Stabilization of a plane channel flow by wall normal controllers

Nonlinear Analysis 67 (2007) 2573–2588 www.elsevier.com/locate/na

Stabilization of a plane channel flow by wall normal controllers Viorel Barbu Department of Mathematics, “Al.I. Cuza” University and “Octav Mayer” Institute of Mathematics of the Romanian Academy, 700506 Ias¸i, Romania Received 17 August 2006; accepted 11 September 2006

Abstract The steady-state flows in a two-dimensional channel with periodic conditions along the axis can be stabilized by boundary controllers with vertical velocity observation which acts on the normal component of velocity only. c 2006 Elsevier Ltd. All rights reserved.

MSC: 76D05; 35Q30; 35B40 Keywords: Stabilization; Normal velocity; Controller; Kalman condition

1. Introduction Consider a laminar flow in a two-dimensional channel with the walls located at y = 0, 1. We shall 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 ∈ (−∞, ∞). Clearly, this is a nonphysical hypothesis which is often, however, assumed for mathematical convenience and does not alter the essential features of the model. (See, e.g., [5] for a discussion on periodic flows in channel flows.) The dynamic of flow is governed by the incompressible 2-D Navier–Stokes equations u t − ν∆u + uu x + vu y = px ,

x ∈ R, y ∈ (0, 1)

vt − ν∆v + uvx + vv y = p y ,

x ∈ R, y ∈ (0, 1)

u x + vy = 0

(1.1)

u(t, x, 0) = u(t, x, 1) = 0, u(t, x + 2π, y) ≡ u(t, x, y),

v(t, x, 0) = ψ(t, x), v(t, x, 1) = ϕ(t, x), v(t, x + 2π, y) ≡ v(t, x, y), y ∈ (0, 1).

∀x ∈ R

Consider a steady-state flow governed by (1.1) with zero vertical velocity component i.e. (U (x, y), 0). (This is stationary flow sustained by a pressure gradient in the x direction.) Since the flow is freely divergent, we have Ux ≡ 0 and so U (x, y) ≡ U (y). Alternatively, substituting into (1.1) gives −νU 000 (y) = px (x, y),

p y (x, y) ≡ 0.

Hence p ≡ p(x) and U 00 ≡ 0. This yields E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.09.024

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Fig. 1.

U (y) = C(y 2 − y)

∀y ∈ (0, 1)

(1.2)

a where C ∈ R − . In the following, we take C = − 2ν where a ∈ R + . This is a parabolic laminar flow profile as in Fig. 1. We recall that the stability property of the stationary flow (U, 0) varies with the Reynold 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 flow profile by a boundary controller

v(t, x, 0) = ψ(t, x),

v(t, x, 1) = φ(t, x),

t ≥ 0, x ∈ R.

Therefore, only the normal velocity v is controlled on the walls y = 0, y = 1. The linearization of (1.1) around steady-state flow (U (y), 0) leads to the following system u t − ν∆u + u x U + vU 0 = px ,

y ∈ (0, 1), x, t ∈ R,

vt − ν∆v + vx U = p y , u x + v y = 0, u(t, x, 0) = u(t, x, 1) = 0, v(t, x, 0) = ψ(t, x), v(t, x, 1) = ϕ(t, x), u(t, x + 2π, y) ≡ u(t, x, y),

(1.3)

v(t, x + 2π, y) ≡ v(t, x, y).

Here the actuator {ϕ, ψ} is a normal velocity boundary controller on the walls y = 1, y = 0. There is no actuation, however, for steamwise or inside the channel. Broadly, the main outcome of this paper is the contention that there is a controller {ϕ, ψ} of the form X X ϕ(t, x) = ϕk (t)eikx , ψ(t, x) = ψk (t)eikx |k|≤M

|k|≤M

(v(t), e−ikx )

