3
Boundary stabilization
3.1
The tangential boundary stabilization of linear systems
We discuss here the exponential feedback stabilization to equation
(1.12) or, equivalently, (1.15).
We look for tangential stabilizable controllers u, that is, u · n = 0
on ∂O.
As in the previous case, we shall study first the stabilization to the
liner system
(3.1)
dy
e = (k + A)u,
e
+ Ay
t ≥ 0,
dt
y(0) = y 0.
Denote as above by λj and ϕj the eigenvalues and eigenfunctions of
A and by ϕ∗j the eigenfunctions of the adjoint operator A∗. Let γ > 0
and N such that Re λj ≤ γ, for j = 1, ..., N.
Assumptions
(i) The eigenvalues {λj }N
j=1 are semisimple.
½ ∗ ¾N
∂ϕj
(ii) The system
is linearly independent on ∂O.
∂ν j=1
As seen earlier (Lemma 2.1), the property (i) is generic with respect
to ye. As regards (ii), it was recently proved in [5] (see also [1]) that
in 2 − D it is also generic with respect to yE . This means that, for
”almost all” steady-state solutions ye, assumptions (i), (ii) hold.
Therefore, as seen earlier, we may choose an eigenfunction system
such as
(ϕi, ϕ∗j ) = δij , i, j = 1, ..., N.
Here, γ is chosen in such a way that Re λ ≤ γ for j = 1, ..., N.
1
Denote, also, by M the number
M = max{mj ; 1 ≤ j ≤ N },
where mj is the multiplicity of the eigenvalue λj . Everywhere in this
section, for simplicity, we shall denote again by D(As) the Sobolev
space H 2s(O). For 0 < s < 14 , this notation is exact, but it might
be confusing for s > 14 because, if A is the operator A = −P ∆
with Dirichlet homogeneous boundary conditions, we have D(As) =
H02s(O) ∩ H. Also, in this case, we use the notation Asy instead of
Q the canonical isomorphism of H s+ε(O) onto H.
Theorem 3.1 Let d = 2, 3 and 0 < ε < 14 . Then, if (i) and (ii)
hold, there is a controller u ∈ C 1([0, ∞); (L2(∂O))d) of the form
M
X
∂ϕ∗j
,
u(t) =
vj (t)
∂n
j=1
(3.2)
t ≥ 0,
such that the corresponding solution y u to system (3.1) satisfies
the following conditions
|y u(t)| ≤ Ce−γt|y 0|,
(3.3)
∀y 0 ∈ H, t ≥ 0.
1
If y 0 ∈ D(A 4 −ε), we have
ky u(t)k
(3.4)
1
D(A 4 −ε )
Z
∞
(3.5)
0
≤ Ce−γtky 0k
1
,
dt ≤ Cky 0k2
1
D(A 4 −ε )
e2γtky u(t)k2
3
D(A 4 −ε )
D(A 4 −ε )
Moreover, {vj }M
j=1 can be chosen in such a way that
(3.6)
(3.7)
vj ∈ C 1[0, ∞),
j = 1, 2, ..., M,
|vj (t)| + |vj0 (t)| ≤ Ce−3γt|y 0|,
2
∀t ≥ 0.
.
Proof. Arguing as in the previous cases, we write system (3.1) as
(3.8)
M
X
∂ϕ∗j
dyu
+ Auyu = PN
vj (t)(Ae + k)
,
dt
∂n
j=1
yu(0) = PN y 0,
(3.9)
M
X
∂ϕ∗j
dys
+ Aesys = (I − PN )
,
vj (t)(Ae + k)
dt
∂n
j=1
ys(0) = (I − PN )y 0.
Here,
eX ,
Aes = A|
s
e Xs = (I − PN )H.
e
= PN H,
Au = A|Xu ,
where Xu = lin span{ϕj }N
j=1
PM
If we set yu = j=1 yiϕi and take into account the biorthogonality
relation, we get

µ ∗
¶
M
∗
X

∂ϕ
∂ϕ
dy

i
j

+ λiyi = −ν
vj (t)
, i
,
dt
∂n
∂n
2
d
(3.10)
(L (∂O))
j=1


 y (0) = y 0, i = 1, ..., N.
i
i
Equivalently,
(3.11)
where
dz
+ Λz = Bv,
dt
z(0) = z0,
t ≥ 0,
N.M
Λ = diagkλj kN
j=1 , B = −νkbij ki=1,j=1 ,
¶
µ ∗
∂ϕj ∂ϕ∗i
and v = {vj }M
bij =
,
j=1 .
∂n ∂n (L2(∂O))d
Taking into account Assumption (ii), we conclude by the same argument as that in the proof of Theorem 2.3 that, for each T > 0, system
3
(3.11) is exactly null controllable on [0, T ]. This implies, however, via
finite-dimensional quadratic stabilization, that the stabilizable controller v can be taken as in (3.6), (3.7). Indeed, in this case there
is a feedback controller v(t) = Rz(t) which stabilizes exponentially
system (3.11) that is z 0 + Λz = BRz on (0, ∞) and
|z(t)| ≤ Ce−3γt|z0|,
∀t ≥ 0.
Now, we come back to the infinite-dimensional system (3.9) which
can be written as
µ ∗¶
M
X
∂ϕj
ys(t) =
vj (t)D
∂n
j=1


µ
¶
M
X
∂ϕ∗j
−As t 

+e
ys(0) −
vj (0)D
∂n
j=1
(3.12)
µ ∗¶
Z t
M
X
∂ϕj
ds
−k
e−As(t−s)
vj (s)D
∂n
0
j=1
µ ∗¶
Z tX
M
∂ϕj
−
vj0 (s)e−As(t−s)D
ds.
