Λ) + wt 2L2 (Ω) + γDwt 2L2 (Ω,Λ) + B(η, η)].
(1.26)
2
By Green’s formula (1.15), the equations (1.25), and the boundary conditions
(1.22) we obtain
(1.27)
d
E(t)
dt d 1
2
2
2
[Wt L2 (Ω,Λ) + wt L2 (Ω) + γDwt L2 (Ω,Λ) + B(η, η)]
=
dt 2
= B(η, ηt ) +
[Wtt , Wt + wtt wt + γDwtt , Dwt ]dx
Ω
=
[Wtt , Wt + wtt wt − γ∆wtt wt ]dx + (Aη, ηt )L2 (Ω,Λ)×L2 (Ω)
Ω
∂wtt
wt dΓ
∂(Aη, ηt ) + γ
+
∂n
Γ
∂wtt
=
wt dΓ
∂(Aη, ηt ) + γ
ηtt − γ(0, ∆wtt ) + Aη, ηt dx +
∂n
Ω
Γ
∂wt
∂wtt
=
wt dΓ.
v1 (η)Wt , n + v2 (η)Wt , τ + v3 (η)
+ v4 (η) + γ
∂n
∂n
Γ1
BOUNDARY FEEDBACK STABILIZATION OF SHALLOW SHELLS
245
For simplicity, we set
(1.28)
ζ̆ =
U, n, U, τ ,
∂u ∂u
,
,u
∂n ∂τ
for any ζ = (U, u). In this paper, we shall consider feedback laws to be defined by

i = 1, 2, 3,
 vi (η) = Ji (ηt ),
(1.29)
 v4 (η) + γ ∂wtt = J4 (ηt ),
∂n
where the feedback operators Ji are given by

i = 1, 2, 3,
 Ji (ζ) = −ζ̆Fiτ ,
(1.30)
∂
 J4 (ζ) = −ζ̆F τ +
(ζ̆F4τ ),
5
∂τ
Fi = (fi1 , fi2 , fi3 , fi4 , fi5 ) for 1 ≤ i ≤ 5, and ζ = (U, u). In the formula (1.30) the
superscript τ denotes a transpose, fij are real L∞ (Γ1 ) functions, and the matrix
F = (F1τ , F2τ , F3τ , F4τ , F5τ ) satisfies
(1.31)
F is symmetric and positive semidefinite on Γ1 .
If we put the feedback laws of the formulas (1.29) and (1.30) into the formula
(1.27), by the assumption (1.31) we obtain
∂
d
∂wt
F4 η˘t τ + wt F4 η˘t τ dΓ
−η˘t F η˘t τ +
E(t) =
dt
∂τ
∂τ
Γ1
=−
(1.32)
η˘t F η˘t τ dΓ ≤ 0
Γ1
so that the resulting closed-loop system under the feedback laws of (1.29) and (1.30)
is dissipative in the sense that E(t) is nonincreasing.
Remark 1.3. When the tangent component W = 0, the feedback laws of (1.29)
and (1.30) are what Lagnese [8] presented for the uniform stabilization of the Kirchhoff
plate.
We now set up some geometric conditions on the middle surface of the shallow
shell which are necessary to get the energy decay.
Assumption (H.1). There is a constant λ0 such that
(1.33)
λ0 B(η, η) ≥ DW 2L2 (Ω,T 2 ) + γD2 w2L2 (Ω,T 2 )
for η = (W, w) ∈ H 1 (Ω, Λ) × H 2 (Ω).
Assumption (H.2). There is a vector field V ∈ X (M ) such that
(1.34)
DV (X, X) = b(x)|X|2 ,
X ∈ Mx ,
x ∈ Ω,
where b is a function on Ω. Set
a(x) =
1
DV, ET 2 ,
x
2
x ∈ Ω,
246
SHUGEN CHAI, YUXIA GUO, AND PENG-FEI YAO
where E is the volume element of M . Moreover, suppose that b and a meet inequality
2 min b(x) > λ0 (1 + µ) max |a(x)|.
(1.35)
x∈Ω
x∈Ω
Assumption (H.3). Γ0 and Γ1 satisfy the following conditions:
(1.36)
Γ0 = ∅,
Γ0 ∩ Γ1 = ∅,
and V (x) · n(x) ≤ 0
on Γ0 .
Assumption (H.4). F ∈ C 1 (Γ1 ) is a positive definite matrix.
Remark 1.4. The assumptions (H.1)–(H.3) are geometric conditions on the middle surface of the shell, while the assumption (H.4) is on the feedback. For a plate the
assumptions (H.1)–(H.2) automatically satisfy, where we set V = x − x0 . For the general case, the assumptions (H.1)–(H.2) can be verified by the geometry method; see,
for example, Yao [21]. Here the geometric assumption (H.3) is, generally, considered
to be much weaker than the following:
V (x) · n(x) ≤ 0
on
Γ0
and V (x) · n(x) > 0
on
Γ1 ,
which is used to avoid the complex trace estimates.
We are now in a position to state our main results.
Theorem 1.1. Assume that the assumptions (H.1)–(H.4) hold. Let the energy
E(t) be defined by (1.26) for the closed-loop system (1.21), (1.22), and (1.29). Then
there are positive constants K and ω such that
E(t) ≤ Ke−ωt E(0),
(1.37)
t ≥ 0,
for any η 0 ∈ HΓ10 (Ω, Λ) × HΓ20 (Ω) and any η 1 ∈ L2 (Ω, Λ) × HΓ10 (Ω).
2. Existence, uniqueness, and properties of solutions. In this section, we
follow the ideas in Lagnese [8] for the Kirchhoff plate to put the shallow shell problem
into a semigroup frame. Then the regularity of solutions we need for the stabilization
is worked out by Agmon, Douglis, and Nirenberg [1].
2.1. Variational formulation. We shall set
(2.1)
W = HΓ10 (Ω, Λ) × HΓ20 (Ω),
V = L2 (Ω, Λ) × HΓ10 (Ω),
Introduce the forms
(2.2)
a0 (η, ζ) =
Ω
[η, ζ + γDw, Du]dx
and
(2.3)
and L = L2 (Ω, Λ) × L2 (Ω).
a1 (η, ζ) =
Γ1
η̆F ζ̆ τ dΓ
for η = (W, w) and ζ = (U, u). It follows from Green’s formula (1.15) that an appropriate variational formulation of the systems (1.21), (1.22), and (1.29) is as follows:
Find a vector field η ∈ C([0, ∞); W) ∩ C 1 ([0, ∞); V) such that