where ϕk , ψk are functions of L 2 (Q) which stabilizes exponentially system (1.3). (See Remark 4.1 below.) Broadly, this means that stabilization can be achieved using a finite number M of Fourier modes which is determined, however, by the Reynold number. There is much literature on the stabilization of systems (1.3) but only some of the results are relevant to our work. In [3] Barbu et al. have established general boundary stabilization results for three-dimensional Navier–Stokes equations but the controller is tangential (see also Fursikov [9], Raymond [11] and Barbu et al. [3] for other results on boundary stabilization of two- and three-dimensional flows). The stabilization of steady-state flows of the form (1.2) was studied recently by Aamo et al. [1], Krstiˇc [7] (the case of wall-normal control) and by Balogh et al. [2] (the case of wall-tangential control). However, most of these stabilization results require sufficiently low values P of the Reynold number. In [12], Triggiani has constructed a wall normal controller acting on y = 1, V (x, t) = Jj=1 ψ j (x)ϕ j (t), which stabilizes system (1.3) subject to boundary value conditions u y (t, x, 0) = 0, v(t, x, 0) = 0,

u y (t, x, 1) = Vx (t, x) v(t, x, 1) = V (t, x).

The construction relies on an elimination procedure of the pressure p via vorticity function ω = vx − u y . A slightly different stabilization control technique is developed by Vasquez and Krstiˇc in [13,14], where the boundary controller

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consists of two parts: the normal velocity v(x, t, 1) = VC (x) and the streamwise velocity u(x, t, 1) = UC (x). Here we shall use a different approach for the elimination of pressure term which uses essentially the periodic structure of the problem along the x axis. 2. The Fourier functional setting Let L 2π (Q), Q = (0, 2π ) × (0, 1) be the space of all functions u ∈ L 2loc (R × (0, 1)) which are 2π -periodic in x. These functions are characterized by their Fourier series X ak (y)eikx , ak = a¯ −k , a0 = 0 u(x, y) = k 1

XZ k

|ak |2 dy < ∞.

0

The norm of this space is denoted by k · k L 2 (Q) . Similarly, Hπ1 (Q), Hπ2 (Q) are defined. For instance ( ) X XZ 1 1 2 ikx 2 2 0 2 Hπ (Q) = u ∈ L π (Q); u ∈ ak e , ak = a¯ −k , a0 = 0, (k |ak | + |ak | )dy < ∞ . k

k

0

We set H = {(u, v) ∈ (L 2π (Q))2 ; u x + v y = 0, v(x, 0) = v(x, 1) = 0}.

(2.1)

(It transpires that if u x + v y = 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 ) (see, e.g., [6,10]).) We also set V = {(u, v) ∈ H ∩ Hπ1 (Q); u(x, 0) = u(x, 1) = 0}.

(2.2)

As defined above, the space L 2π (Q) is in fact the factor space L 2π (Q)/Z . Furthermore, H ⊥ = {(ξ, η) ∈ (L 2π (Q))2 ; ξ = px , η = p y , p ∈ Hπ1 (Q)}

(2.3)

where H ⊥ is the orthogonal complement of H into (L 2π (Q))2 . In terms of Fourier series the space H can be defined equally as ( X X u k (y)eikx , v = vk (y)eikx , vk (0) = vk (1) = 0, H = u= k6=0

XZ k6=0 0

1

k6=0

) (|u k |2 + |vk |2 )dy < ∞, iku k (y) + vk0 (y) = 0, a.e. y ∈ (0, 1), k ∈ R .

(2.4)

We now return to system (1.3) and rewrite it in terms of Fourier coefficients {(u k , vk )}∞ k=−∞ . We have (u k )t − νu 00k + (νk 2 + ikU )u k + U 0 vk = ikpk ,

a.e. in (0, 1)

(vk )t − νvk00 + (νk 2 + ikU )vk = pk0 iku k + vk0 = 0,

(2.5)

a.e. on (0, 1), k 6= 0

u k (0) = u k (1) = 0,

vk (0) = ψk (t),

vk (1) = ϕk (t).

Here p=

X k6=0

pk (y)eikx ,

ψ=

X k6=0

ψk (y)eikx ,

ϕ=

X

ϕk (y)eikx .

k6=0

In order to homogenize the boundary conditions on the wall y = 0, y = 1 in system (2.5), consider two arbitrary functions µ j ∈ C 3 [0, 1], j = 1, 2, such that

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µ1 (0) = 0, µ2 (0) = 1,

(k) µ2 (0)

= 0,

(k)

µ1 (1) = 1,

k = 0, 1, 2, 3;

k = 1, 2, 3;

µ1 (1) = 0, (k) µ2 (1)

= 0,

k = 1, 2, 3.

