2
Internal stabilization
2.1
Internal stabilization of linear Oseen–Stokes systems
We shall study here the stabilization of the linear equation
(2.1)
dy
+ Ay = P (1lO0 u), t ≥ 0,
dt
y(0) = y0,
where A is the Oseen–Stokes operator (1.9), as a first step toward stabilization of system (1.8). In a few words, the strategy is the following:
one designs an appropriate stabilizable feedback controller u = φ(y)
for (2.1), which is also stabilizable in equation (1.8) in a neighborhood
of the origin.
We denote again by A the extension of the operator A into the
e = H + iH. Since (λI − A)−1 is compact for
complexified space H
each λ ∈ ρ(A) (the resolvent of A) the spectrum σ(A) consists of a
countable set of eigenvalues {λj }∞
j=1 and for each γ there exists a finite
number N of eigenvalues such that Re λj ≤ γ. In the following, each
λj is repeated according to each algebraic multiplicity mj . Let {ϕj }∞
j=1
k
be the eigenfunction system, that is, Aϕj = λj ϕj or (A − λj ) ϕj = 0,
k = 1, ..., mj . The eigenvalue λj is said to be simple if mj = 1
and semi-simple if the algebraic multiplicity coincides with geometric
multiplicity.
Denote by ϕ∗j the dual eigenfunctions system, that is, A∗ϕ∗j = λj ϕ∗j ,
where A∗ is the dual Oseen–Stokes operator.
It turns out that the property of an eigenvalue λj to be simple
(and, in particular, semi-simple) is generic. More precisely, we have
the following lemmas.
1
Lemma 2.1 ([2], [3]) The set M = {ye ∈ D(A), all eigenvalues
λj are simple} it is a countable intersection of open and dense
sets.
We have also the following unique continuation property of eigenfunctions system (see [2], p. 164).
∗ N
Lemma 2.2 Let O0 be an open subset of O and {ϕj }N
j=1 , {ϕj }j=1
be eigenfunctions of A and A∗, respectively. Then, {ϕ∗j }N
j=1 and
{ϕ∗j }N
j=1 are linearly independent on O0 .
Everywhere in the following, {ϕj }N
j=1 is a system of eigenfunctions
mj
for the operator A, that is, ϕj = {ϕjk }k=1
, where
(2.2)
(A − λj I)k ϕjk = 0 for k = 1, ..., mj , j = 1, ..., `,
where ` is the number of distinct eigenvalues and mk is the algebraic
multiplicity of eigenvalue λk (repeated according its algebraic multiplicity). Here, N is chosen by the condition
(2.3)
Re λj ≤ γ,
j = 1, ..., N,
and γ > 0 is arbitrary but fixed.
In the following, we denote by M the number
(2.4)
M = max{mj ; j = 1, 2, ..., `},
which has an important role in the stabilization procedure to be
described.
We set m = 1lO0 and | · |M the norm in RM .
2
Theorem 2.3 There is a controller u of the form
(2.5)
u(t) =
M
X
P (mφi)vi(t),
∀t ≥ 0,
i=1
which stabilizes exponentially system (2.1).
e to system (2.1)
More precisely, the solution y ∈ C([0, ∞); H)
with control u given by (2.5) satisfies
|y(t)|He ≤ Ce−γt|y 0|,
(2.6)
t ≥ 0.
2
M
The controller v = {vi}M
i=1 can be chosen in L (0, T ; C ) and such
that
Z T
(2.7)
|vi(t)|2M dt ≤ C|y0|2, vi(t) = 0
for t ≥ T.
0
Here, [0, T ] is an arbitrary interval and {φi}M
i=1 ⊂ D(A) is a system of functions which is made precise below.
1
The controller v = {vj }M
j=1 can also be found as a C -function
on [0, ∞) such that
(2.8)
|vj (t)| + |vj0 (t)| ≤ Ce−γt|y0|, ∀t ≥ 0, j = 1, ..., M.
Proof. First, we prove the theorem in the special case where
1◦ the eigenvalues λj , j = 1, ..., N , are semisimple.
∗ N
As seen earlier, in this case the systems {ϕj }N
j=1 and {ϕj }j=1 can
be chosen biorthogonal, that is,
(2.9)
(ϕi, ϕ∗j ) = δij ,
i, j = 1, ..., N.