 d [a0 (ηt , ζ) + a1 (η, ζ)] + B(η, ζ) = 0
∀ ζ ∈ W,
dt
(2.4)


η(0) = η 0 ∈ W, ηt (0) = η 1 ∈ V.
BOUNDARY FEEDBACK STABILIZATION OF SHALLOW SHELLS
247
2.2. Well-posedness of (1.21), (1.22), and (1.29). The bilinear forms a0 (·, ·),
a1 (·, ·), and B(·, ·) are continuous, symmetric, and nonnegative on V and W, respectively, and if we set
a0 (η) = a0 (η, η)
(2.5)
and a1 (η) = a1 (η, η),
then we have
a0 (η) ≥ γη2L2 (Ω,Λ)×H 1 (Ω)
(2.6)
and a1 (η) ≥ 0.
The form a0 (·, ·) defines a scalar product on V and so does B(·, ·) on W because
of the ellipticity (1.33). Those scalar products are equivalent to the ones previously
introduced in those spaces. We identify L with its dual L so that we have the dense
and continuous embeddings
W ⊂ V ⊂ L ⊂ V ⊂ W .
(2.7)
Let A0 (respectively, P) denote the canonical isomorphism of V (respectively, W)
endowed with the scalar product a0 (·, ·) (respectively, B(·, ·)) onto V (respectively,
W ). Then
a0 (η, ζ) = A0 η, ζ
∀η, ζ ∈ V,
B(η, ζ) = P η, ζ
∀η, ζ ∈ W,
where ·, · refers to (·, ·)L2 (Ω,Λ)×L2 (Ω) . Furthermore, there is a nonnegative operator
A1 ∈ B(W, W ) such that
a1 (η, ζ) = A1 η, ζ
∀η, ζ ∈ W.
We write (2.4) as
d
(A0 ηt + A1 η) + P η = 0
dt
(2.8)
in W .
Let us formally rewrite (2.4) as the system
P
0
0
A0
η
ηt
+
0
P
−P
A1
η
ηt
=0
or
CY + QY = 0,
(2.9)
where
C=
P
0
0
A0
,
Q=
0
P
−P
A1
t ≥ 0,
,
and Y =
η
ηt
.
We wish to solve (2.9) in the space W × V. In order to make sense of (2.9) in that
space it is natural to introduce
D(Q) = {(η, ζ)|η ∈ W, ζ ∈ W, P η + A1 ζ ∈ V }.
248
SHUGEN CHAI, YUXIA GUO, AND PENG-FEI YAO
Then Q : D(Q) → W × V . Since C is the canonical isomorphism of W × V onto
W × V , we rewrite (2.9) in the form
Y + C−1 QY = 0
(2.10)
in W × V.
Solutions of the system (1.21), (1.22), and (1.29) are therefore defined via (2.10).
Theorem 2.1. −C−1 Q is the infinitesimal generator of a C0 -semigroup of contraction on W × V.
Proof. (i) D(Q) is dense in W × V.
By the definition of P and A1 , for ς = (U, u) ∈ W, we obtain
P η + A1 ζ, ς = B(η, ς) + a1 (ζ, ς)
=−
∆µ W + (1 − µ)kW + F(w), U dx − γ
d∆w, dudx
Ω
Ω
+
[−(1 − µ)δ(kdw) + G(W ) + (H 2 − 2(1 − µ)k)w]udx
Ω
∂u
+
dΓ
v1 (η)U, n + v2 (η)U, τ + v3 (η)
∂n
Γ1
∂ 2
∂w
− (1 − µ)γ
D w(n, τ ) + k
u dΓ
∂n
Γ1 ∂τ
∂u
−
+ J4 (ζ)u dΓ.
J1 (ζ)U, n + J2 (ζ)U, τ + J3 (ζ)
(2.11)
∂n
Γ1
The expression on the right-hand side of the formula (2.11) implies the relation
D(Q) ⊃ D0 = { (η, ζ) | η ∈ W ∩ H 2 (Ω, Λ) × H 4 (Ω), ζ ∈ W, v1 (η) = J1 (ζ),
v2 (η) = J2 (ζ), and v3 (η) = J3 (ζ) on Γ1 }.
Indeed, if (η, ζ) ∈ D0 , then
|P η + A1 ζ, ς| ≤ C(W H 2 (Ω,Λ) + wH 3 (Ω) )(U L2 (Ω,Λ) + uH 1 (Ω) )
∂ 2
∂w D w(n, τ ) + k
+ C J4 (ζ) + (1 − µ)γ
∂τ
∂n − 1
H
2
(Γ1 )
u
1
H 2 (Γ1 )
≤ Cη,ζ ςV ,
that is,