(2.6)

k = 0, 1, 2, 3.

We set Φk (t, y) = µ1 (y)ϕk (t) + µ2 (y)ψk (t), t ≥ 0, y ∈ [0, 1] 1 1 ηk = Φk , ξk = − ηk0 = − (Φk ) y . ik ik

(2.7)

We subtract now {ξk , ηk } from {u k , vk } and set U k = u k − ξk ,

Vk = vk − ηk ,

k ∈ Z.

(2.8)

We obtain the system (Uk )t − νUk00 + (νk 2 + ikU )Uk + U 0 Vk = ikpk + Wk ,

a.e. in (0, 1)

(Vk )t − νVk00 + (νk 2 + ikU )Vk = pk0 + Θk Uk (0) = Uk (1) = 0; ikUk + Vk0 = 0,

(2.9)

Vk (0) = Vk (1) = 0

a.e. on (0, 1)

where ν (νk 2 + ikU ) 1 (Φk ) y − (Φk ) yy + (Φk ) y − U 0 Φk ik ik ik

Wk =

(2.10)

Θk = −(Φk )t + ν(Φk ) yy − (νk 2 + ikU )Φk .

(2.11)

Now, in (2.10) we take the projection P on H in order to get rid of pressure { pk }. Note first that the sequence pair { f k , gk }k6=0 is in H if and only if (see (2.4)) i k f k (y) + gk0 (y) = 0,

a.e. y ∈ (0, 1); gk (0) = gk (1) = 0.

(2.12)

By (2.9)–(2.11) (see also (2.6)) we see that condition (2.12) is satisfied by all pairs arising there but {U 0 Φk + 2U 0 Vk − νUk00 , −νVk00 }k6=0 . We set { f k , gk } = P{U 0 Φk + 2U 0 Vk − νUk00 , −νVk00 }

(2.13)

and note that the pair { f k , gk } is the solution to optimization problem 1

Z Min 0

(| f k + νUk00 − U 0 Φk − 2U 0 Vk |2 + |gk + νVk00 |2 )dy;

gk0 + ik f k = 0,

gk (0) = gk (1) = 0.

(2.14)

We see that { f k , gk } ∈ L 2 (0, 1) × H01 (0, 1) to (2.14) is given by gk00 − k 2 gk = ik(νUk00 − Φk U 0 − 2U 0 Vk )0 + νk 2 Vk00 ,

a.e. on (0, 1)

gk (0) = gk (1) = 0 f k = −(ik)−1 gk0 ,

a.e. in (0, 1).

Recalling that ikUk + Vk0 = 0, we obtain gk00 − k 2 gk = νk 2 Vk00 − νVkiv − ik(Φk U 0 + 2U 0 Vk )0 gk (0) = gk (1) = 0 f k = −(ik)−1 gk0

on (0, 1).

in (0, 1)

(2.15) (2.16)

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We set H02 (0, 1) = {v ∈ H 2 (0, 1); v(0) = v(1) = 0; v 0 (0) = v 0 (1) = 0} and note that Vk ∈ H02 (0, 1) and Vk000 ∈ L 2 (0, 1), ∀k. For each V ∈ H02 (0, 1) ∩ H 3 (0, 1), we set L k (V ) = −g,

(2.17)

k ∈ R,

where g ∈ H01 (0, 1) = {v ∈ H 1 (0, 1); v(0) = v(1) = 0} is the solution to equation g 00 − k 2 g = νk 2 V 00 − νV iv − 2ik(U 0 V )0 ,

y ∈ (0, 1)

(2.18)

g(0) = g(1) = 0. Denote also by G k ∈ H01 (0, 1) ∩ H 2 (0, 1) the solution to equation G 00k − k 2 G k = −ik(Φk U 0 )0 ,

y ∈ (0, 1)

(2.19)

G k (0) = G k (1) = 0. As a matter of fact, we may equivalently write G k as G k (y) = G 1k (y)ϕk (t) + G 2k (y)ψk (t) where j

j

(G k )00 − k 2 G k = −ik(µ j U 0 )0 , j

j = 1, 2

(2.190 )

j

G k (0) = G k (1) = 0. Then by (2.15) we have gk = −L k (Vk ) + G k ,

a.e. y ∈ (0, 1).

