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4
Stabilization by noise of the Navier–Stokes equation
Consider the Navier-Stokes equation

Xt − ν0∆X + (X · ∇)X = fe + ∇p,



 ∇ · X = 0, in (0, ∞) × O,
(4.1)

X = 0, on (0, ∞) × ∂O,



X(0) = x0, in O.
in (0, ∞) × O,
where O is an open and bounded subset of Rd, d = 2, 3, with smooth
boundary ∂O. Here fe ∈ (L2(O))d is given. Let Xe = ye be an
equilibrium solution to (4.1), i.e.,


 −ν0∆Xe + (Xe · ∇)Xe = fe + ∇pe, in O,
(4.2)
∇ · Xe = 0 in O,

 X = 0 on (0, ∞) × ∂O.
e
As seen earlier, if we replace X by X − Xe equation (4.1) reduces to


Xt − ν0∆X + (X·∇)Xe+(Xe·∇)X+(X·∇)X = ∇p





in (0, ∞) × O,

(4.3)
∇ · X = 0 in O,



X = 0 on (0, ∞) × ∂O,



 X(0) = x in O,
where x = x0 − Xe. We can rewrite system (4.3) as
(
Ẋ(t) + AX(t) + B(X(t)) = 0, t ≥ 0,
(4.4)
X(0) = x,
where A and B were defined earlier in (1.9).
1
To stabilize (4.4), we associate the controlled stochastic system

N

X

 dX(t) + (AX(t) + B(X(t))dt =
Vj (t)ψj dβj (t),
(4.5)
j=1


 X(0) = x,
where {βj }N
j=1 is an independent system of real Brownian motions in
a filtered probability space (Ω, P, F, {Ft}t>0).
The main result, Theorems 4.1 below, amounts to saying that,
e associated with H, under appropriate
in the complexified space H
assumptions on A (and, implicitly, on Xe), for each γ > 0 there
e
exist N ∈ N, {ψj }N
j=1 ⊂ H, and an N -dimensional adapted process
{Vj = Vj (t, ω)}N
j=1 , ω ∈ Ω, such that for all x in a sufficiently small
γt
neighbourhood of the origin, t → e 4 X(t, ω) is decaying to zero for
t → ∞ in a set Ω∗x of positive probability which is precisely estimated.
Moreover, it turns out that the stabilizable controller arising in the
right hand side of (4.5) is a linear feedback controller of the form
(4.6)
Vj (t) = ηhX(t), ϕ∗j iHe ,
ψj = P (mφj ),
j = 1, ..., N,
where |η| > 0 and ϕ∗j are the eigenfunctions of the dual Stokes-Oseen
operator A∗ corresponding to eigenvalues λj with Re λj ≤ γ, {φj }N
j=1
∗
is a system of functions related to ϕj and m = 1lO0 is the characteristic
function of O0 where O0 is a given arbitrary open subset of O.
We may view (4.5) as the deterministic system (4.4) perturbed by
P
the white noise controller N
j=1 Vj (t)ψj β̇j with the support in O0 .
Notations
Throughout in the following βj , j = 1, ..., N are independent real
Brownian motions in a filtered probability space (Ω, P, F, {Ft}t>0)
and we shall refer to [7] and [8] for definition and basic results on
2