∂n
0 j=1
Recalling (3.7) and that σ(As) ⊂ {λ; Re λ > γ}, we have, for some
δ > 0,
(3.13)
while
(3.14)
ke−AstkL(H,H) ≤ Ce−(γ+δ)t,
∀t ≥ 0,
° ∗°
° µ ∗ ¶°
° ∂ϕj °
°
∂ϕj °
°
°D
°
≤C°
,
°
°
°
° 3
∂n
∂ν
2
d
d
(H (O))
(H 2 (∂O))
and so, (3.3) holds.
4
We have, also, by (3.12)
¯
µ ∗ ¶¯
¯
∂ϕj ¯
1 −ε
1 −ε
¯
¯
4
4
|vj (t)| ¯A D
|A ys(t)| ≤
¯
∂n
j=1
¯

¯
¯
¯
µ
¶
M
X
¯ 1 −ε −A t
¯
∂ϕ∗j
s
¯
¯


4
+ ¯A e
ys(0) −
vj (0)D
¯
∂n
¯
¯
j=1
µ ∗ ¶¯
Z t
M ¯
X
¯
∂ϕj ¯
1
¯ ds.
¯A 4 −εe−As(t−s)D
+C
e−3γs
¯
∂n ¯
0
j=1
M
X
(3.15)
Taking into account (3.15), we obtain that
1
1
|A 4 −εys(t)| ≤ Ce−γt|A 4 −εys(0)|
µ ∗ ¶¯
Z t
M ¯
X
¯
(3.16)
∂ϕj ¯
1
¯ ds, ∀t ≥ 0.
¯A 4 −εe−As(t−s)D
+C e−3γs
¯
∂n ¯
0
j=1
By (3.14), we have also that
µ ∗ ¶¯
M ¯
X
¯
∂ϕj ¯
3 −ε
−2γt
¯
¯
4
|A y (t)| ≤ Ce
¯A D ∂n ¯
j=1
¯
¯

¯
¯
¶
µ
M
∗
X
¯ 3 −ε
∂ϕ
j ¯¯
−A
t
s
¯

+ ¯A 4
e
(ys(0)) −
vj (0)D
¯
∂n
¯
¯
j=1
µ ∗ ¶¯
Z t
M ¯
X
¯
∂ϕj ¯
3 −ε −A (t−s)
−3γs
s
¯
¯ ds.
4
+
e
D
¯A e
¯
∂n
0
j=1
3 −ε
4
s
(3.17)
On the other hand, if we denote z(t) = eγte−Astz0, we have
z 0 + νAz + A0z − γz = 0,
1
∀t ≥ 0,
and this yields, by multiplying with A 2 −2εz and using the interpolation
inequality,
5
1 d
|z(t)|21 −ε + ν|z(t)|23 −ε ≤ C|z(t)|2, ∀t ≥ 0.
4
4
2 dt
Since, by (3.13), |z|L2(0,∞;H) ≤ C|z0|, we have
Z ∞
3
|z(t)|21 −ε +
|A 4 −εz(t)|2dt ≤ C|z0|21 −ε
4
and, therefore,
|e
−As t
Z
2
z0| 1 −εe
We have also
2γt
4
4
0
∞
+
0
3
e2γt|A 4 −εe−Astz0|2dt ≤ C|z0|21 −ε.
4
¯
µ ∗ ¶¯ 2
∂ϕj ¯ 2γt
3
¯ e dt ≤ C.
¯e−AstA 4 −εD
¯
∂n ¯
0
Then, we obtain that
Z ∞
1
1
eγt|A 4 −εys(t)| +
e−2γt|A 4 −εys(t)|2dt ≤ Ckys(0)k2
Z
∞¯
1
D(A 4 −ε )
0
,
which implies (3.4), (3.5). This completes the proof.
Now, we design a stabilizable feedback controller u in the absence of
Hypothesis (ii). To this purpose, we orthogonalize the system {ϕj }N
j=1 ,
that is, (ϕj , ϕj ) = δij , i, j = 1, ..., N, and set as above
e = lin span{ϕj }N , Xs = (I − PN )H,
e
Xu = PN (H)
j=1
where PN : H → Xu is the algebraic projection on Xu, and denote
by PN∗ its dual. We assume that
n oN
∂ϕ
(3.18) The system ∂νj
is linearly independent in (L2(∂O))d.
j=1
2
d
Then, as noticed earlier, there is a system {Φj }N
j=1 ⊂ (L (∂O)) of
the form
N
X
∂ϕk
, j = 1, ..., N.
(3.19)
Φj =
αjk
∂ν
k=1
6
3.2
The tangential boundary stabilization of Navier–Stokes
equations
Here, we design a feedback stabilizable controller starting from a
low-gain observation (cost). Namely, we consider the cost functional
Z ∞
J(v) =
(|yv (t)|2 + |v(t)|2M ∗ )dt,
0
∗
where |v|M ∗ is the norm in the Euclidean space RM , while yv is the
solution to the controlled system
µ ∗¶
M∗
X
dy
∂ψi
e − γy =
+ Ay
vi(t)(Ae + k)D
, t ≥ 0,
dt
∂n
(3.20)
i=1
y(0) = y0.
By Theorem 3.1, we know that, under Assumptions (i) and (ii), the
minimization problem
(3.21)
∗
inf{J(v); v ∈ L2(0, ∞; RM )}
has a unique solution v ∗ and, by standard theory of linear quadratic
optimal control problems, we know that there is R0 ∈ L(H, H),
R0 = R0∗ such that
∗
(3.22)
(R0y0, y0) = inf{J(v); v ∈ L2(0, ∞; RM ))
Z ∞
=
(|y ∗(t)|2 + |v ∗(t)|2M ∗ )dt,
0
where y ∗ = yv∗ .
In Theorem 3.2, we collect together the main properties of the operator R0 and we prove that the optimal controller v ∗(t) = v(t, y0) in
(3.21) is a stabilizing feedback controller of the form
(3.23)
v ∗(t) = ν
∂
R0y ∗(t),
∂n
7
t ≥ 0.