We take in (2.5)
(2.10)
φi = ϕ∗i ,
i = 1, ..., N,
3
and we are decoupling the system
M
X
dy
+ Ay =
P (mϕ∗i )vi,
dt
i=1
t ≥ 0,
y(0) = y0,
in the finite-dimensional part
M
X
dyu
+ Auyu = PN
P (mϕ∗i )vi,
dt
i=1
(2.11)
yu(0) = PN y0,
and the infinite-dimensional γ-stable part
M
(2.12)
X
dys
+ Asys = (I − PN )
P (mϕ∗i )vi,
dt
i=1
ys(0) = (I − PN )y0.
¯
¯
e =
¯
Here, yu = i=1 yiϕi and Au = A Xu , As = A¯Xs , Xu = PN (H)
e
lin span{ϕj }N , Xs = (I − PN )H.
PN
j=1
By (2.9), system (2.11) can be rewritten as
M
(2.13)
X
dyj
+ λj yj =
(ϕ∗i , ϕ∗j )0vi,
dt
i=1
j = 1, ..., N,
yj (0) = yj0 = (PN y0, ϕ∗j ),
where (·, ·)0 is the scalar product in (L2(O0))d. This yields
(2.14)
z 0(t) + Λz(t) = Bv(t),
t ≥ 0,
z(0) = z0,
where
M,N
0 N
∗
∗
z(t) = {yj (t)}N
j=1 , z0 = {(y)j }j=1 and B = k(ϕi , ϕj )0 ki,j=1
4
while Λ is the diagonal N × N matrix
°
°
°J
°
° 1
°
°
°
°
J2
0 °
°
Λ=°
° 0
°
...
°
°
°
°
°
J` °
where Jj is the mj × mj diagonal matrix
°
°
°λ
°
° j
°
°
°
°
λj
0 °
°,
Jj = °
j = 1, ..., `.
° 0
°
...
°
°
°
°
°
λj °
Recall that ` is the number of distinct eigenvalues λj with Re λj ≤γ.
Then, by Kalman’s controllability theorems, it follows that system
(2.14) is exactly null controllable on (0, T ), where T > 0 is arbitrary
but fixed. Indeed, the equation
B ∗e−Λtx = 0,
t ≥ 0,
implies that
(2.15)
m1
X
i=1
bij xi = 0,
m2
X
bij xi = 0, ...,
i=m1 +1
m
X̀
bij xi = 0,
i=m`−1 +1
∗
∗
where x = {xi}N
i=1 , bij = (ϕj , ϕi )0 , i = 1, ..., N, j = 1, ..., M.
Since the system {ϕ∗j }M
j=1 is linearly independent on O0 , we have
that rankkbij kN,M
i,j=1 = M and so, we conclude by (2.15) that x = 0,
as claimed. Hence, by the Kalman controllability theorem there is a
M
control input {vj }M
j=1 ⊂ C([0, T ]; C ) such that yu (T ) = 0. Moreover, by the linear finite-dimensional controllability theory, we know
5
that {vj }M
j=1 can be chosen in such a way that
Z T
|vj (t)|2dt ≤ C|z0|2,
∀t ∈ [0, T ], j = 1, ..., M.
0
From the exact null controllability of (2.14) on [0, T ] it follows also,
via the linear quadratic stabilization technique, the existence of a stabilizable controller vj ∈ C 1([0, ∞)), j = 1, ..., M, satisfying (2.8).
Now, recalling that σ(As) = {λj ; Re λj > γ} and that −As
generates a C0-analytic semigroup, we have that
−γt
ke−AstkL(H,
,
e H)
e ≤ Ce
∀t ≥ 0,
and so, by (2.12) we have
|y(t, y0)He
(2.16)

≤ C e−γt|y0| +
Z tX
M
0 j=1

|e−A2(t−s)P (mϕ∗j )vj (s)|ds
≤ Ce−γt|y0|,
as claimed. This completes the proof.
2◦ We consider now the general case of non-semisimple eigenvalues
λj , j = 1, ..., N. By the Gram–Schmidt orthogonalization algorithm,
we may replace {ϕj }N
j=1 by an orthonormal system again denoted by
{ϕj }N
j=1 . Then, we take in (2.1) the controller u again of the form
(2.5), where {φi}M
i=1 is specified later on.