P η + A1 ζ ∈ V .
We mention that in the above inequality the following result is used: η ∈ W ∩
∂
− 12
D2 w(n, τ )+k ∂w
(Γ1 ).
H 2 (Ω, Λ)×H 4 (Ω) and ζ ∈ W imply J4 (ζ)+(1−µ)γ[ ∂τ
∂n ] ∈ H
D(Q) is then dense in W × V since D0 is in W × V.
(ii) −C−1 Q is dissipative. This is shown by
C−1 Q(η, ζ), (η, ζ) = (−ζ, A−1
0 (P η + A1 ζ)), (η, ζ)
= −B(ζ, η) + B(η, ζ) + a1 (ζ, ζ)
(2.12)
for (η, ζ) ∈ D(Q).
= a1 (ζ, ζ) ≥ 0
BOUNDARY FEEDBACK STABILIZATION OF SHALLOW SHELLS
249
(iii) We also have Range(λI + C−1 Q) = W × V, for λ > 0, (η, ζ) ∈ D(Q). In fact,
this is equivalent to
Range(λ2 A0 + λA1 + P ) = V .
But, by the Lax–Milgram theorem, it is actually true.
As a consequence of Theorem 2.1, we have the following result.
Theorem 2.2. Assume that (1.29) and (1.31) hold and
(2.13)
η 0 ∈ W, η 1 ∈ W, P η 0 + A1 η 1 ∈ V .
Then the problem (2.4) admits a unique solution with
η ∈ C 1 ([0, ∞); W) ∩ C 2 ([0, ∞); V),
ηtt ∈ C([0, ∞); V),
(2.14)
A0 ηtt + A1 ηt + P η = 0,
η(0) = η 0 ,
t ≥ 0,
ηt (0) = η 1 .
2.3. Regularity of solutions. In the following, we shall use local coordinate
systems to obtain the regularity of variational solutions to the system (1.21), (1.22),
and (1.29). Let us consider the system


 η ∈ W, Aη ∈ V ,
1
i = 1, 2, 3,
vi (η) ∈ H 2 (Γ1 ),
(2.15)


− 12
v4 (η) ∈ H (Γ1 ).
First, we have the following lemma.
Lemma 2.3. Let η satisfy the problem (2.15). Then
(2.16)
η ∈ H 2 (Ω, Λ) × H 3 (Ω) ∩ W.
Proof. First, we prove that, for any ϕ ∈ C ∞ (Ω), ϕη still satisfies the problem
(2.15), so our analysis of η on Ω can be localized.
Note that
(2.17)
A(ϕη) = ϕAη + [A, ϕ]η,
where the commutator [A, ϕ] is a first-order differential operator on the component
W and a third-order differential operator on the component w. Then the hypothesis
Aη ∈ V , together with η ∈ W, gives A(ϕη) ∈ V . Similarly, it is easy to check
from the formulas in (1.20) that when η satisfies the problem (2.15) all the boundary
conditions in the problem (2.15) for ϕη are still true.
So suppose η, satisfying (2.15), is supported on a coordinate chart (U, ψ) where
(U, ψ) is chosen in such a way that there exists a positive smooth function Θ on U to
meet
(2.18)
g = Θ(dx21 + dx22 )
on U,
250
SHUGEN CHAI, YUXIA GUO, AND PENG-FEI YAO
where g is the induced metric of the Riemannian manifold M; see Wu [19]. It is
noticeable that the expression in (2.18) does not hold in general when the dimension
of the manifold is larger than 2.
In addition, the formulas in (1.19) and the hypothesis η ∈ W imply
F(w) ∈ HΓ10 (Ω)
(2.19)
and G(W ) ∈ L2 (Ω, Λ).
Set
(2.20)
W = u1
∂
∂
+ u2
.
∂x1
∂x2
By the relations (2.18)–(2.20) and through a computation, we separate the problem (2.15) into the two following ones:

1
(u , u ) ∈ H∂O
(O),

0
 1 2






∂ 2 u1
1 − µ ∂ 2 u1
1 + µ ∂ 2 u2


+
+
∈ L2 (O),

2
2

∂x
2
∂x
2
∂x
∂x

1
2
1
2



 1 − µ ∂2u
∂ 2 u2
1 + µ ∂ 2 u1
2
+
+
∈ L2 (O),
(2.21)
2
2
2
∂x
∂x
2
∂x
∂x

1
2
1
2





1
∂u
∂u
∂u
∂u
1
2
1
2
n

+ n2
− µn2
+ µn1
∈ H 2 (∂O1 ),
1


∂n
∂n
∂τ
∂τ






 −n ∂u1 + n ∂u2 + n ∂u1 + n ∂u2 ∈ H 12 (∂O ),
2
1
1
2
1
∂n
∂n
∂τ
∂τ
and

2
w ∈ H∂O
(O),

0




2
−1

∆0 w ∈ H (O),




1
∂2w
∆0 w − (1 − µ)Θ 2 ∈ H 2 (∂O1 ),


∂τ





1
∂w
∂3w
∂∆0 w


+ (1 − µ)Θ k
+ 2
∈ H − 2 (∂O1 ),

∂n
∂n
∂τ ∂n
(2.22)
where
O = ψ(U ∩ Ω),
∂O1 = ψ(Γ1 ∩ U),
∂
∂
+ n2
,
∂x1
∂x2
τ = −n2
n = n1
∂O0 = ∂O/∂O1 ,
∂
∂
+ n1
,
∂x1
∂x2
∆0 =
∂2
∂2
+
.
2
∂x1
∂x22
It is clear that the problem (2.22) is a classical elliptic boundary value problem
since Γ0 ∩ Γ1 = ∅ and Γ is smooth enough. We therefore obtain
2
(O).
w ∈ H 3 (O) ∩ H∂O
0
Next, since the determinant of coefficients of {∂u1 /∂n, ∂u1 /∂n} in the boundary
conditions of the problem (2.21) is 1/Θ > 0, the classical theory of Agmon, Douglis,
and Nirenberg [1] yields
2
(u1 , u2 ) ∈ H 2 (O) .
BOUNDARY FEEDBACK STABILIZATION OF SHALLOW SHELLS
251
Finally, the partition of unity subject to a coordinate cover of Ω completes the
proof.
In order to apply Green’s formula to solutions of the system (1.21), (1.22), and
(1.29), we have to get more regularity than what is given by Theorem 2.2. To this
end, we need to require more regularity of the initial data than what is supposed in
(2.13). The requisite assumptions are obtained as follows. We introduce
(2.23)
Ã0 = A0 |W ,
H = RangeÃ0 ,
and we assume
η 0 ∈ W,
(2.24)
η 1 ∈ W ∩ H 2 (Ω, Λ) × H 3 (Ω),
and A1 η 1 + P η 0 ∈ H.
From (2.14) we have A0 ηtt (0) = −(A1 η 1 + P η 0 ) ∈ H and, therefore,
1
0
ηtt (0) = −Ã−1
0 (A1 η + P η ) ∈ W.
(2.25)
We further assume
vi (η 1 ) = Ji (ηtt (0)),
(2.26)
i = 1, 2, 3,
on
Γ1 .
Then we conclude that
A1 ηtt (0) + P η 1 ∈ V .
(2.27)
Indeed, for ς = (U, u) ∈ V, from (2.3), (1.31), and (2.26) we have
(2.28)
∂u
+ J4 (ηtt (0))u dΓ
v1 (η 1 )U, n + v2 (η 1 )U, τ + v3 (η 1 )
∂n
Γ1
a1 (ηtt (0), ς) = −
and, therefore, by Green’s formula (1.15),
A1 ηtt (0) + P η 1 , ς = a1 (ηtt (0), ς) + B(η 1 , ς)
=−
∆µ W 1 + (1 − µ)kW 1 + F(w1 ), U dx − γ
d∆w1 , du dx
+
Ω
Ω
Ω
γ[−(1 − µ)δ(kdw1 ) + (H 2 − 2(1 − µ)k)w1 + G(W 1 )]u dx
(2.29)
−
Γ1
∂ 2 1
∂w1
J4 (ηtt (0)) + γ(1 − µ)
D w (n, τ ) + k
∂τ
∂n
u dΓ.
Now the expression (2.29) produces
|A1 ηtt (0) + P η 1 , ς|
≤C
(2.30)
W 1 H 2 (Ω,Λ) + w1 H 3 (Ω)
1 ∂
∂w
2
1
ςV .
+
J4 (ηtt (0)) + γ(1 − µ) ∂τ D w (n, τ ) + k ∂n − 1
H 2 (Γ1 )
Consequently, (2.27) holds as claimed.
252
SHUGEN CHAI, YUXIA GUO, AND PENG-FEI YAO
We have shown that if (2.24) and (2.26) hold, then
η 1 ∈ W,
(2.31)
ηtt (0) ∈ W,
A1 ηtt (0) + P η 1 ∈ V ,
that is, {η 1 , ηtt (0)} ∈ D(C−1 Q). It follows from Theorem 2.2 that η satisfies
ηt ∈ C 1 ([0, ∞); W)
(2.32)
and ηtt ∈ C([0, ∞); W).
Then, as a consequence of A1 ηt + P η = −A0 ηtt we obtain that η satisfies
(2.33)