Theorem 3.2 Assume that Hypotheses (i) and (ii) hold. Then
the operator R0 ∈ L(H, H) is the unique self-adjoint and positive
solution to the algebraic Riccati equation
(R0y, A∗z − γz) + (Ay − γy, R0z)
¶
µ
∂
∂
R0y,
R0z
+ν 2
= (y, z),
∂n
∂n
(L2 (∂O))2
(3.24)
∀y, z ∈ D(A).
We have also
∂
(j) R0y ∈ D(A), ∀y ∈ H and the operator F = ν ∂n
R0 is conti1
nuous from H to (H 2 (∂O))2.
(jj) The operator AF : D(AF ) ⊂ H → H, −AF y = A(y − DF y) −
kDF y, ∀y ∈ D(AF ) = {y ∈ H; y − DF y ∈ D(A)} is the
infinitesimal generator of a C0-analytic semigroup e−AF t in
H and
ke−AF tkL(H,H) ≤ Ce−γt,
(3.25)
∀t ≥ 0.
1
(jjj) e−AF t is a C0-analytic semigroup in W = D(A 4 −ε) and
Z
ke−AF tkL(W,W ) ≤ Ce−γt
∞
3
|A 4 −εe−AF ty0|2e2γtdt ≤ Cky0kW ,
∀y0 ∈ W.
0
Moreover, the optimal controller v ∗ in Problem (3.21) is expressed
in the feedback form (3.23) or, equivalently,
(3.26)
v ∗(t, y0) = ν
∂
R0e−AF ty0,
∂n
By Theorem 3.2, we have
8
∀t ≥ 0.
Corollary 3.3 Under Assumptions (i) and (ii), the feedback controller
M∗
X
∂ψi∗
∂
(R0y(t))
, t ≥ 0,
u(t) = ν
∂n
∂n
i=1
exponentially stabilizes system (3.1) in H and W with exponent
decay −γ. In other words, the solution y to the closed-loop system
∂y
− ν∆y + (y · ∇)ye + (ye · ∇)y = ∇p
∂ν
y(0, x) = y0(x),
in (0, ∞) × O,
∀x ∈ O,
∗
M
X
∂
∂ψi∗
y=ν
(R0y)
∂n
∂n
i=1
on (0, ∞) × ∂O,
satisfies
ky(t)kW ≤ Ce−γtky0kW ,
∀t ≥ 0, y0 ∈ W,
|y(t)| ≤ Ce−γt|y0|,
∀t ≥ 0, y0 ∈ H.
∗
e 0 the
Proof of Theorem 3.2. We denote by Γ : RM → (D(A))
operator
µ ∗¶
M∗
X
∂ψi
∗
, v ∈ RM .
Γv =
vi(Ae + k)D
∂n
i=1
Then, system (3.20) can be rewritten as
(3.27)
dy
e − γy = Γv,
+ Ay
dt
y(0) = y0,
t ≥ 0,
and so, by standard maximum principle for infinite time horizon linear quadratic optimal control problems, the optimal controller v ∗ in
Problem (3.21) is expressed as
9
½
v ∗(t) = Γ∗p(t) =
(3.28)
¾M ∗
∂p ∂ ∂ψi∗
−ν
·
dx
,
∂n
∂n
∂n
∂O
i=1
Z
where Γ∗ is the dual of Γ and p is the solution to the dual backward
system
dp
− A∗p + γp = y ∗, t ≥ 0,
(3.29)
dt
p(∞) = 0.
(Since y ∗ ∈ L2(0, ∞; H), the dual equation to (3.27) involves A∗
only.)
By the dynamic programming principle, we have that v ∗ is still
optimal in the problem
½Z t
¾
Min
(|yv (t)|2 + |v(t)|2M ∗ )dt + (R0y(t), y(t)) , ∀t ≥ 0,
0
and this yields
(3.30)
p(t) = −R0y ∗(t),
∀t ≥ 0,
which, by virtue of (3.28), implies that v ∗ has the feedback representation
½Z
¾M ∗
∂
∂
(3.31) v ∗(t) = −Γ∗R0y ∗(t) = ν
R0y ∗(t)
ψi∗dx
.
∂n
∂n
∂O
i=1
Since p(t) ∈ D(A), ∀t ≥ 0, we have that R0 ∈ L(H, D(A)) and so,
by the trace theorem,
1
∂
R0y ∗(t) ∈ (H 2 (∂O))2, ∀t ≥ 0.
∂n
We note, also, that by
Z ∞
(R0y ∗(t), y ∗(t)) =
(|y ∗(s)|2 + |v ∗(s)|2)ds, ∀t ≥ 0,
t
10
it follows that
d
(R0y ∗(t), y ∗(t)) = −(|y ∗(t)|2 + |Γ∗R0y ∗(t)|2M ∗ ), t ≥ 0.
dt
This yields
1
1
(Ay ∗(t) − γy ∗(t), R0y ∗(t)) + |Γ∗R0y ∗(t)|2M ∗ = |y ∗(t)|2, ∀t ≥ 0,
2
2
which, clearly, implies (3.24), as claimed.
Now, to prove (jj) and (jjj), consider the closed-loop system
dy
e − γy + ΓΓ∗R0y = 0,
+ Ay
dt
y(0) = y0.
(3.32)
t ≥ 0,
Equivalently,
dy
+ AF y − γy = 0,
dt
y(0) = y0.
(3.33)
t ≥ 0,
As seen above, for each y0 ∈ H, equation (3.33) has a unique
solution y ∗ = y(t, y0) ∈ C([0, ∞); H) and the dynamics t → y ∗(t) is
a C0-semigroup on H, that is, y(t, y0) = e−AF ty0, ∀y0 ∈ H. Moreover,
by (3.33) and (3.24), we see that
1
1 d
(R0y(t), y(t)) +γ(R0y(t), y(t)) + |Γ∗R0y(t)|2M ∗
2 dt
2
1
+ |y(t)|2 = 0,
a.e., t > 0,
2
and, therefore,
(3.34)
Z
(R0y(t), y(t)) ≤ e−2γt(R0y0, y0),
∞
|y(s)|2e2γsds ≤ C|y 0|2,
0
The latter implies (jj).