N
X
Setting as above yu =
yiϕi, we have also in this case y = yu+ys,
i=1
where
N
(2.17)
M
X
dyi X
+
aij yi =
(P (mφj ), PN∗ ϕi)vj , i = 1, ..., N,
dt
j=1
j=1
yi(0) = yi0 = (y0, ϕi),
6
(2.18)
M
X
dys
+ Asys = (I − PN )
P (mφj )vj ,
dt
j=1
ys(0) = (I − PN )y0.
As in the previous case, it suffices to show that the finite-dimensional system (2.17) is exactly null controllable on some interval [0, T ].
Though in this case the matrix A0 = kaij kN
i,j=1 is not the Jordan
matrix associated with {λj }N
j=1 there is, however, a nonsingular matrix
0 0 0 −1
Λ0 = kγij kN
is. Then, system (2.17)
i,j=1 such that J = Λ A (Λ )
can be written as
dz
+ Jz = Dv,
t ≥ 0,
(2.19)
dt
z(0) = z0,
where z = Λ0yu and
∗
∗ ∗
D = Λ0B, B = kbij kN,M
i,j=1 , bij = (P (mφj ), PN ϕi ) = (φj , PN ϕi )0 .
We have the following lemma.
Lemma 2.4 There is a system {φj }M
j=1 of the form
(2.20)
φj =
N
X
αjk PN∗ ϕk ,
j = 1, ..., M,
k=1
such that the finite-dimensional system (2.17) (equivalently, (2.19))
is exactly null controllable on each interval [0, T ].
Proof. If M = N , it suffices to take φj = PN∗ ϕj , j = 1, ..., N.
Indeed, since the system {PN∗ ϕj }N
j=1 is linearly independent on O0 we
have in this case that det B 6= 0 and so det D 6= 0, too. This, clearly,
implies that system (2.19) is exactly null controllable on [0, T ].
7
Assume now that M given by (2.4) is less than N . The Jordan
matrix J has, therefore, the following form
J = {λ1Em̃1 + Hm̃1 , ..., λ`Em̃` + Hm̃` },
where Em̃i is a unitary matrix of order m̃i ≤ mi and Hm̃i is a m̃i × m̃i
matrix of the form
°
°
° 0 1 0 ··· 0 °
°
°
°
°
° 0 0 1 ··· 0 °
Hm̃i = °
°
° 0 0 0 ··· 1 °
°
°
° 0 0 0 ··· 0 °
(Some of λj might be repeated.)
If we set Ji = λiEm̃i + Hm̃i , then J is the matrix
°
°
°
°J
°
° 1
°
°
°
J2
0 °
°
(2.21)
J =°
°
° 0
...
°
°
°
°
°
J` °
For simplicity, we assume that the blocks J` are distinct, that is,
λj 6= λi and m̃i = mi. (The general case follows in a similar way.)
We start with the equation
(2.22)
∗
D∗e−J tx = 0,
∀t ≥ 0,
where ∗ stands for adjoint operation.
Taking into account that
°
°
° e−J1∗tx1 °>
°
°
−J ∗ t
°
e
x = ° .................. °
° ,
∗
° e−J` tx` °
8
where
and
(2.23)




x1
xm`−1
x1 =  ...  , ..., x` =  ... 
xm1
xm`
°
°M,N
N
°X
°
°
°
∗
∗ ∗
D =B Λ =°
bjk γki°
= kd∗ij kM,N
i=1,j=1 .
°
°
k=1
We have
i,j=1
°
° xm
° k−1
°
° txmk−1 + xmk−2
°
−Jk∗ t k
e
x = ° ...............................................
° m
° t k−1
°
° (mk−1) xmk−1 + · · · + x(mk−1+mk )
°
°
°
°
°
°
°
°
°
°
°
for all k = 1, 2, ..., `. Then, by (2.22) we have that
(2.24)
m1
X
d∗kj xj
= 0,
mX
1 +m2
d∗kj xj
j=m1 +1
j=1
= 0, ...,
X̀
d∗kj xj = 0,
j=m`−1
k = 1, ..., M, and this implies, once again, x = 0.
For future convenience, we reformulate Theorem 2.3 in the real
space H. Taking into account that
y(t) = Re y(t)+i Im y(t), φj = Re φj +i Im φj , vj = Re vj +i Im vj ,
j = 1, ..., M ∗, we infer by Theorem 2.3 that there is a controller
u∗ : [0, ∞) → H of the form
∗
(2.25)
u∗(t) =
M
X
P (m Re φj )Re vj (t) − P (m Im φj )Im vj (t),
j=1
9
which exponentially stabilizes the real system
dy
+ Ay = u∗, t ≥ 0,
dt
y(0) = y0.