(2.20)

For the time being it is convenient to endow the space H01 (0, 1) with the scalar product Z hW, V i1,k = Re

1

0

(W 0 V + k 2 W V )dy

(2.21)

0

and regard L k : D(L k ) ⊂ H01 (0, 1) → H01 (0, 1) as the linear operator on H01 (0, 1) defined by (see (2.17)) L k (V ) = −g, D(L k ) =

∀V ∈ D(L k )

H02 (0, 1) ∩

(2.22)

H 3 (0, 1).

Lemma 2.1. For all k ∈ R we have Z 1 Z 00 2 2 0 2 hL k (V ), V i1,k = −ν (|V | + k |V | )dy + 2Re ik 0

1

0

! ∀V ∈ D(L k ).

U 0 V V dy

(2.23)

0

Proof. By (2.21), (2.17) and (2.18), and Z 1 0 hL k (V ), V i1,k = −Re (g 0 V + k 2 gV )dy 0 1

Z

(|V | + k |V | )dy + 2Re ik

= −ν

00 2

2

0 2

0

1

Z

0

!

U V V dy . 0

0

Finally, we consider the operator Ak : D(Ak ) ⊂ H01 (0, 1) → H01 (0, 1), Ak (V ) = L k (V ) − (νk 2 + ikU )V D(Ak ) = D(L k ) = H02 (0, 1) ∩ H 3 (0, 1).

(2.24)

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The operator Ak is closed and densely defined in H01 (0, 1). Moreover, we have ! Z 1 Z 1 0 U 0 V V dy (|V 00 |2 + k 2 |V 0 |2 )dy + 2Re ik hAk (V ), V i1,k = −ν 0

0 1

Z − Re

0

((νk 2 + ikU )V )0 V + k 2 ((νk 2 + ikU )|V |2 )dy

0 1

Z

(|V 00 |2 + k|V 0 |2 )dy − νk 4

= −ν

1

Z

|V |2 dy − νk 2

|V 0 |2 dy + Re

0

0

0

1

Z

1

Z

0

ikU 0 V V dy.

0

This yields hAk (V ), V i1,k ≤ −ν

1

Z

(|V | + 2k |V | + k |V | )dy + C 00 2

2

0 2

4

2

1

Z

k(|V |2 + |V 0 |2 )dy,

∀k ∈ R

(2.25)

0

0

where C = |U 0 | L ∞ (0,1) . In particular, it follows that Z ν 1 hAk (V ), V i1,k ≤ − (|V 00 |2 + 2k 2 |V 0 |2 + k 4 |V |2 )dy, 2 0

∀|k| ≥ k0 .



(2.26)

Lemma 2.2. For each k ∈ R, the operator Ak generates a C0 -analytic semigroup on H01 (0, 1). Moreover, for each λ ∈ ρ(Ak ), (λI − Ak )−1 is compact in H01 (0, 1) and Ak satisfies dissipativity inequalities (2.25) and (2.26). Proof. By (2.25) we see that for each k, and λ ∈ ρ(Ak ), k(λI − Ak )−1 f k H 2 (0,1) ≤ Ck f k H 1 (0,1) , 0

∀ f ∈ H01 (0, 1)

which implies that (λI − Ak )−1 is compact in H01 (0, 1). We note also that (see (2.24)) Ak = L 1k + L 2k where L 1k : D(L 1k ) = D(L k ) ⊂ H01 (0, 1) → H01 (0, 1) is a self-adjoint positive definite operator in H01 (0, 1) and L 2k is a continuous operator on H01 (0, 1). More precisely, L 1k (V ) = −g1 where g1 ∈ H01 (0, 1) is the solution to equation g100 − k 2 g1 = ν(k 2 V 00 − V iv )

in (0, 1)

g1 (0) = g1 (1) = 0 while L 2k (V ) = −g2 − (νk 2 + ikU )V and g200 − k 2 g2 = −2ik(U 0 V )0

in (0, 1)

g2 (0) = g2 (1) = 0. This implies that Ak generates a C0 -analytic semigroup on H01 (0, 1) as claimed. Next, we return to system (2.9), taking into account (2.13), (2.15), (2.20) and (2.24) we may rewrite the second equation in (2.9) as a linear Cauchy problem in H01 (0, 1) (Vk )t = Ak (Vk ) − (Φk )t + ν(Φk ) yy − (νk 2 + ikU )Φk − G k , Vk (0) =

t ≥ 0, k ∈ R,

Vk0 .