11
∀t > 0,
∀y0 > H.
3.3
The tangential feedback stabilization of Navier–Stokes
equations
One might suspect that the stabilizable feedback controllers for the
Stokes–Oseen system found in the previous section would stabilize the
Navier–Stokes equation in a neighborhood of the origin (respectively,
of equilibrium solution). We see below that this is, indeed, the case
and we prove this by a fixed-point argument.
We come back to the Navier–Stokes equation (1.11) with f ≡ fe
and a boundary tangential controller u, that is
∂y
− ν∆y + (y · ∇)y = fe + ∇p in (0, ∞) × O,
∂t
y(0) = y0
in O,
(3.35)
∇·y =0
in (0, ∞) × O,
y=u
on (0, ∞) × ∂O.
Equivalently,
∂y
− ν∆y + (y · ∇)y + (y · ∇)ye + (ye · ∇)y = ∇p in (0, ∞) × O,
∂t
y(0) = y0 − ye in O,
∇ · y = 0 in (0, ∞) × O,
y = u on (0, ∞) × ∂O.
The latter reduces as above to
dy
e
y(t) + Ay(t)
+ By(t) = (Ae + k)Du,
(3.36)
dt
y(0) = y0.
t ≥ 0,
(We have denoted y0 − ye again by y0.)
We study the stabilization of the controlled equation (3.36) via the
abstract feedback controller
(3.37)
u = F y,
12
where F ∈ L(W, (L2(∂O))d), and
1
1
1
1
W = D(A 4 −ε) = H 2 −2ε(O) ∩ H if d = 2, and
W = D(A 4 +ε) = H 2 +2ε(O) ∩ H if d = 3.
We also set
Z = H 2 −2ε(O)
3
if d = 2,
3
if d = 3.
Z = H 2 +2ε(O)
Consider the operator
AF y = A(y − DF y) − kDF y,
∀y ∈ D(AF ),
D(AF ) = {y ∈ H; y − DF y ∈ D(A)}.
If we plug u into (3.36), we are lead to the closed-loop system
(3.38)
dy
+ AF y + By = 0 in (0, ∞),
dt
y(0) = y0.
Everywhere in the following, the solution y to (3.38) is considered in
the following “mild” sense
Z t
(3.39)
y(t) = e−AF ty0 −
e−AF (t−s)By(s)ds.
0
We assume for the time being that the following assumptions hold.
(k) −AF generates a C0-analytic semigroup e−AF t in W and H.
Moreover, there are C, c, δ > 0, such that
√
(3.40) ke−AF ty0kW ≤ c e−(γ+δ)tky0kW , ∀t ≥ 0, y0 ∈ W.
Z ∞
(kk)
ke−AF ty0k2Z e2γtdt ≤ Cky0k2W , ∀y0 ∈ W.
0
Theorem 3.4 is the main stabilization result.
13
Theorem 3.4 Assume that Assumptions (k) and (kk) hold. Then
there is ρ > 0 such that, for all ky0kW ≤ ρ, equation (3.38) has a
unique solution y ∈ C([0, ∞); W ) such that
(3.41)
ky(t)kW ≤ Ce−γtky0kW ,
Z
∞
(3.42)
0
∀t ≥ 0,
e2γtky(s)k2Z ds ≤ Cky0k2W .
Here,
1
3
1
3
W = D(A 4 −ε), Z = D(A 4 −ε), if d = 2 and
W = D(A 4 +ε), Z = D(A 4 +ε), if d = 3.
Later on, we present some significant examples of such feedback
laws u = F y.
In particular, we obtain the following stabilization result by Theorem 3.4.
Corollary 3.5 Under Assumptions (k) and (kk), the boundary
feedback controller
(3.43)
u(t) = −F (y(t) − ye),
∀t ≥ 0,
exponentially stabilizes the equilibrium solution ye for ky0−yekW ≤ρ,
that is,
(3.44)
ky(t) − yekW ≤ Ce−γtky0 − yekW , ∀t ≥ 0.
Proof of Theorem 3.4. For simplicity, we take γ = 0 as the
general case follows by substituting y(t) by y(t)eγt.
For any r > 0, we introduce the ball Kr of radius r, centered at the
origin, of the space L2(0, ∞; Z):
(
)
½Z ∞
¾ 12
Kr ≡ z ∈ L2(0, ∞; Z) : kzkL2(0,∞;Z) =
kz(t)k2Z dt ≤ r .
0
14
Next, for any η0 ∈ W and z ∈ L2(0, ∞; Z), we introduce the map
(Λz)(t) ≡ e−AF ty0 − (N z)(t);
Z t
(N z)(t) ≡
e−AF (t−τ )(Bz)(τ )dτ.
0
Clearly, equation (3.38) reduces to y = Λy and, therefore, to existence
of a fixed-point to operator Λ on Kr .