η ∈ C 2 ([0, ∞); W),





Aη = −(ηtt − γ(0, ∆wtt )) ∈ C([0, ∞); L2 (Ω, Λ) × L2 (Ω)),



1

vi (η) = Ji (ηt ) ∈ H 2 (Γ1 ),
i = 1, 2,
3


v3 (η) = J3 (ηt ) ∈ H 2 (Γ1 ),






 v (η) + γ ∂wtt = J (η ) ∈ H 12 (Γ ).
4
4 t
1
∂n
Elliptic theory then yields
w ∈ H 4 (Ω).
We now write the above analysis into the following result.
Theorem 2.4. Assume that Γ is smooth enough, Γ1 ∩ Γ0 = ∅, and conditions
(2.24) and (2.26) hold. Then variational solutions of the system (1.21), (1.22), and
(1.29) satisfy
(2.34) η ∈ C([0, ∞); H 2 (Ω, Λ) × H 4 (Ω) ∩ W) ∩ C 1 ([0, ∞); H 1 (Ω, Λ) × H 3 (Ω) ∩ V).
Remark 2.1. If η 0 ∈ H 3 (Ω, Λ) × H 4 (Ω) ∩ W and η 1 ∈ H 2 (Ω, Λ) × H 3 (Ω) ∩ W,
then conditions (2.24) and (2.26) hold.
3. Proof of Theorem 1.1. We assume that the initial data satisfy the assumptions of Theorem 2.4 and, therefore, the solution η = (W, w) of the system (1.21),
(1.22), and (1.29) meets the regularity of (2.34).
Let a vector field V be given to satisfy the assumption (H.2). Set
η1 = (W, 0),
η2 = (0, w),
m(η) = (DV W, V (w)),
L(t) = W 2L2 (Ω,Λ) + w2L2 (Ω) + γwt 2L2 (Ω) + Dw2L2 (Ω,Λ) ,
σ0 = max |V |,
x∈Ω
QT = (0, T ) × Ω,
σ1 = min b(x) −
x∈Ω
ΣT = (0, T ) × Γ,
λ0 (1 + µ)
max |a(x)|,
2
x∈Ω
ΣT0 = (0, T ) × Γ0 ,
and
ΣT1 = (0, T ) × Γ1 ,
where T > 0 is given.
Lemma 3.1. Let the assumptions (H.1) and (H.2) hold. If we denote
(3.1)
σ1 b
η1 − η2 ,
p(η) = b −
2
2
BOUNDARY FEEDBACK STABILIZATION OF SHALLOW SHELLS
253
then we have
T
1
σ1
E(t)dt ≤
[|ηt |2 + γ|Dwt |2 − B(η, η)]V, n dΣ
2 ΣT1
0
T
1
B(η, η)V, n dΣ −
a1 (ηt , m(η) + p(η)) dt
+
2 ΣT0
0
+ CT E(0) + E(T ) +
(3.2)
T
0
L(t) dt .
Proof. By the embedding theorem there is CT > 0 such that
L(0) ≤ CT E(0)
(3.3)
and L(T ) ≤ CT E(T ).
Then, from (3.3), Theorem 1.1 of Yao [21] gives the following inequality:
T
T
E(t) dt ≤ (SB)1 |ΣT + (SB)2 |ΣT + CT E(0) + E(T ) +
(3.4) σ1
L(t) dt ,
0
0
where
(3.5)
(SB)1 |ΣT
1
=
2
(3.6)
(SB)2 |ΣT =
ΣT
ΣT
[|ηt |2 + γ|Dwt |2 − B(η, η)]V, n dΣ,
∂wtt
1
dΣ.
∂ (Aη, m(η) + p(η)) + γ V (w) − bw
2
∂n
Let us examine the integrals over ΣT0 in (3.5) and (3.6). By the boundary conditions of (1.22) on Γ0 we have, from Proposition 2.12(ii) of Yao [21],
(3.7)
(SB)1 |ΣT0
1
=−
2
ΣT
0
B(η, η)V, n dΣ
and
(SB)2 |ΣT0 =
ΣT
0
B(η, η, )V, n dΣ.
First, we consider the calculation of (SB)2 |ΣT1 .
From the formula (1.16) and the feedback law of (1.29) and (1.30) we obtain, for
any ς = (U, u) ∈ L2 (Ω, Λ) × H 1 (Ω),
∂wtt
∂
τ
τ ∂u
dΓ = −a1 (ηt , ς) +
dΓ
∂(Aη, ς) + γu
u (η˘t F4 ) + η˘t F4
∂n
∂τ
∂τ
Γ1
Γ1
= −a1 (ηt , ς)
(3.8)
since Γ0 ∩ Γ1 = ∅. By applying the formula (3.8) to the expression (3.6) with ς =
m(η) + p(η), one finds out
T
(SB)2 |ΣT1 = −
(3.9)
a1 (ηt , m(η) + p(η)) dt.
0
Substituting (3.7) and (3.9) into (3.4) yields the inequality (3.2).
Next, we consider a trace result which is a consequence of Horn [6]. Let U be a
vector field on Ω. Set
1
S(U ) = (DU + D∗ U ),
(3.10)
2
a two-order tensor on Ω.
254
SHUGEN CHAI, YUXIA GUO, AND PENG-FEI YAO
Lemma 3.2. Let T > 0 and 1/2 > ? > 0 be given. Suppose that a vector field U
satisfies the problem