stochastic analysis of differential systems and spaces of stochastic processes adapted to filtration {Ft}t>0. The scalar product of H is dee the complexnoted h·, ·iH and the norm | · |H . We shall denote by H
ified space H + iH with scalar product denoted by h·, ·iHe and norm
e is the space of all adapted square-mean
by | · |He . CW ([0, T ]; L2(Ω, H))
e
H-valued
continuous processes on [0, T ].
The main result
To begin with, let us briefly recall a few elementary spectral properties of the Stokes-Oseen operator A already mentioned in Section 1.
e The
Denote again by A the extension of A to the complex space H.
operator A has a compact resolvent (λI − A)−1 and −A generates a
e Consequently, A has a countable
C0-analytic semigroup e−At in H.
number of eigenvalues {λj }∞
j=1 with corresponding eigenfunctions ϕj
each with finite algebraic multiplicity mj . Of course, certain eigenfunctions ϕj might be generalized and so, in general, A is not diagonalizable, i.e., the algebraic multiplicity of λj might not coincide with
its geometric multiplicity. Also, each eigenvalue λj will be repeated
according to its algebraic multiplicity mj .
We shall denote by N the number of eigenvalues λj with Re λj ≤ γ,
j = 1, ..., N, where γ is a fixed positive number.
Denote by PN the projector on the finite dimensional subspace
Xu = lin span{ϕj }N
j=1 .
e and
We have Xu = PN H
(4.7)
1
PN = −
2πi
Z
(λI − A)−1dλ,
Γ
where Γ is a closed smooth curve in C which is the boundary of a
domain containing in interior the eigenvalues {λj }N
j=1 .
3
Let Au = PN A, As = (I − PN )A. Then Au, As leave invariant
e and the spectra σ(Au), σ(As)
the spaces Xu and Xs = (I − PN )H
are given by
∞
σ(Au) = {λj }N
j=1 , σ(As ) = {λj }j=N +1 .
Since σ(As) ⊂ {λ ∈ C; Re λ > γ} and As generates an analytic
e we have
C0-semigroup on H,
(4.8)
e t ≥ 0.
|e−Astx|He ≤ Ce−γt|x|He , ∀x ∈ H,
Herein, we shall assume that the following hypothesis holds.
(H1) All eigenvalues λj , j = 1, ..., N, are semi-simple.
Denote by ϕ∗j the eigenfunction of A∗ corresponding to the eigenvalue λj . We have, therefore,
(4.9)
Aϕj = λj ϕj , A∗ϕ∗j = λj ϕ∗j , j ∈ N.
Since the eigenvalues {λj }N
j=1 are semi-simple, it turns out that the
∗ N
system consisting of {ϕj }N
j=1 , {ϕj }j=1 can be chosen to form a
e i.e,
bi-orthonormal sequence in H,
(4.10)
hϕj , ϕ∗k iHe = δjk ,
j, k = 1, ..., N,
where δjk is the Kronecker symbol. We notice also that the functions
ϕj and ϕ∗j have the unique continuation property, i.e.,
(4.11)
ϕj 6≡ 0, ϕ∗j 6≡ 0 on O0 for all j = 1, ..., N.
Consider the following stochastic perturbation of the system (4.4)
considered in the complex space