First, we prove the following inequalities
½Z ∞
¾ 21
kN zkL2(0,∞;Z) ≡
k(N z)(t)k2Z dt
0
(Z
(3.45)
°
°
°
∞ °Z t
=
0
0
°2 ) 21
°
−AF (t−τ )
e
(Bz)(τ )dτ °
°
√
ckBzkL1(0,∞;W )
Z ∞
√
= c
k(Bz)(τ )kW dτ,
Z
≤
0
To this end, we proceed by duality. Let ζ ∈ L2(0, ∞; Z 0), Z 0 is the
dual of Z with H as a pivot space. Then, starting from (3.45) on N ,
we perform a change in the order of integration via Fubini’s Theorem,
we use Schwarz inequality and we invoke
15
Z
Z
∞
∞ µZ t
¶
e−AF (t−τ )(Bz)(τ )dτ, ζ(t) dt
((N z)(t), ζ(t))dt =
0
0
0
Z ∞Z t
≤
ke−AF (t−τ )(Bz)(τ )kZ kζ(t)kZ 0 dτ dt
¾
Z 0 ∞ ½0Z ∞
=
ke−AF (t−τ )(Bz)(τ )kZ kζ(t)kZ 0 dt dτ
0
τ
¸ 1 ·Z
Z (·Z
∞
∞
≤
0
τ
ke−AF (t−τ )(Bz)(τ )k2Z dtdt
Z
∞
2
0
∞ ·Z ∞
= kζkL2(0,∞;Z 0)
ke−AF σ (Bz)(τ )k2Z dσ
0
0
√
≤ c kζkL2(0,∞;Z 0)kBzkL1(0,∞;W ).
¸ 21
kζ(t)k2Z 0 dt
¸ 21 )
dτ
We recall that
kBzkW ≤ Kkzk2Z ,
(3.46)
∀z ∈ Z.
Using estimate (3.46), the inequality yields
kN zkL2(0,∞;Z)
Z ∞
√
√
≤ ckBzkL1(0,∞;W ) ≡ c
k(Bz)(t)kW dt
0
Z ∞
√
√
≤ cK
kz(t)k2Z dt = c Kkzk2L2(0,∞;Z)
0
and so,
kΛzkL2(0,∞;Z)
·
Z
√
≤ c ky0kW + K
∞
0
We obtain, therefore, that, if
Z ∞
kz(t)k2Z dt ≤ r2,
0
16
¸
kz(t)k2Z dt .
r
ky0kW ≤ √ ,
2 c
dτ
where r > 0 is chosen to satisfy the constraints
1
r≤ √
,
2 cK
we have
Z
kΛzk2L2(0,∞;Z)
≡
0
∞
k(Λz)(t)k2Z dt ≤
1 2 1 2
r + r = r2,
2
2
that is, Λz ∈ Kr . Hence, the ball Kr is invariant under the action of
the operator Λ. We have, also,
(3.47)
kΛz1 − Λz2kL2(0,∞;Z) = kN z1 − N z2kL2(0,∞;Z)
√
≤ 2 c Krkz1 − z2kL1(0,∞;Z),
∀z1, z2 ∈ S(0, r).
Indeed, let z1, z2 ∈ L2(0, ∞; Z). Then, we have as above (see (3.45))
√ R∞
kN z1 − N z2kL2(0,∞;Z) ≤ c 0 k(Bz1)(t) − (Bz2)(t)kW dt
√
= ckBz1 − Bz2kL1(0,∞;W ).
On the other hand, for z1, z2 ∈ Kr , we have by (3.46)
kBz1 − Bz2kW = kP [(z1 · ∇)z1 − (z2 · ∇)z2]kW
= kP [((z1 − z2) · ∇)z1 + (z2 · ∇)(z1 − z2)]kW
≤ K(kz1kZ + kz2kZ )kz1 − z2kZ .
17
This yields
kN z1−N z2kL2(0,∞;Z)
Z ∞
√
≤ cK
[kz1(τ )kZ + kz2(τ kZ ]kz1(τ ) − z2(τ )kZ dτ
0
√
≤2 cK
µZ
∞
0
[kz1(τ )k2Z + kz2(τ )k2Z ]dτ
µZ
×
√
≤4 cK
0
µZ
∞
0
∞
¶ 21
×
kz1(τ ) − z2(τ )k2Z dτ
kz1(τ ) − z2(τ )k2Z dτ
¶ 21
¶ 21
.
√
Then, by the Banach fixed-point theorem, we infer that for 4Kr c < 1,
the operator Λ has a unique fixed-point
z ∈ Kr . This implies that,
√
√
r c
for any y0 ∈ W , ky0kW ≤ 2 , where 0 < r < (2 ck)−1 there is
a unique solution y ∈ Kr ∈ L2(0, ∞; Z) to equation (3.38). Clearly,
such a solution, which can be seen as a “mild” solution to (3.38),
satisfies also y ∈ C([0, ∞); W ) ∩ L2(0, ∞; Z) and
√
√
(3.48)
ky(t)kW ≤ c e−δtky0kW + c K|y|2L2(0,∞;Z).
Moreover, we have the estimate (recall that y = Λy)
kyk2L2(0,∞;Z) ≤ 2cky0k2W + 2cK 2kyk4L2(0,∞;Z
≤ 2cky0k2W + 2cK 2r2kyk2L2(0,∞;Z).
Hence,
kyk2L2(0,∞;Z) ≤ 2(1 − cK 2r2)−1cky0k2W ,
and so,
(3.49)
ky(t)kW
√
√ −δt
2c c K
ky0k2W ,
≤ c e ky0kW +
2
2
1 − 2cK r
18
Now, if we take t ≥ T sufficiently large and ky0kW ≤ ρ sufficiently
small, we see that
(3.50)
ky(t)kW ≤ ηky0kW ,
∀t ≥ T,
where 0 < η < 1. Taking into account that the flow t → y(t, y0) is a
semigroup, it follows that
ky(nT )kW ≤ ηky((n − 1)T )kW ≤ η nky0kW .
This yields
ky(t)kW ≤ Ce−δ0tky0kW ,
∀t ≥ 0,
where δ0 > 0. This completes the proof.
We consider now a few special cases of feedback operators of the
form u = F y for which assumptions (k) and (kk) hold and so, Theorem 3.4 is applicable.
1◦ Let u = F y be the feedback controller constructed in Theorem
3.2 and, more precisely, in Corollary 3.3, that is,
∗
(3.51)
M
X
∂ψi∗
∂
(F y)(t) = ν
(R0y(t))
,
∂n
∂n
i=1
t ≥ 0,
where R0 is the solution to the algebraic Riccati equation (3.24).