The stabilization of system (2.27) is our principal goal in the next section.

(2.27)



3. Open loop stabilization of system (2.27) Let M be the first number k0 for which (2.26) holds. By (2.25) we see that it can be estimated as M≥

1 0 a |U | L ∞ = 2 . ν 2ν

(3.1)

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V. Barbu / Nonlinear Analysis 67 (2007) 2573–2588

For |k| ≥ M we have 2

keAk t k1,k ≤ e−νk t ,

∀t > 0.

(3.2)

Equivalently, Z 1 Z 1 ek (0, y)|2 )dy, e0 (0, y)|2 + k 2 |V ek (t, y)|2 )dy ≤ e−2νk 2 t e0 (t, y)|2 + k 2 |V (|V (|V k k

(3.3)

0

0

ek is the solution to uncontrolled (homogeneous) system (2.27); i.e. for ϕk , ψk ≡ 0. where V This means that if we want to stabilize (2.27) we must concentrate on the first equations; i.e. for |k| < M. For fixed k consider the eigenvalues {λkj } and the corresponding eigenfunctions {χ kj } of the operator Ak ; i.e., Ak χ kj = λkj χ kj , |k| ≤ M. Taking into account (2.24) and (2.17) we see that λkj and χ kj are the eigenvalues and, respectively, eigenfunctions to −(νk 2 + ikU )χ j + θ j = λ j χ j , θ 00j

− k θj = 2

ν(χ iv j

j = 1, . . . ,

− k 2 χ 00j ) + 2ik(U 0 χ j )0

(3.4)

θ j (0) = θ j (1) = 0. Equivalently, 2 00 3 4 2 00 νχ iv j − (λ j + 2νk + ikU )χ j + (ik U + νk + λ j k + ikU )χ j = 0,

χ j (0) = χ j (1) = 0,

χ 0j (0)

=

χ 0j (1)

y ∈ (0, 1)

= 0.

(3.5)

(Here we have dropped k from λkj and χ kj .) k with Recalling Lemma 2.2, it follows that for each k, |k| ≤ M there is a finite number Nk of eigenvalues {λkj } Nj=1 k Re λ j ≥ 0. (We shall call these unstable eigenvalues.) We set

N = max{Nk ; |k| ≤ M} and consider the linear space X uN ,k = span{χ kj } Nj=1 ,

|k| ≤ M.

This is the unstable space of system (2.27); note that X uN ,k ⊂ H02 (0, 1). We recall that the dimension of the eigenfunction space for λ j is called the geometric multiplicity of λ j . Lemma 3.1. The geometric multiplicity of each unstable eigenvalue λ j is ≤ 2. Proof. Let {α` }4`=1 be a fundamental system of solutions for Eq. (3.5). Then any solution χ j to problem (3.5) is given P P P (k) by χ j = 4`=1 C` α` , where {C j }4j=1 are determined by system 4j=1 C j α j (0) = 0, k = 0, 1; 4j=1 C j α kj (1) = 0, k = 0, 1. Since rank kχ kj (0)k4j=1 1k=0 = 2, the space of all such {C j } is two-dimensional, as claimed. The algebraic multiplicity of λ j is the dimension m j of the range space R(P j ) of the operator Z 1 Pj = − (λI − Ak )−1 dλ 2πi Γ j where Γ j is a smooth closed contour encircling λ j (see e.g. [8], p. 181). Note that the algebraic multiplicity is greater or equal to the geometric multiplicity.  We have Lemma 3.2. The algebraic multiplicity of each unstable eigenvalue λ j is ≤ 2.