(2.26)
Here, 1 < M ∗ ≤ N and
φj = ϕ∗j , j = 1, ..., M ∗,
if the spectrum is semisimple, while
∗
φj ∈ lin span{PN∗ ϕi}N
i=1 , j = 1, ..., M ,
in the general case. (See (2.20).)
In particular, if all the eigenvalues λj , j = 1, ..., N, are real, then
M ∗ = M , while, if all the eigenvalues λj are simple, then M ∗ = 2
(It follows also that M ∗ = 1 if all the eigenvalues are real and simple).
Therefore, by Theorem 2.3 we have the following result.
Theorem 2.5 There is a real controller u∗ of the form
∗
(2.27)
u∗(t) =
M
X
P (mψj )vj∗(t)
j=1
such that the corresponding solution
(2.28) y ∗ ∈ C([0, T ; H) ∩ L2(0, T ; V ) ∩ L2loc(0, T ; D(A)), ∀T > 0,
to (2.26) satisfies the estimate
(2.29)
|y ∗(t)| ≤ Ce−γt|y0|, ∀t ≥ 0,
and
vj ∈ C 1[0, ∞),
|vj (t)|+|vj0 (t)| ≤ Ce−γt|y0|,
10
j = 1, ..., M ∗, t > 0.
Here,
∗
2
d
1
d
∗
N
{ψj }M
j=1 ∈ (H (O)) ∩ (H0 (O)) ∩ H, ψj ∈ lin span{PN ϕk }k=1 ,
j = 1, ..., N.
If all λj are semisimple, then
ψj = Re ϕ∗j for 1 ≤ j ≤
M∗
2 ,
ψj = Im ϕ∗j for
M∗
2
< j < M ∗.
Remark 2.6 We see that the structure of the stabilizing controller
is quite simple if the spectrum {λj } is semisimple and if all the eigen∗
values {λj }N
j=1 are simple, then the controller u is two-dimensional,
that is,
u∗(t) = P (m Re ϕ∗1 )v1∗(t) + P (m Im ϕ∗1 )v2∗(t).
Taking into account that the property of eigenvalues λj to be simple
is a generic one, we may conclude that for “almost all” steady-state
solutions ye the stabilizable controller u∗ is of the above form and this
fact allows us to simplify the numerical construction of the controller.
The stabilizable controller found in Theorem 2.3 is an open-loop
controller only, but it will be used below to design a stabilizable feedback control.
Everywhere in the following, (·, ·) is the scalar product in the space
1
H or in a duality pair with H as pivot space. We set V = D(A 2 ),
1
W = D(A 4 ) and denote by k · k the norm in V . By (·, ·)0 denote the
scalar product in (L2(O0))d.
11
Theorem 2.7 Let γ > 0 and M ∗ and N as in Theorems 2.3 or
2.5. Then there is a linear self-adjoint operator R : D(R) ⊂ H→H
such that for some constants 0 < a1 < a2 < ∞ and C1 > 0,
1
1
1
a1|A 4 y|2 ≤ (Ry, y) ≤ a2|A 4 y|2, ∀y ∈ D(A 4 );
(2.30)
(2.31)
|Ry| ≤ C1kyk, ∀y ∈ V ;
∗
(2.32)
M
1X
(νAy + A0y − γy, Ry) +
2
(ψi, Ry)20 =
i=1
1 3 2
|A 4 y| ,
2
∀y ∈ D(A).
Moreover, the feedback controller


∗
M
X
∗
(2.33)
u (t) = −P m
(Ry(t), ψi)0ψi
j=1
exponentially stabilizes the linear system (2.1), that is, the solution
y to the corresponding closed-loop system satisfies
ky(t)kW ≤ e−γtky0kW ,
∀y0 ∈ W
Z ∞
3
e2γt|A 4 y(t)|2dt ≤ Cky0kW .
(2.34)
(2.35)
0
M∗
Here, {ψi}i=1 are as in Theorem 2.5.
Proof. We consider the optimization problem
Z
1 ∞ 3
(2.36)
ϕ(y0) = Min
(|A 4 y(t)|2 + |u(t)|2M ∗ )dt,
2 0
∗
subject to u ∈ L2(0, ∞; RM ) and
(2.37)
y 0 + νAy + A0y − γy = P
Ã
m
∗
M
X
i=1
12
!