Namely, we have the following theorem.
19
Theorem 3.6 Assume that d = 2, Assumptions (i) and (ii) hold
and that ky0 − yekW ≤ ρ. If ρ is sufficiently small, then the closedloop system
∂y
− ν∆y + (y · ∇)y = fe + ∇p
∂t
y(0, x) = y0(x)
(3.52)
∇·y =0
in (0, ∞) × O,
in O,
in (0, ∞) × O,
∗
M
X
∂ψi∗
∂
y=ν
(R0(y − ye)
∂n
∂n
i=1
on (0, ∞) × ∂O,
has a unique solution y ∈ C([0, ∞); W ), which satisfies
(3.53)
ky(t) − yekW ≤ Ce−γtky0 − yekW ,
Z
(3.54)
0
where
∞
∀t > 0,
ky(t) − yek2Z e2γtdt ≤ Cky0 − yekW ,
1
1
W = D(A 4 −ε) = (H 2 −2ε(O))d ∩ H
3
and
3
Z = D(A 4 −ε) = (H 2 −2ε(O))d.
Of course, the solution y is considered in the mild sense
Z t
y(t) − ye = e−AF t(y0 − ye) −
e−AF (t−τ )B(y(τ ) − ye)dτ.
0
Theorem 3.6 is the main boundary stabilization result for the Navier–
Stokes equation (3.35) and it amounts to saying that in 2 − D, under
quite reasonable assumptions on the spectrum of the corresponding
linearized (Stokes–Navier) system, the equilibrium solutions are exponentially stabilizable by tangential feedback controllers having a
finite-dimensional structure. As seen earlier in 3 − D, such a result,
20
at least in this form, is not possible due to the fact that the analysis
of the nonlinear term P (y · ∇)y (see the key estimate (3.46)) requires
1
1
3
W = (H 2 +ε(O))d, that is, (H 2 +ε(O))d → (H 2 +ε(O))d functional
setting, which is in contradiction with the finite-dimensional structure
of the boundary control. Of course, one might expect to have also
in 3 − D a similar result, but for feedback operators F which have a
more general structure.
3.4
Normal stabilization of a plane-periodic channel flow
Consider the model of a laminar flow in a two-dimensional channel with the walls located at y = 0, 1. We assume that the velocity
field (u(t, x, y), v(t, x, y)) and the pressure p(t, x, y) are 2π periodic in
x ∈ (−∞, ∞). The dynamic of flow is governed by the incompressible
2 − D Navier–Stokes equation
(3.55)
ut − ν∆u + uux + vuy = px, x ∈ R, y ∈ (0, 1), t ≥ 0,
vt − ν∆v + uvx + vvy = py , x ∈ R, y ∈ (0, 1), t ≥ 0,
ux + vy = 0
u(t, x, 0) = u(t, x, 1) = 0, x ∈ R, t ≥ 0,
v(t, x, 0) = 0, v(t, x, 1) = v ∗, ∀x ∈ R, t ≥ 0,
u(t, x + 2π, y) ≡ u(t, x, y), v(t, x + 2π, y) ≡ v(t, x, y),
y ∈ (0, 1), t ≥ 0.
Consider a steady-state flow with zero vertical velocity component,
that is, (U (x, y), 0). (This is a stationary flow sustained by a pressure
gradient in the x direction.)
We have U (y) = C(y 2 − y), ∀y ∈ (0, 1), where C < 0. In the
following, we take
a
C=− ,
2ν
21
where a ∈ R+.
We recall that the stability property of the stationary flow (U, 0)
varies with the Reynolds number ν1 ; there is ν0 > 0 such that for
ν > ν0 the flow is stable while for ν < ν0 it is unstable. Our aim here
is the stabilization of this parabolic laminar flow profile by a boundary
controller v(t, x, 1) = v ∗(t, x), t ≥ 0, x ∈ R, that is, only the normal
velocity v is controlled on the wall y = 1.
The linearization of (3.55) around steady-state parabolic flow profile
(U (y), 0) leads to the following system
(3.56)
ut − ν∆u + uxU + vU 0 = px, y ∈ (0, 1), x, t ∈ R,
vt − ν∆v + vxU = py ,
ux + vy = 0, u(t, x, 0) = u(t, x, 1) = 0,
v(t, x, 0) = 0, v(t, x, 1) = v ∗(t, x),
u(t, x + 2π, y) ≡ u(t, x, y), v(t, x + 2π, y) ≡ v(t, x, y),
which governs the small perturbations to this equilibrium profile. Here
the actuator v ∗ is a normal velocity boundary controller on the wall
y = 1. However, there is no actuation in x = 0 or inside the channel.
The main advantage of the periodic control problem (3.56) is that
it can be reduced, via Fourier analysis, to an infinite system of 1−D
parabolic problems, which greatly reduces the complexity of the control system.
For this purpose, let us briefly describe the Fourier functional setting
for Problem (3.56).
Let L2π (Q), Q = (0, 2π) × (0, 1) be the space of all functions
u ∈ L2loc(R × (0, 1)) which are 2π-periodic in x. These functions
22
are characterized by their Fourier series
X
u(x, y) =
ak (y)eikx, ak = ā−k , a0 = 0,
k
XZ 1
|ak |2dy < ∞.
0
k
We set
Hπ = {(u, v) ∈ (L2π (Q))2; ux + vy = 0, v(x, 0) = v(x, 1) = 0}.
(If ux + vy = 0, then the trace of (u, v) at y = 0, 1 is well-defined as
an element of H −1(0, 2π) × H −1(0, 2π)). We have
(
X
X
ikx
Hπ = u =
uk (y)e , v =
vk (y)eikx, vk (0) = vk (1) = 0,
k6=0
1
XZ
k6=0
0
k6=0
(|uk |2 + |vk |2)dy < ∞, ikuk (y) + vk0 (y) = 0,
)
a.e., y ∈ (0, 1), k ∈ R .