N

X

 dX + (AX + B(X))dt = η
hX, ϕ∗j iHe P (mφj )dβj ,
(4.12)
j=1


 X(0) = x,
4
where |η| > 0 and m = 1lO0 is the characteristic function of the open
e
subset O0 ⊂ O. Here {φj }N
j=1 ⊂ H is a system of functions to be
specified later on.
In two dimensions the stochastic differential equation (4.12) has a
e for all T > 0.
global solution X ∈ CW ([0, T ]; L2(Ω, H))
The closed loop system (4.12) can be, equivalently, written as


dX(t) − ν0∆X(t)dt+(X(t) · ∇)Xedt





+(Xe · ∇)X(t)dt+(X(t) · ∇)X(t)dt




N

X



hX(t), ϕ∗j iHe φj dβj (t)
= ηm
(4.13)
j=1



+∇p(t)dt in (0, ∞) × O, P-a.s.


¯


¯


∇
·
X(t)
=
0
in
O,
X(t)
¯ = 0, ∀t ≥ 0, P-a.s.


∂O


 X(0) = x in O.
Hence, in the space (L2(O))d, the feedback controller
{uj = ηmhX, ϕ∗j iHe φj }N
j=1
has the support in O0.
We shall define now φj , j = 1, ..., N, as follows.
(4.14)
φj (ξ) =
N
X
αlj ϕ∗l (ξ), ξ ∈ O,
l=1
where αlj are chosen in such a way that
N
X
αlj hϕ∗l , ϕ∗k i0 = δjk ,
j, k = 1, ..., N.
l=1
(By Lemma 2.2, the system {ϕ∗` }N
`=1 is linear independent on O0 and
so, the Gram matrix {hϕ∗l , ϕ∗k i0}N
l,k=1 is not singular, which makes this
5
possible.) With this choice, we have
hφj , ϕ∗k i0 = δkj , k, j = 1, ..., N.
R
Here, we have used the notation hu, vi0 = O0 u(ξ)v̄(ξ)dξ.
In the following, we shall denote by Aα , α ∈ (0, 1), the fractional
power of order α of A, by D(Aα ) its domain and set |x|α = |Aα x| for
1
all x ∈ D(Aα ). Moreover, we shall denote by W the space D(A 4 ) if
1
d = 2 and D(A 4 +²) if d = 3 where ² > 0 is small.
(4.15)
Theorem 4.1 Let d = 2, 3, Xe ∈ C 2(O) and
p
(4.16)
|η| ≥ max 6γ − 2Re λj .
1≤j≤N
Then, there is C ∗ > 0 independent of ω such that, for each x ∈ W ,
|x|W ≤ (C ∗)2, there is Ω∗x ⊂ Ω with
³
´− γ 2
1
−
2(ηN )
(4.17)
P(Ω∗x) ≥ 1 − 2 C ∗|x|W2 − 1
,
the solution X(t, x) to (4.12) satisfies
´
³ γt
(4.18)
lim e 4 |X(t, x)|He = 0,
t→∞
P-a.s. in Ω∗x.
In particular, Theorem 4.1 implies that, if |x|W ≤ ρ0 < (C ∗)−2,
then X = X(t, x) is exponentially decaying to 0 on a set Ω∗x of probability greater than
³
´− γ 2
− 21
2(ηN )
∗
1 − 2 C |x|W − 1
.
The constant C ∗ depends of Xe only.
For the linearized Navier–Stokes equation, that is, if one takes
B = 0, the exponential decay in (4.12) occurs with probability one.
In fact, as seen from the proof of Theorem 4.1, the constant C ∗ comes
6
out from estimates on the nonlinear inertial term B and so, it is zero
if this term is absent from the equation.
As mentioned earlier, system (4.13) is written here in the complex
e
space H.
If we set X1(t) = Re X(t), X2(t) = Im X(t), it can be rewritten as
a real system in (X1, X2). In this case, the feedback controller is an
implicit stabilizable feedback controller with support in O0 for the real
Navier–Stokes equation (4.3). Of course, if λj , j = 1, ..., N , are real,
then we may view X(t) as a real valued function and so, in (4.17),
|X|He = |X|H .
In particular, by Theorem 4.1 we have the following corollary.
Corollary 4.2 Under the assumptions of Theorem 4.1 the feedback controller
(4.19)
ηm
N
X
hX − Xe, ϕ∗j iHe φj
j=1
stabilizes exponentially the stationary solution Xe, P-a.e in Ω∗x.
Proof of Theorem 4.1
The idea of the proof is to transform equation (4.12) in a deterministic equation with random coefficients via substitution
(4.20)
y(t) =
N
Y
e−βj (t)Γj X(t),
t ≥ 0,
j=1
e →H
e is the linear operator
where Γj : H
(4.21)
Γj x := ηhx, ϕ∗j iHe P (mφj ),
7
e j = 1, ..., N
x ∈ H,
e H)
e is the C0-group generated by Γj , i.e.,
and esΓj ∈ L(H,
(4.22)
d sΓj
e x − Γj esΓj x = 0,
ds
e
∀ s ∈ R, x ∈ H.
We have by (4.21) and by (4.14) that
(4.23)
Γj Γk x = η 2hx, ϕ∗j iHe P (mφj )δjk ,
∀ j, k = 1, ..., N
and therefore the operators Γ1, ..., ΓN commute, because the Leray
operator P is self-adjoint.
Then, (4.12) reduces to

N

X

1
dy(t)

2

+
Ay(t)
+
Γ

j y(t) + F (t)y(t)

dt
2


j=1

´
³
PN
PN
β
(t)Γ
−
β
(t)Γ
j
j
j
j
(4.24)
y(t) = 0,
e j=1
B e j=1





∀ t ≥ 0, P-a.s.