ψiui ,
y(0) = y0.
1
Let us show first that ϕ(y0) < ∞, ∀y0 ∈ D(A 4 ). We set
à M∗
!
X
∗
Du = P m
ψiui , u ∈ RM .
i=1
∗
(Here, | · |M ∗ is the Euclidean norm in the space RM .) By Theorem
2.5 there is an admissible pair (y, u) such that y ∈ L2(0, ∞; H) ∩
L2loc(0, ∞; D(A)). For such a pair, we have
1 d
|y(t)|2 + νky(t)k2 ≤ |b(y, ye, y)(t)| + |Du| |y(t)| + γ|y|2
2 dt
≤ C(|y(t)| ky(t)k + |y(t)|2 + |u(t)|2M ∗ )
ν
≤ ky(t)k2 + C1(|y(t)|2 + |u(t)|2M ∗ ), ∀t > 0.
2
1
Hence, y ∈ L2(0, ∞; V ). Next, we multiply equation (2.37) by A 2 y(t)
to obtain that
3
1 d 1
|A 4 y(t)|2 + ν|A 4 y(t)|2
2 dt
1
1
≤ |b(y(t), ye, A 2 y(t))| + |b(ye, y(t), A 2 y(t))|
1
1
+|Du(t)| |A 2 y(t)| + γ|A 2 y(t)| |y(t)|
1
1
≤ C(ky(t)k kyek 3 |A 2 y(t)| + kyek2ky(t)k |A 2 y(t)|
2
1
2
1
+|u(t)|M ∗ |A y(t)| + γ|A 2 y(t)| |y(t)|)
≤ C(ky(t)k2 + |u(t)|M ∗ ky(t)k), t > 0.
1
Integrating on (0, ∞), we see that ϕ(y0) < ∞, ∀y0 ∈ D(A 4 ).
Moreover, it follows from the previous equality that
1
1
1
α1|A 4 y0|2 ≤ ϕ(y0) ≤ α2|A 4 y0|2, ∀y0 ∈ D(A 4 ),
where αi > 0, i = 1, 2. Thus, there is a linear self-adjoint operator
R : D(R) ⊂ H → H such that R ∈ L(W, W 0) and
1
1
ϕ(y0) = (Ry0, y0), ∀y0 ∈ W = D(A 4 ).
2
13
In other words, ∇ϕ = R and (2.30) follows.
Let us prove (2.31) and (2.32). By the dynamic programming principle, for each T > 0, the solution (u∗, y ∗) to (2.36) is also the solution
to the optimization problem
½ Z T
¾
3
1
Min
(|A 4 y(s)|2+|u(s)|2M ∗ )ds+ϕ(y(T )), subject to (2.37)
2 0
and so, by the maximum principle,
(2.38)
∗
u∗(t) = {(qT (t), ψi)0}M
i=1 , a.e., t ∈ (0, T ),
where qT is the solution to the dual backward equation
3
(2.39)
qT0 − (νA + A0)∗qT + γqT = A 2 y ∗,
∀t ∈ (0, T ),
qT (T ) = −Ry ∗(T ).
Since T is arbitrary, we have
Ry ∗(t) = −qT (t),
(2.40)
∀t > 0,
and, therefore,
(2.41)
∗
u∗(t) = −{(Ry ∗(t), ψi)0}M
i=1 ,
∀t ≥ 0.
Now, let y0 ∈ V be arbitrary but fixed. By (2.37), multiplying by
Ay ∗, we have as above that
d ∗
ky (t)k2 + 2ν|Ay ∗(t)|2
dt
≤ 2(|b(ye, y ∗(t), Ay ∗(t))| + |b(y ∗(t), ye, Ay ∗(t))|)
+2|Du∗(t)| |Ay ∗(t)| + γky ∗(t)k2
≤ C(kyek2ky ∗(t)k |Ay ∗(t)| + |y ∗(t)| 1 kyek2|Ay ∗(t)|
2
+|Du∗(t)| |Ay ∗(t)| + γky ∗(t)k2).
14
This implies that
(2.42)
∗
Z
2
ky (t)k +
t
|Ay ∗(s)|2ds ≤ C(1 + ky0k2), ∀t ≥ 0.