1

U ∈ L2 ([0, T ]; H 2 + (Ω, Λ)), Ut ∈ L2 (ΣT , Λ),




 U − ∆ U ∈ H − 12 (QT , Λ),
tt
µ
(3.11)

(1 − µ)S(U )(n, n) − µδU ∈ L2 (ΣT , Λ),





S(U )(n, τ ) ∈ L2 (ΣT , Λ).
Then, for T /2 > α > 0, there is CT,α, > 0 such that
(3.12)
Dτ U 2L2 ([α, T −α];L2 (Γ,Λ))
≤ CT,α, (Ut 2L2 (ΣT ,Λ) + (1 − µ)S(U )(n, n) − µδU 2L2 (ΣT ) + S(U )(n, τ )2L2 (ΣT ) )
+ CT,α, (Wtt − ∆µ W 2H −1/2 (QT ,Λ) + U 2L2 ([0,T ];H 1/2+ (Ω,Λ)) ).
Proof. It is easy to check that, if U satisfies the problem (3.11), then, for any
ϕ ∈ C ∞ (Ω), ϕU still does, which means the problem (3.11) can be localized. Suppose
that U is supported in a coordinate chart (U, φ) with the metric g of (2.18). Set
U = u1
∂
∂
+ u2
.
∂x1
∂x2
On the chart (U, φ), the problem (3.11) changes into the problem

ui ∈ L2 ([0, T ]; H 1/2+ (O)), uit ∈ L2 ((0, T ) × ∂O),
i = 1, 2,




2

2
2

∂ u1
1 − µ ∂ u1
1 + µ ∂ u2

−1/2

((0, T ) × O),

 u1tt − ∂x2 + 2 ∂x2 + 2 ∂x1 ∂x2 ∈ H
1
2
(3.13)

1 − µ ∂ 2 u2
∂ 2 u2
1 + µ ∂ 2 u1



u
∈ H −1/2 ((0, T ) × O),
−
+
+

2
2
 2tt
2
∂x
∂x
2
∂x
∂x
1
2

1
2



f1 , f2 ∈ L2 ([0, T ] × ∂O),
where
(3.14)

∂u1
∂u2
∂u1
∂u2


 f1 = n1 ∂n + n2 ∂n − µn2 ∂τ + µn1 ∂τ ,


 f2 = −n2 ∂u1 + n1 ∂u2 + n1 ∂u1 + n2 ∂u2 ,
∂n
∂n
∂τ
∂τ
O = φ(U), and ∂O = ∂φ(U).
Next we want to show that the problem (3.13) is a special case of some three
dimensional dynamic elasticity, so Theorem 1.2 of Horn [6] can be applied. To this
end, we set
u = (u1 , u2 , 0)
and O = O × (0, 1).
If we let the Lamé coefficients λ and µ in Horn [6] be µ and
have
1 − µ ∂ui
∂uj
(3.15)
,
+
σij = µ(divu)δij +
2
∂xj
∂xi
1−µ
2 ,
respectively, we
255
BOUNDARY FEEDBACK STABILIZATION OF SHALLOW SHELLS
∂ui
=0
∂x3
(3.16)
for any 1 ≤ i, j ≤ 3. By the notation of Horn [6], we have

 
n1 n2 0
f1
σ(u)n =  −n2 n1 0   f2 
(3.17)
0
0
0 1
and the problem (3.13) is then equivalent to the three dimensional problem