y(0) = x,
where
F (t)y(t) = e
−
PN
j=1 βj (t)Γj
³ PN
´
β
(t)Γ
A e j=1 j j y(t) − Ay(t).
By a solution to (4.24) we mean a function
1
y ∈ C([0, ∞); D(A 4 )) ∩ L2(0, ∞; D(A)),
which fulfills (4.24) P-a.s. in the mild sense (see Lemma 4.5 below).
Conversely, if y is a solution to (4.24), then it is an adapted process
and so
(4.25)
X(t) =
N
Y
eβj (t)Γj y(t),
j=1
8
t ≥ 0,
3
1
belongs to CW ([0, T ]; L2(Ω, P; D(A 4 )) ∩ L2(Ω, P, C([0, T ]; D(A 4 )))
and satisfies equation (4.12).
Then we shall confine in the following to study existence and exponential convergence in probability to solutions y to equation to (4.24).
We notice first that, as easily follows by (4.21) and (4.23), we have
(4.26)
esΓj y=η −1Γj y(eηs−1)+y=(eηs−1)hy, ϕ∗j iHe P (mφj )+y,
∀ s > 0, j = 1, ..., N, y ∈ H.
respectively
e−sΓj y = η −1Γj y(e−ηs − 1) + y = (e−ηs − 1)hy, ϕ∗j iHe P (mφj ) + y,
∀ s > 0, j = 1, ..., N, y ∈ H.
This yields
N ³
´
X
βj (t)
(4.27) F (t)y =
e
− 1 hy, ϕ∗j iHe (AP (mφj ) − λj P (mφj )).
j=1
We consider the operator
(4.28)
N
1 X 2
AΓy := Ay +
Γ y,
2 j=1 j
∀ y ∈ D(A)
e is
and notice that the C0-semigroup e−AΓt generated by −AΓ on H
analytic. The operator AΓ + F (t) generates an evolution operator
e that is
U (t, τ ) on H,

d


U (t, τ ) + (AΓ + F (t))U (t, τ ) = 0, 0 ≤ τ ≤ t

dt


 U (τ, τ ) = I.
9
Lemma 4.3 Let
p γ the number fixed as in Notation. We have for
η ≥ max1≤j≤N 6γ − 2Re λj
µ Z t
¶
−γ(t−τ )
kU (t, τ )kL(H,
(1+η 2)|x| 1+ e−γ(τ +2s)ζ(s)ds ,
e H)
e ≤Ce
(4.29)
τ
∀ t ≥ τ, P-a.s.,
P
βj (t)
.
where C is independent of ω and ζ(t) = N
j=1 e
Proof. We shall use the spectral decomposition of the system

dy


+ AΓy + F (t)y = 0, t ≥ τ

dt
(4.30)


 y(τ ) = x.
in the direct sum Xu ⊕ Xs of γ-unstable and γ-stable spaces of the
operator A. Namely we set
yu = PN y,
ys = (I − PN )y
and, since by (4.27), PN F (t)y = 0, we may rewrite system (4.30) as

N

X
dy
1

 u +A y + P
Γ2j yu = 0, t ≥ τ,
u u
N
dt
2
(4.31)
j=1


 y (τ ) = P x.
u
and
(4.32)
N

N
X

dy
1

s
2


 dt + Asys + 2 (I−PN ) Γj yu+(I−PN )F (t)yu=0,
j=1

t≥τ




ys(τ ) = (I − PN )x.
10
We have y = yu + ys, yu =
PN
j=1 yj ϕj
Auϕj = λj ϕj ,
and, by (4.9),
j = 1, ..., N.
Recalling that
Γ2j y = ηΓj y = η 2hy, ϕ∗j iHe P (mφj ),
we may rewrite (4.31) as

1
 dyj
+ λj yj + η 2yj hP (mφj ), ϕ∗j iHe = 0,
dt
2
 y (τ ) = hx, ϕ∗i .
j
e
j H
t ≥ τ, j = 1, ..., N,
Taking into account (4.15), it follows that

1
 dyj
+ λj yj + η 2yj = 0, t ≥ τ, j = 1, ..., N,
dt
2
 y (τ ) = hx, ϕ∗i .
j
e
j H
This yields
1
2
yj (t) = e−(λj + 2 η )thx, ϕ∗j iHe ,
j = 1, ..., N, t ≥ 0.
Hence, for η 2 ≥ 6γ − 2 Re λj , j = 1, ..., N, we have
(4.33)
|yu(t)|He ≤ Ce−3γ(t−τ )|x|He ,
∀ t ≥ τ.
Now, coming back to system (4.32), we shall rewrite it as