We now return to system (3.56) and rewrite it in terms of the Fourier
∞
coefficients {uk }∞
k=−∞ , {vk }k=−∞ .
We have
(3.57)
(uk )t − νu00k + (νk 2 + ikU )uk + U 0vk = ikpk ,
y ∈ (0, 1), t ≥ 0,
(vk )t − νvk00 + (νk 2 + ikU )vk = p0k
ikuk + vk0 = 0, y ∈ (0, 1), k 6= 0, t > 0
uk (t, 0) = uk (t, 1) = 0,
vk (t, 0) = 0, vk (t, 1) = vk∗(t), t ≥ 0,
uk (0, y) = u0k (y), vk (0, y) = vk0(y),
23
where
p=
X
pk (t, y)e
ikx
,
u=
k6=0
v=
X
vk (t, y)eikx,
X
uk (t, y)eikx,
k6=0
2
u00 =
k6=0
∂
u,
∂y 2
u0 =
∂
u.
∂y
Here, k is the wave number in streamwise direction and (3.57) is a
completely decoupled system in state variables uk , vk and the boundary controller vk∗(t). This yields
ik(vk )t − ikνvk00 + ik 2(νk + iU )vk − (u0k )t + νu000
k
0
0
0 0
00
−k(νk + iU )uk − ikU uk − U vk − U vk = 0.
Taking uk = − ik1 vk0 , we obtain that
1
ik(vk )t − ikνvk00 + ik 2(νk + iU )vk + (vk00)t
ik
ν iv 1
− vk + (νk + iU )vk00 − U 00vk = 0,
ik
i
Finally,
(3.58)
t ≥ 0, y ∈ (0, 1).
(vk00 − k 2vk )t − νvkiv + (2νk 2 + ikU )vk00
−k(νk 3 + ik 2U + iU 00)vk = 0, t ≥ 0, y ∈ (0, 1)
vk0 (t, 0) = vk0 (t, 1) = 0,
vk (t, 0) = 0, vk (t, 1) = vk∗(t),
vk (0, y) = vk0(y), y ∈ (0, 1).
system (3.58) is a linear parabolic control system in variable vk on
(0, ∞) × (0, 1) with the boundary controller vk∗ on y = 1.
In the following, we denote by H the complexified space L2(0, 1)
with the norm | · | and product scalar denoted by (·, ·). We denote by
H m(0, 1), m = 1, 2, 3, the standard Sobolev spaces on (0, 1) and
H01(0, 1) = {v ∈ H 1(0, 1); v(0) = v(1) = 0}
H02(0, 1) = {v ∈ H 2(0, 1) ∩ H01(0, 1); v 0(0) = v 0(1) = 0}.
24
We set H=H 4(0, 1) ∩ H02(0, 1) and denote by H0 the dual of H in the
pairing with pivot space H, that is H⊂H⊂H0 algebraically and topologically. Denote by (H 2(0, 1))0 the dual of H 2(0, 1) and by H −1(0, 1)
the dual of H01(0, 1) with the norm denoted k · k−1. Denote also by
Hπ−1(Q) the space L2(0, 2π; H −1(0, 1)) with the norm k · kHπ−1(Q).
For each k ∈ R, we denote by Lk : D(Lk ) ⊂ H→H and Fk :
D(Fk ) ⊂ H → H the second order differential operators
(3.59) Lk v = −v 00 + k 2v,
v ∈ D(Lk ) = H01(0, 1) ∩ H 2(0, 1)
Fk v = νv iv − (2νk 2 + ikU )v 00 + k(νk 3 + ik 2U + iU 00)v,
∀v ∈ D(Fk ) = H 4(0, 1) ∩ H02(0, 1).
(3.60)
We set
Fk v = νv iv − (2νk 2 + ikU )v 00 + k(νk 3 + ik 2U + iU 00)v
and consider the solution Vk of the equation
θVk + Fk Vk = 0, y ∈ (0, 1),
Vk0(0) = Vk0(1) = 0, Vk (0) = 0, Vk (1) = vk∗(t).
(3.61)
(As easily seen, for θ positive and sufficiently large, there is a unique
solution Vk .) Then, subtracting (3.58) and (3.61), we obtain that
(Lk vk )t + Fk (vk − Vk ) − θVk = 0, t ≥ 0.
Equivalently,
(3.62)
(Lk (vk −Vk ))t+Fk (vk −Vk )=θVk −(Lk Vk )t, vk −Vk ∈D(Fk ),
vk (0) = vk0 .
In order to represent (3.62) as an abstract boundary control system,
we consider the operator Ak : D(Ak ) ⊂ H → H defined by
(3.63)
Ak = Fk L−1
k ,
D(Ak ) = {u ∈ H; L−1
k u ∈ D(Fk )}.
We have the following lemma.
25
Lemma 3.7 The operator −Ak generates a C0-analytic semigroup
on H and for each λ ∈ ρ(−Ak ) (the resolvent set of −Ak ),
(λI + Ak )−1 is compact. Moreover, one has for each γ > 0
σ(−Ak ) ⊂ {λ ∈ C; Re λ ≤ −γ},
µ
¶ 12
(3.64)
1
a
∀|k| ≥ M = √
γ+1+ √
,
2ν
2ν
where σ(−Ak ) is the spectrum of −Ak .
In particular, it follows by Lemma 3.7 that, for |k| ≥ M , we have
ke−Ak tkL(H,H) ≤ Ce−γt, ∀t ≥ 0.
Let γ > 0 and let k ∈ R, |k| ≤ M , be arbitrary but fixed. Here, M
is given in Lemma 3.7. Then, the operator −Ak has a finite number
N = Nk of the eigenvalues λj = λkj with Re λj ≥ −γ. (In the
following, since k is fixed, we omit the index k from Ak and λkj .)