2
1/2+
(O ))3 ), ut ∈ (L2 ([0, T ] × ∂O ))3 ,

 u ∈ L ([0, T ]; (H
(3.18)
utt − ∇ · σ(u) ∈ (H −1/2 ([0, T ] × O ))3 ,


σ(u) · n ∈ (L2 ([0, T ] × ∂O ))3 ,
where n = (n1 , n2 , 0) and σ(u) = (σij )3×3 .
Applying Theorem 1.2 of Horn [6] locally and using the partition of unity complete
our proof.
By using the same ideas as in Lemma 3.2 and applying Theorem 2.1 of Lasiecka
and Triggiani [11], we get the following lemma.
Lemma 3.3. Let T /2 > α > 0 and 1/2 > ? > 0 and 1/2 > s0 . Suppose that w
satisfies the problem

wtt − γ∆wtt + γ∆2 w ∈ H −s0 (QT ),





∂w


w =
= 0 on ΣT0 ,

∂n
(3.19)
 ∆w − (1 − µ)D2 w(τ, τ ) ∈ L2 (ΣT1 ),






∂
∂∆w
∂w ∂wtt

2

+ (1 − µ)
(D w(τ, n)) + k(x)
−
∈ H −1 (ΣT1 ).
∂n
∂τ
∂n
∂n
Then there is Cα,,T such that
(3.20)
D2 w2L2 ([α, T −α];L2 (Γ1 ,T 2 ))
≤ Cα,,T wtt − γ∆wtt + γ∆2 w2H −s0 (QT ) + ∆w − (1 − µ)D2 w(τ, τ )2L2 (ΣT )
1
2
∂∆w
∂
∂w ∂wtt 2
+
(1
−
µ)
(D
−
+
w(τ,
n))
+
k(x)
+ Dwt 2L2 (ΣT ,Λ)
∂n
1
∂τ
∂n
∂n H −1 (ΣT )
1
+ w2L2 ([0,T ];H 3/2+ (Ω)) + wt 2L2 (ΣT ) .
1
Lemma 3.4. Let η be a solution of the system (1.21), (1.22), and (1.29) with the
regularity (2.34). Let T /2 > α > 0 and 1/2 > ? > 0 and 1/2 > s0 > 0 be given. Then
T −α 2
2
2
(|DW | + |D w| ) dΣ ≤ Cα,T,s0 ,
(3.21)
(|ηt |2 + |Dwt |2 ) dΣ + lot(η).
α
Γ1
ΣT
1
Proof. From the formulas in (1.20), (1.29), and (1.30),
(1 − µ)Υ(η)(n, n) + µ(wH − δW ) = −η˘t F1τ
(3.22)
(1 − µ)Υ(η)(n, τ ) = −η˘t F2τ on Γ1 ,
on
Γ1 ,
256
SHUGEN CHAI, YUXIA GUO, AND PENG-FEI YAO
and we obtain