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Proof. Let f ∈ H01 (0, 1) and X (λ, y) = (λI − Ak )−1 f . By (2.17) and (2.24) we see that X is the solution to equation (see (3.4) and (3.5)) −(νk 2 + ikU )X + θ = λX + f

on (0, 1)

θ − k θ = ν(X − k X ) + 2ik(U 0 X )0 θ (0) = θ (1) = 0. 00

2

IV

2

on (0, 1)

00

Equivalently, ν X IV − (λ + 2νk 2 + ikU )X 00 + (ik 3 U + νk 4 + λk 2 + ikU 00 )X = f 00 − k 2 f X (0) = X (1) = 0,

on (0, 1)

(3.6)

X (0) = X (1) = 0. 0

0

Equivalently, X 0 − Y = 0, y ∈ (0, 1), 1 1 Y 0 − Z = f, ν ν Z 0 − W = 0,

(3.7) 

1 1 W 0 − (λ + 2νk 2 + ikU )Z + (ik 3 U + νk 4 + λk 2 + ikU 00 )X = − (λ + 2νk 2 + ikU ) + k 2 ν ν X (0) = X (1) = 0, Y (0) = Y (1) = 0.

 f,

By the variation of constant formula we may therefore represent the solution X to (3.6) as (assuming λ is not eigenvalue)   Z y 4 X X (λ, y) = X k (λ, y) Ck + Wk (ξ, λ) f (ξ )dξ 0

k=1

where {X k , Yk , Z k , Wk }4k=1 is a fundamental system of solutions for homogeneous system (3.7), Wk are analytic functions in λ and Ck are constants. We may choose {X k , Yk , Z k , Wk }4k=1 in such a way that det(X k , Yk , Z k , Wk )4k=1 ≥ ρ > 0, ∀λ ∈ C and X 1 (λ, 0) = 0,

X 2 (λ, 0) = 0,

det(X k (λ, 0), X k0 (λ, 0))4k=3 6= 0,

X 10 (λ, 0) = 0, ∀λ ∈ C.

Then, by (3.7), we see that 4 X

4 X

X k (λ, 0)Ck = 0,

k=3 2 X

X k (λ, 1) Ck +

1

Z

! Wk (ξ, λ) f (ξ )dξ

=0

0

k=1 2 X

X k0 (λ, 0)Ck = 0

k=3

X k0 (λ, 1)

k=1

1

Z Ck +

! Wk (ξ, λ) f (ξ )dξ

0

because f ∈ H01 (0, 1). Hence Z 1 Ck = − Wk (ξ, λ) f (ξ )dξ, 0

k = 1, 2,

=0

X 20 (λ, 0) = 0

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and therefore ! Z 1 Z 2 X 1 Wk (ξ, λ) f (ξ )dξ dλ Pj ( f ) = X k (λ, y) 2π i Γ j k=1 y Z 1 2 Z 1 X = f (ξ )dξ X k (λ, y)Wk (ξ, λ)dλ 2π i k=1 Γ j y =

2 X

ηk Φk (y)

k=1

where ηk ∈ C and Φk ∈ H01 (0, 1) are given functions. This implies that rank P j ≤ 2 as claimed.



Proposition 3.1. Let |k| ≤ M. Then there is µ = {µ1 , µ2 } satisfying (2.6) and a controller {ϕk , ψk } ∈ C 1 [0, ∞) with compact support in [0, T ] such that the solution Vk to (2.27) satisfies the estimate   (3.8) kVk (t)k H 1 (0,1) ≤ Ce−γ t kVk (0)k H 1 (0,1) + 1 , ∀t ≥ 0, 0

0

for some γ > 0. Proof. Taking into account (2.7), we may write (2.27) as d Vk = Ak (Vk ) − µ1 ϕk0 − µ2 ψk0 + ϕk (νµ001 − (νk 2 + ikU )µ1 − G 1k ) dt + ψk (νµ002 − (νk 2 + ikU )µ2 − G 2k ), t ≥ 0 Vk (0) =

(3.9)

V0k .

We shall prove that, for a certain µ = (µ1 , µ2 ) satisfying conditions (2.26), system (3.9) is exponentially stabilizable. In order to make the treatment more transparent, we shall assume first that the unstable eigenvalues λ j are semisimple, i.e. the algebraic multiplicity coincides with geometric multiplicity. Then there is an adjoint bi-orthogonal system {χ ∗j } such that A∗k χ ∗j = λ j χ ∗j and hχ j , χ`∗ i1,k = δ j` . We set

J1 0



λ ` 0

k ∗ N J

, 2 A N = khAk χ j , χ` i1,k k j,`=1 =

, J` = 0 λ`

. .