0
Finally, to show that R is a solution to the Riccati equation (2.32),
we first notice that, again by the dynamic programming principle, we
have
Z ∞
3
1
ϕ(y ∗(t)) =
(|A 4 y ∗(s)|2 + |u∗(s)|2M ∗ )ds, ∀t ≥ 0.
2 t
³
´
d
∗
∗ dy ∗
This yields, by virtue of (2.41) that (recall that dt ϕ(y ) = Ry , dt )
µ
¶
M∗
∗
X
dy
1 3
1
Ry ∗(t),
(t) + |A 4 y ∗(t)|2 +
(Ry ∗(t), ψi)20 = 0, ∀t ≥ 0.
dt
2
2 i=1
This leads, via a standard device involving (2.37), to
−(Ry ∗(t), νAy ∗(t) + A0y ∗(t) − γy ∗(t))
∗
M
1X ∗
1 3
−
(Ry (t), ψi)20+ |A 4 y ∗(t)|2 = 0, ∀t ≥ 0,
2 i=1
2
which implies (2.32). In order to prove (2.34)–(2.35), it suffices to
multiply the closed-loop equation


∗
M
X
dy
+ Ay + P m
(Ry, ψi)0ψi = 0, t ≥ 0,
dt
j=1
by Ry and use equation (2.32). We get
1 d
1 3
(Ry(t), y(t)) + |A 4 y(t)|2 + γ(Ry(t), y(t))
2 dt
2
∗
M
1X
+
(Ry(t), ψi)20 = 0, ∀t ≥ 0,
2 i=1
which, by integration, yields (2.34), (2.35). This completes the proof.
15
Theorem 2.8 The feedback controller
∗
(2.43)
u=−
M
X
(R(y − ye), ψi)0ψi,
i=1
exponentially stabilizes the steady-state solution ye to (1.1) in a
neighborhood
Uρ = {y0 ∈ W ; k(y0 − ye)kW < ρ}
of ye for suitable ρ > 0. More precisely, if ρ > 0 is sufficiently
small, then, for each y0 ∈ Uρ, there exists a strong solution
y ∈ C([0, ∞); W ) ∩ C((0, ∞); V ) to the closed-loop system
à M∗
!
X
dy
+ νAy + By + P m
(R(y − ye), ψi)0ψi = P fe,
dt
i=1
(2.44)
t ≥ 0;
y(0) = y0,
√
√
2
such that t A(y − ye) ∈ L2(0, ∞; H), t dy
dt ∈ L (0, ∞; H) and
Z ∞
3
(2.45)
e2γt|A 4 (y(t) − ye)|2dt ≤ C2ky0 − yek2W ,
0
(2.46)
ky(t) − yekW ≤ C3e−γtky0 − yekW ,
∀t ≥ 0,
where C2, C3 > 0.
In particular, it follows that, for all y0 ∈ Uρ, equation (2.44) has a
unique strong solution y.
Proof of Theorem 2.8. By the substitution y − ye → y and
y 0 = y0 − ye, we reduce the closed-loop system (2.44) to
(2.47)
dy
+ νAy + A0y + By + P
dt
y(0) = y 0.
16
∗
M
X
i=1
(Ry, ψi)0ψi = 0, t > 0,
We are going to show that ϕ(y) = 12 (Ry, y) is a Lyapunov function
for system (2.47) in a neighborhood of the origin.
Equation (2.47) has at least one weak solution y ∈ Cw ([0, T ]; H) ∩
L2(0, T ; V ), ∀T > 0, given as limit of strong solutions yε ∈ C((0, T ); H)
∩L2(δ, T ; D(A)), ∀δ > 0, to the equation
à M∗
!
X
dyε
+νAyε + A0yε+Bεyε+P m (Ryε, ψi)0ψi =0,
dt
(2.48)
i=1
t > 0,
yε(0) = y 0,
where Bε is the truncated operator

if kyk ≤ 1ε ,
 By
Bεy =
By
 2
if kyk > 1ε .
2
ε kyk
In fact, one has
(2.49)
yε → y strongly in L2(0, T ; H), weakly in L2(0, T ; V ),
∀t > 0, and the following estimate holds
°
°4 !
Z tÃ
° dyε ° 3
2
2
|yε(t)| +
kyε(s)k + °
(s)°
°
° 0 ds
ds
(2.50)
0
V
≤ |y 0|2 + CT , ∀ε > 0.