τ

 DW (n, n) = −µDW (τ, τ ) − w[(1 − µ)Π(n, n) + µH] − η˘t F1 ,
(3.23)
2

η˘t F2τ .
 DW (τ, n) = −DW (n, τ ) − 2wΠ(n, τ ) −
1−µ
It follows from (3.23) that
|DW |2 = |Dτ W |2 + [DW (n, n)]2 + [DW (τ, n)]2
(3.24)
≤ C(|Dτ W |2 + |ηt |2 + |Dwt |2 + w2 )
on
Γ1 .
Next, note Υ(η) = S(W ) + wΠ and the formulas (3.22) and, after applying the
inequality (3.12) with U = W , we have
Dτ W 2L2 ([α, T −α];L2 (Γ,Λ))
(3.25)
≤ CT,α, (Wt 2L2 (ΣT ,Λ) + wt 2L2 (ΣT ) + Dwt 2L2 (ΣT ,Λ) ) + lot(η),
1
where the following inequality is used:
Wtt − ∆µ W 2H −1/2 (QT ,Λ) = (1 − µ)kW + F(w)2H −1/2 (QT ,Λ)
≤ C(1 − µ)kW + F(w)2L2 (QT ,Λ) = lot(η).
(3.26)
Let us consider the component w. We now prove that there is a constant C > 0
such that
T
D2 w2H −s0 (QT ,T 2 ) ≤ C
w2H 2−s0 (Ω) dt
(3.27)
∀ w ∈ H −s0 (QT ).
0
For simplicity, we assume that Ω is a coordinate U with the coordinate system x =
(x1 , x2 ). Then QT = (0, T ) × U. Denote the Fourier transform variable of (t, x) by
(ζ0 , ζ). By definition,
H −s0 (QT ) = (H0s0 (Q))∗ ,
H0s0 (QT ) = { w | w ∈ H s0 (R3 ), supp w ⊂ QT }.
For u ∈ H −s0 (QT ) given, we then have
u2H −s0 (QT ) = (1 + |ζ0 |2 + |ζ|2 )−s0 /2 û2L2 (R3 )
≤ (1 + |ζ|2 )−s0 /2 û2L2 (R3 )
=
(1 + |ζ|2 )−s0
|û|2 dζ0 dζ1 ζ2
R2
R
=
R2
(3.28)
=
0
T
(1 + |ζ|2 )−s0
T
0
|ûx |2 dt dζ1 ζ2
u2H −s0 (Ω) dt
since supp u ⊂ QT , where ûx denotes the Fourier transform of u with respect to the
variable x. In addition, for i = 1, 2,
2
2
2ˆw ∂ w 2
s0
∂
= (1 + |ζ|2 )− 2
∂xi ∂xj −s
∂x
∂x
i
j
H 0 (Ω)
(3.29)
≤ (1 + |ζ|2 )
2−s0
2
ŵ2 = w2H 2−s0 (Ω) .
BOUNDARY FEEDBACK STABILIZATION OF SHALLOW SHELLS
257
The inequality (3.27) follows from the inequalities (3.28) and (3.29). The same argument leads us to the following:
T
2
DW H −s0 (QT ,T 2 ) ≤ C
(3.30)
W 2H 1−s0 (Ω,Λ) dt,
0
and
2
∂
(η˘t F5τ )
∂τ
(3.31)
H −1 (QT )
≤ Cη˘t F5τ 2L2 (QT ) .
Apply the inequality (3.20), together with the inequalities (3.27), (3.30), and
(3.31), to obtain
2
2
(3.32)
(|ηt |2 + |Dwt |2 ) dΣ + lot(η).
D wL2 ([α,T −α];L2 (Γ1 ,T 2 )) ≤ Cα,,T
ΣT
1
The inequality (3.21) follows from the inequalities (3.24), (3.25), and (3.32).
Proof of Theorem 1.1. First, we make some estimates for the terms in the righthand side of the inequality (3.2).
Let s1 , s2 > 0 be such that
s1 |ν|2 ≤ νF ν τ ≤ s2 |ν|2
(3.33)
∀ ν ∈ R5 .
Then the definition (2.3) of a1 (·, ·) gives, for ς = (U, u),
(3.34)
(|ς|2 + |Du|2 ) dΓ ≤ a1 (ς, ς) ≤ s2
(|ς|2 + |Du|2 ) dΓ.
s1
Γ1
Γ1
Use of the right-hand side of the inequality (3.34), therefore, yields
|a1 (ηt , m(η) + p(η))|
≤ [a1 (ηt , ηt )]1/2 [a1 (m(η) + p(η), m(η) + p(η))]1/2
≤C
(|ηt |2 + |Dwt |2 + |m(η) + p(η)|2 + |D(V (w) − b/2w)|2 ) dΓ
Γ1
≤C
Γ1
(3.35)
=C
Γ1
[|ηt |2 + |Dwt |2 + |DW |2 + |W |2 + |D2 w|2 + |Dw|2 + w2 ] Γ
(|ηt |2 + |Dwt |2 + |DW |2 + |D2 w|2 ) dΓ + l(η).
In addition, we have
B(η, η) ≤ C(|DW |2 + |D2 w|2 + w2 )
(3.36)
on
Γ1
and, from the geometrical condition (H.3),
(3.37)
B(η, η)V, n dΣ ≤ 0.
ΣT
0
Now we substitute the inequalities (3.35)–(3.37) into the inequality (3.2) to obtain
T
σ1
E(t) dt ≤ C
(|ηt |2 + |Dwt |2 + |DW |2 + |D2 w|2 ) dΣ
0
(3.38)
ΣT
1
+ CT (E(0) + E(T )) + lot(η).
258
SHUGEN CHAI, YUXIA GUO, AND PENG-FEI YAO
Next, change the integral domain ΣT1 into [α, T − α] × Γ1 in both sides of the
inequalities (3.38) and use the inequality (3.21) to give
(3.39)
σ1
T −α
α
E(t) dt ≤ CT
E(α) + E(T − α) +
ΣT
1
2
2
(|ηt | + |Dwt | ) dΣ
+ lot(η).
Note the relation E (t) = −a1 (ηt , ηt ) and the right-hand side of the inequality
(3.34) again, and we find, for any T > β > 0,
(3.40)
E(β) = E(T ) +
T
β
a1 (ηt , ηt ) dt ≤ E(T ) + C
ΣT
1
(|ηt |2 + |Dwt |2 ) dΣ.
Using the inequality (3.40) in the inequality (3.39), we obtain, for T > 0 suitably
large,
E(T ) ≤ CT
(3.41)
(|ηt |2 + |Dwt |2 ) dΣ + lot(η).
ΣT
By the compactness and uniqueness (Proposition 2.13 of Yao [21]) approach, we now
have
(3.42)
(|ηt |2 + |Dwt |2 ) dΣ.
E(T ) ≤ CT
ΣT
1
Finally, using the inequality (3.42) and the left-hand side of the inequality (3.34)
leads to
T
(3.43)
a1 (ηt , ηt ) dt = CT (E(0) − E(T )),
E(T ) ≤ CT
0
that is,
(3.44)
E(T ) ≤
CT
E(0).
1 + CT
Theorem 1.1 follows from the inequality (3.44).
Acknowledgments. The authors would like to thank Professors I. Lasiecka and
R. Triggiani for picking up two mistakes in the first version of this paper and for some
advice.
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