0 . and so the restriction of system (3.9) to X uN ,k (unstable space) is given by    0 ϕk ϕk z 0N (t) = AkN z N (t) + B N1 , t ∈ (0, T ), − B N2 ψk ψk0 z N (0) =

(3.10)

PN Vk0 ,

where z N = PN Vk and PN is the projection on X uN ,k , and B N1 , B N2 are the matrices B N1 = colkhνµ001 − (νk 2 + ikU )µ1 − G 1k , χ ∗j i1,k , hνµ002 − (νk 2 + ikU )µ2 − G 2k , χ ∗j i1,k k.

(3.11)

B N2 = colkhµ1 , χ ∗j i1,k , hµ2 , χ ∗j i1,k k. (The operator PN is the algebraic projection on X uN ,k , i.e. Z 1 (λI − Ak )−1 dλ, PN = − 2πi Γ where Γ is a closed smooth contour which contains inside the eigenvalues λkj , 1 ≤ j ≤ N , |k| ≤ M.) Then we have Vk (t) =

N X j=1

j

z N (t)χ j + (I − PN )Vk (t),

(3.12)

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where Y N = (I − PN )Vk is the solution to Y N0 = (I − PN )Ak Vk + ϕk (t)(I − PN )(νµ001 − (νk 2 + ikU )µ1 − G 1k ) + ψk (t)(I − PN )(νµ002 − (νk 2 + ikU )µ2 − G 2k ) − ϕk0 (t)(I − PN )µ1 − ψk0 (I − PN )µ2 .

(3.13)

We shall prove first null-controllability of system (3.10) on some interval [0, T ], i.e. z N (T ) = 0 for {ϕk , ψk } suitably chosen. To this end, we need the following controllability result which perhaps is not new but we could find no explicit reference to it in the literature.  Lemma 3.3. Consider the linear control system y 0 = Ay + B1 u − B2 u 0 ,

t ≥0

(3.14)

y(0) = y0 where A is a N × N matrix and B1 , B2 are N × m matrices. Assume that the Kalman condition holds rankkB1 − AB2 , A(B1 − AB2 ), . . . , A N −1 (B1 − AB2 )k = N .

(3.15)

Then for each y0 ∈ R N and T > 0 there is a controller u ∈ H01 (0, T ; R m ) such that y(T ) = 0 (equivalently y(T ) = y1 where y1 is arbitrary in R N ). This means that system (3.14) is exactly controllable if y 0 = Ay + (B1 − AB2 )u is controllable. Proof. We notice that for u ∈ H01 (0, T ; R N ) y(T ) = e AT y0 +

T

Z

e A(T −t) (B1 − AB2 )u(t)

(3.16)

0

and so we may apply the Kalman’s rank condition to find the result. Now we shall apply Lemma 3.3 to system (3.10). Note that B N1 − AkN B N2 = colkhχ ∗j , −λ j µ1 + νµ001 − (νk 2 + ikU )µ1 − G 1k i1,k , hχ ∗j , −λ j µ2 + νµ002 − (νk 2 + ikU )µ2 − G 2k i1,k k Nj=1 . We set Γ j` = hχ ∗j , −λ j µ` + νµ00` − (νk 2 + ikU )µ` − G `k i1,k ,

∀ j = 1, . . . , N , ` = 1, 2

and therefore B N1 − AkN B N2 = colkΓ j1 , Γ j2 k Nj=1 . The Kalman controllability condition for system (3.10) reads as rankkB N1 − AkN B N2 , AkN (B N1 − AkN B N2 ), . . . , (AkN ) N −1 (B N1 − AkN B N2 )k = N ,

(3.17)

or equivalently, (B N1 − AkN B N2 )∗ e−(A N ) t p0 = 0, k ∗

∀t ∈ [0, T ] H⇒ p0 = 0.

Taking into account the structure of matrix AkN , condition (3.17) reduces to

(3.18)

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..

1

Γ 1

Γ12 λ1 Γ11 λ1 Γ12 . λ1N −1 Γ11 λ1N −1 Γ12



.