N
X

dy
1
s


+ Asys + η 2
yj (I − PN )P (mφj )


dt
2


j=1


 X
N ³
´
β
(t)
j
(4.34)
+
e −1 yj (I−PN )(AP (mφj )−λj P (mφj ))=0,



j=1




t ≥ τ,



ys(τ ) = (I − PN )x.
11
Then by (4.33) and (4.8) we have that
|ys(t)|He ≤ |e−As(t−τ )(I − PN )x|He
Z N
´
1 2 t X ³ βj (s)
+ η
e
− 1 |e−As(t−s)yj (s)(I − PN )
2
τ j=1
×|P (mφj ) + AP (mφj ) − λj P (mφj )|He ds
Z tX
N
2
η
−γ(t−τ )
≤ Ce
|x| + C
|x|H
|e−3γse−γ(t−s)ζ(s)ds
2
0 j=1
Z t
≤ Ce−γ(t−τ )(1+η 2)|x|H e−γ(τ +2s)ζ(s)ds, ∀ t ≥ 0, P-a.s.,
τ
for some constant C independent of x and ω ∈ Ω. This completes the
proof.
Now, we fix η.
Lemma 4.4 We have
Z ∞
eγ(t−τ )|U (t, τ )x|2W dt
τ
µ
¶2
Z ∞
(4.35)
≤ C|x|2W 1 +
e−γ(τ +2t)ζ(t)dt , ∀ x ∈ W,
τ
where C is independent of ω ∈ Ω, 0 ≤ ² < 14 .
Proof. We set
z(t) := e
γ
2 (t−τ )
U (t, τ )x,
Then, by Lemma 4.3, we have
µ
Z ∞
Z
|z(t)|2He dt ≤ C|x|2 1 +
τ
∞
τ
12
0 < τ < t.
¶2
e−γ(τ +2t)ζ(t)dt ,
∀ x ∈ H,
while
N
dz
1 X 2
γ
+ ν0Az + A0z +
Γj z + F (t)z = z,
dt
2 j=1
2
t ≥ τ.
1
e we get (4.35).
Multiplying the latter by z and A 2 +2²z (scalarly in H)
We come back to (4.24) and set
G(t, y) := e
−
PN
j=1 βj (t)Γj
B(e
PN
j=1 βj (t)Γj
e t ≥ 0.
∀ y ∈ H,
y),
Recalling (4.21) and (4.26), we see that
(4.36)
B(eβj (t)Γj y) = B(y) + hy, ϕ∗j i2He (eηβj − 1)2B(P (mφj ))
= (eηβj (t) − 1)hy, ϕ∗j iHe [B1(y, P (mφj )) + B2(y, P (mφj ))],
where B(y) = P ((y · ∇)y) and
(4.37)
B1(y, z) = P ((y · ∇)z),
B2(y, z) = P ((z · ∇)y),
∀ y, z ∈ D(A).
Then by (4.26),(4.27) and (4.36) we have, for all j, k = 1, ..., N ,
e−βk (t)Γk B(eβj (t)Γj y)
= e−βk (t)Γk [B(y) + hy, ϕ∗j i2He (eηβj − 1)2B(P (mφj ))
+(eηβj (t) − 1)hy, ϕ∗j iHe [B1(y, P (mφj )) + B2(y, P (mφj ))].
We write (4.24) as
dy(t)
+ AΓy(t) + G(t, y(t)) + F (t)y(t) = 0,
dt
1
∀ t ≥ 0, P-a.s.
We set z(t) = e 2 γty(t) and rewrite it as