We denote by {ϕj }N
j=1 the corresponding eigenfunctions and repeat
each λj according to its algebraic multiplicity mj . We have
(3.65)
Ak ϕj = −λj ϕj , j = 1, ..., N,
and recall that the geometric multiplicity of λj is the dimension of
eigenfunction space corresponding to λj . (The eigenfunctions ϕj = ϕkj
depend, of course, on k but, in agreement with the above convention,
we omit k from ϕkj .) Here, we assume that the following assumption
holds.
(A1) All the eigenvalues λj with 1 ≤ j ≤ N are simple.
In each case, such a condition can be checked in part by taking into
account that λj are eigenvalues λ of the boundary value problem
λ(−v 00 + k 2v) + νv iv − (2νk 2 + ikU )v 00
+k(νk 3 + ik 2U + iU 00)v = 0, y ∈ (0, 1),
v(0) = v(1) = 0, v 0(0) = v 0(1) = 0.
26
We denote by ϕ∗j the eigenfunctions to the dual operator −A∗k , that is,
(3.66)
A∗k ϕ∗j = −λj ϕ∗j , j = 1, ..., N.
As seen earlier, it follows by Assumption (A1) that the system {ϕ∗j }
can be chosen in such a way that
­
®
(3.67)
ϕ`, ϕ∗j = δ`j , `, j = 1, ..., N.
For the time being, we prove the following lemma.
Lemma 3.8 Under Assumption (A1), all the eigenvalues λj ,
j = 1, ..., N, are simple and
(3.68)
(ϕ∗j )000(1) 6= 0,
∀j = 1, ..., N.
Proposition 3.9 For each |k| ≤ M , there is vk∗ ∈ C 1([0, ∞))
such that the corresponding solution vk to (3.62) satisfies
(3.69)
and
(3.70)
|vk (t)| ≤ Ce−γt|vk0|,
∀t ≥ 0,
¯
¯
¯
¯
d
|vk∗(t)| + ¯¯ vk∗(t)¯¯ ≤ Ce−γtkvk0 kH 1(0,1).
0
dt
Proof. We proceed as in the previous cases. Namely, we set y = Lk vk
and rewrite (3.62) (taking into account the biorthogonality relation
(3.67)) as
y = PN y + (I − PN )y,
(3.71)
y=
N
X
y i ϕi ,
ys = (I − PN )y,
i=1
dyj
+ λj yj = ((Fek + θ)Dk )∗ϕ∗j vk∗,
dt
yj (0) = PN (Lk vk0 ),
27
j = 1, ..., N,
dys
+ Aesk ys = (I − PN )(Fek + θ)Dk vk∗, t ≥ 0,
(3.72)
dt
ys(0) = (I − PN )(Lk vk0 ),
¯
e
where Ak = Ak ¯Xs , Xs = (I − PN )H, and PN is the projection on
s
Xu = lin{ϕj }N
j=1 = PN (H). (In the following, we simply write Ak
instead of Aesk .)
We have
d
yj + λj yj = µj vk∗, j = 1, ..., N,
(3.73)
dt
yj (0) = PN (Lk vk0),
where µj = ν(ϕ∗j )000(1) and, by Lemma 3.8, µj 6= 0 for all j. Taking
into account that λi 6= λj for i 6= j, we infer by (3.73) via Kalman’s
controllability criterion that there is a function vk∗ satisfying (3.70)
and such that the solution {yj }N
j=1 to (3.73) satisfies
(3.74)
|yj (t)| ≤ Ce−(γ+δ)t|yj (0)|,
∀j = 1, ..., N, t ≥ 0,
for some δ > 0.
Corollary 3.10 Let {vk∗}M
k=1 be as in Proposition 3.9. Then the
boundary controller
X
∗
v (t, x) =
eikxvk∗(t), x ∈ R, t ≥ 0,
|k|≤M
stabilizes exponentially system (3.56), that is,
ku(t)kL2π (Q) + kv(t)kL2(Q) ≤ Ce−γt(ku(0)kL2π (Q) + kv(0)kL2(Q)).
28
Comments
The results of this section, for which we used the book [1], are closely
related to that given in the works [3], [4] by V. Barbu, I. Lasiecka and
R. Triggiani. Other results of the same type were given by A. Fursikov
[6] and J.P. Raymond [9]. As regards the stabilization of periodic flows
in a 2 − D channel, we refer to [2], [7], [9].
References
[1] V. Barbu, Stabilization of Navier–Stokes Flows, Communications and Control Engineering, Springer, New York, 2010.
[2] V. Barbu, Stabilization of a plane channel flow by wall normal controllers, Nonlinear Anal. Theory – Methods. Appl., 56
(2007), 145-168.
[3] V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc.,
852 (2006), 1-145.
[4] V. Barbu, I. Lasiecka, R. Triggiani, Abstract setting for tangential
boundary stabilization of Navier–Stokes equations by high and
law gain feedback controllers, Nonlinear Anal., 64 (2006), 27042746.
[5] V. Barbu, I. Lasiecka, R. Triggiani, The unique continuation
property of eigenfunctions to Stokes–Oseen operator, Nonlinear
Anal. (to appear).
[6] A. Fursikov, Stabilization for the 3−D Nav ier–Stokes systems by
feedback boundary controllers, Discrete Contin. Dyn. Systems,
10 (2004), 289-314.
29
[7] I. Munteanu, Normal feedback stabilization of periodic flows in a
2−D channel (to appear).
[8] J.P. Raymond, Feedback boundary stabilization of the twodimensional Navier-Stokes equations, SIAM J. Control Optim.,
45 (2006), 790-828.
[9] R. Vázquez, E. Trélat, M. Coron, Control for fast and stable laminar-to-high-Reynolds-numbers transfer in a 2D NavierStokes channel flow, Discrete Cont. Dyn. Syst., 10 (2008), 925956.
30