Using Riccati equation (2.32), we obtain by (2.48) that
M
(2.51)
X
3
d
(Ryε, yε) +
(Ryε, ψi)20 + |A 4 yε|2 + 2γ(Ryε, yε)
dt
i=1
= −2(Bεyε, Ryε),
17
a.e. t > 0.
On the other hand, we have, by standard estimate on Navier–Stokes
equations in 3 − D
|(Bεyε, Ryε)| ≤ |b(yε, yε, Ryε)| ≤ Ckyεk |yε| 3 |Ryε|
4
3
4 ε
3
4 ε 2
≤ Ckyεk |A y | |Ryε| ≤ C|A y | kyεk2
(2.52)
3
1
≤ C|A 4 yε|2(Ryε, yε) 2 ,
where the various constants C are independent of ε.
We obtain for (Ry 0, y 0) ≤ ρ and ρ sufficiently small that on the
maximal interval (0, Tε), where (Ryε(t), yε(t)) ≤ ρ, we have
1 3
d
(Ryε(t), yε(t)) + 2γ(Ryε(t), yε(t)) + |A 4 yε(t)|2 ≤ 0.
dt
2
We have, therefore, by (2.30) that
Z Tε
1
3
|A 4 yε(t)|2e2γtdt ≤ C|A 4 y 0|2
0
and
1
1
kyε(t)k2W = |A 4 yε(t)|2 ≤ Ce−2γt|A 4 y 0|2, ∀t ∈ (0, Tε),
where C is independent of ε. Recalling (2.30), the latter implies that,
for ρ sufficiently small, Tε = ∞ for all ε > 0 and so, the previous
estimates extend to (0, ∞).
On the other hand, if we multiply (2.48) by t Ayε(t) and integrate
on (0, t), we obtain that
Z t
Z t
1
1
t(Ayε(t), yε(t)) +
(Ayε(s), yε(s))ds
s|Ayε(s)|2ds ≤
2
2
0
0
Z t
+
s(b(yε(s), ye, Ayε(s)) + b(ye, yε(s), Ayε(s)))ds
Z0 t
Z t
+
sb(yε(s), yε(s), Ayε)ds + C
s|Ayε(s)| |Ryε(s)|ds, ∀t ≥ 0.
0
0
18
Recalling that
|b(yε, ye, Ayε)| + |b(ye, yε, Ayε)| + |b(yε, yε, Ayε)|
≤ C|Ayε|(|ye| 1 |yε|1 + |ye|1|yε| 1 + |yε| 1 |yε| 3 ),
2
2
2
4
where |y|α = |Aα y|, we obtain by the previous estimates that
Z t
tkyε(t)k2 +
s|Ayε(s)|2ds ≤ Cky 0k2W , ∀ε > 0, t ≥ 0,
0
and so, letting ε tend to zero, we obtain that
Z t
tky(t)k2 +
s|Ay(s)|2ds ≤ Cky 0k2W ,
Z ∞ 0
1
|A 4 y(t)|2e2γtdt ≤ Cky 0k2W ,
∀t ≥ 0,
0
kykW ≤ Ce−γtky 0kW , ∀t > 0,
for all y 0 ∈ W such that (Ry 0, y 0) ≤ ρ0.
This implies also that y is strong solution to (2.47) with
y ∈ C([0, ∞); W ) ∩ C((0, ∞); V ).
√ dy
√
t Ay ∈ L2(0, ∞; H),
t
∈ L2(0, ∞; H).
dt
Then y − ye satisfies the conditions of Theorem 2.8. This completes
the proof.
3
Comments
Theorems 2.3, 2.5, 2.8 are taken from [3], Chap. 3, but in a slight
different form were established in [2], where a more precise description
of the dimension M ∗ is given. The idea of the proof of Theorem 2.8
was earlier used in [1].
19
References
[1] V. Barbu, Feedback stabilization of Navier–Stokes equations,
ESAIM COCV, 9 (2003), 197-206.
[2] V. Barbu, Stabilization of Navier–Stokes Flows, Springer, New
York, 2010.
[3] V. Barbu, R. Triggiani, Internal stabilization of Navier–Stokes
equations with finite dimensional controllers, Indiana Univ.
Math. J., 53 (2004), 1443-1494.
[4] V. Barbu, I. Lasiecka, R. Triggiani, The unique continuation
property of eigenfunctions to Stokes–Oseen operator, Nonlinear
Analysis (to appear).
20