γ
1
 dz(t)
+ (AΓ − γ)z(t) + e− 2 tG(t, z(t)) + F (t)z(t) = 0
(4.38)
dt
2

z(0) = x.
13
Equivalently,
Z
(4.39)
t
z(t) = S(t, 0)x −
γ
S(t, s)e− 2 sG(s, z(s))ds,
∀ t ≥ 0,
0
where
1
S(t, τ ) = U (t, τ )e− 2 γ(t−τ ).
Lemma 4.5 There is Ωx ⊂ Ω, with
´− γ 2
³
1
−
8(ηN )
P(Ωx) ≥ 1 − C ∗|x|W2 − 1
,
with C ∗ > 0 independent of ω and x such that for each x ∈ X
with |x|W ≤ C ∗ equation (4.39) has a unique solution
z ∈ C([0, ∞); W )) ∩ L2(0, ∞; Z).
Here
1
W = D(A 4 ),
3
Z = D(A 4 )
1
if d = 2 and
3
W = D(A 4 +²), Z = D(A 4 +²) if d = 3.
Proof. We shall proceed as in the proof of Theorem 3.6. Namely, we
rewrite (4.39) as
z(t) = S(t, 0)x + N z(t) := Λz(t),
t ≥ 0,
where N : L2(0, ∞; Z) is the integral operator
Z t
− γ2 s
N z(t) = −
S(t, s)e G(s, z(s))ds.
0
We set
U(ω) := {z ∈ L2(0, ∞; Z) : |z|L2(0,∞;Z) ≤ R(ω)},
14
where R : Ω → R+ is a random variable such that
2C1∗|x|W
p
1 + 1 − 4(C1∗)2|x|W (1 + µ)2
(4.40)
2C1∗|x|W
p
≤ R(ω) ≤
,
∗
2
2
1 − 1 − 4(C1 ) |x|W (1 + µ)
Then, as easily follows for
ω ∈ Ω.
|x|W ≤ ρ1(ω) := [8(1 + µ(ω)2)(C1∗)2]−1,
(4.41)
we have
ΛU(ω) ⊂ U(ω).
Now, we shall apply the Banach fixed point theorem to Λ on the set
U(ω). Let z1, z2 ∈ U (ω). We find that
|N z1 − N z2|L2(0,∞;Z)
µ Z ∞
¶
Z ∞
≤ C1∗ e−γt|G(t, z1) − G(t, z2)| 1 dt 1 + e−2γsζ(s)ds
4
0 Z
0
∞
≤ C1∗C2∗ (1 + δ(t))e−γt|z1(t)
0
µ Z ∞
¶
−z2(t)|Z (|z1(t)|Z + |z2(t)|Z )dt 1 + e−2γsζ(s)ds
µZ
≤ C1∗C2∗
µZ
0
∞
×
¶ 21
∞
|z1(t) − z2(t)|2Z dt
e−γt(|z1(t)|2Z + |z2(t)|2Z )dt
0
¶ 12
(1 + µ(ω))2
0
∗ ∗
≤ 2C1 C2 (1 + µ(ω))2R(ω)|z1 − z2|L2(0,∞;Z),
where C1∗, C2∗ are independent of ω.
Now, we take
(4.42)
|x|W ≤ ρ(ω) := min{ρ1(ω), ρ2(ω)} = ((C ∗)2(1 + µ)2)−1,
15
where C ∗ is a suitable chosen constant independent of ω. Then, for
x satisfying (4.42), N is a contraction on U(ω) and maps U(ω) on
itself.
We set
(4.43)
Ωx = {ω ∈ Ω : |x|W ≤ ρ(ω)}.
Hence, for each ω ∈ Ωx, the equation (4.39) has a unique solution z
satisfying conditions in Lemma 4.5. On the other hand, by (4.42) and
(4.43) we see that
³
´− γ 2
1
−
8(ηN )
P(Ωx) ≥ 1 − C ∗|x|W2 − 1
,
as claimed.
Comments
The main result, Theorem 4.1, is taken from [5]. A similar result for
the linearized Oseen–Stokes system was earlier obtained in [1]. Noise
stabilizable feedback controller were designed in [3] for periodic flows
in a 2 − D channel and also in [4] for controllers with support in a
discrete set of points. For the Oseen-Stokes systems is designed in [6]
a stabilizable Stratonovich controller of the form
u = 1lO0
M
X
Rk (X) · β̇k ,
k=1
which has the advantage to be stable to the perturbation of βk .
16
References
[1] V. Barbu, The internal stabilization by noise of the linearized
Navier-Stokes equations, ESAIM COCV, (2011).
[2] V. Barbu, Stabilization of Navier–Stokes Flows, SpringerVerlag, London, 2010.
[3] V. Barbu, Stabilization of a plane periodic channel flow by noise
wall normal controllers, Systems & Control Letters, vol. 59
(2010), 608-618.
[4] V. Barbu, Exponential stabilization of the linearlized Navier–
Stokes equation by pointwise feedback controllers, Automatica,
vol. 46 (2010), 2022-2027.
[5] V. Barbu, Internal stabilization of the Oseen-Stokes equations by
Stratonovich noise, Systems & Control Letters (to appear).
[6] V. Barbu, G. Da Prato, Internal stabilization by noise of the
Navier-Stokes equations, SIAM J. Control & Optim., vol. 49
(2011), 1-21.
[7] G. Da Prato, An Introduction to Infinite Dimensional Analysis, Springer-Verlag, Berlin, 2006.
[8] G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional
Systems, London Mathematical Society Lecture Notes, n. 229,
Cambridge University Press, 1996.
17