1

.. λ N −1 Γ 1 Γ22 λ1 Γ21 λ1 Γ22 λ1N −1 Γ22

Γ 2

1 2



. N −1 1 N −1 2

Γ 1

. 2 1 2 Γ3 λ2 Γ 3 λ2 Γ3 . λ2 Γ 3 λ2 Γ3

3

rank

=N . . N −1 1 N −1 2 . 2 1 2

Γ 1 Γ4 λ2 Γ 4 λ2 Γ4 . λ2 Γ 4 λ2 Γ4

4

........................................................................................................



.. 2 1 2 1 2 N −1 N −1

Γ 1

2 p−1 Γ2 p−1 λ p Γ2 p−1 λ p Γ2 p−1 . λ p Γ2 p−1 λ p Γ2 p−1



..

Γ 2

2 1 2 1 2 N −1 N −1 Γ λ Γ λ Γ . λ Γ λ Γ 2p

2p

p 2p

p 2p

p

2p

p

2p

We consider for simplicity the case where all eigenvalues have multiplicity 2 and N = 2 p. Other possible situations are treated similarly. It transpires that if 1 1 Γ 1 2 2 2 Γ Γ Γ Γ Γ 2 p−1 3 6= 0, . . . , det 2 p−1 1 6= 0, det 3 (3.19) det 11 6= 0, 1 2 2 1 2 Γ Γ2 p Γ2 Γ2 Γ2 p Γ4 4 then the above rank condition holds. Indeed, if (3.19) holds, then the following determinant

..

2

λ1 Γ11 λ21 Γ12 . λ1N −2 Γ12 λ1N −1 Γ11

Γ 1



.. N −2 2 N −1 1

Γ 2

1 2 2 λ1 Γ 2 λ1 Γ 2 . λ1 Γ 2 λ1 Γ 2

2

......................................................................................... D = det

..

Γ 2

1 2 2 1 2 N −2 N −1

2 p−1 λ1 Γ2 p−1 λ p Γ2 p−1 . λ p Γ2 p−1 λ p Γ2 p−1



..

Γ 2

1 2 2 1 2 N −2 N −1 λ Γ λ Γ . λ Γ λ Γ 1 p p p 2p 2p 2p 2p 2p

1

2 M

Γ

Y

2 j−1 Γ2 j−1 =C det 1

Γ Γ22j j=1 2j where C 6= 0. Taking into account the expression of Γ j` , it is readily seen that there is a pair µ = {µ1 , µ2 } (as a matter of fact, infinitely many), which satisfies conditions (2.6) and such that for all |k| ≤ M,

Γ 1

2

2 j−1 Γ2 j−1 det 1

6= 0, ∀ j = 1, . . . , N . 2

Γ 2 j Γ2 j Hence, by Lemma 3.2, there exist {ϕk , ψk }|k|≤M ∈ (H01 (0, T ))2 such that z N (T ) = 0. This implies that system (3.10) is null controllable with a controller {ϕk , ψk } ∈ (H01 (0, T ))2 . The general case. If the eigenvalues λ j , 1 ≤ j ≤ M, are not semisimple, then, for each λ j we may choose a linear `j independent system of generalized eigenfunctions {ϕ ji } M , ` j — the algebraic multiplicity of λ j , such that the j=1 i=1

matrix AkN corresponding to Ak in this basis has the Jordan canonical form; i.e. i i A{ϕi j }`j=1 = Ji (ϕi j )`j=1 ,

i = 1, . . . , M,

where Ji is the Jordan block of dimension `i ≤ 2 corresponding to λi and so we may represent AkN as

J1 0



λi 1

0 J2

k 0

, i = 1, 2, . . . , M AN =

, Ji = 0 λi

. .

0 . JM

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and B N1 , B N2 in the form (3.11) where {χ ∗j } is a biorthogonal system in H01 (0, 1) associated with {ϕi j }. For simplicity we shall assume as above that ` j = 2 ∀ j = 1, . . . , M = 2 p. Then condition (3.17) reduces to

1 Γ12 λ1 Γ11 + Γ21 λ1 Γ12 + Γ22

Γ 1

1

Γ2 Γ22 λ1 Γ21 λ1 Γ22

1

Γ3 Γ32 λ2 Γ31 + Γ41 λ2 Γ32 + Γ42 rank

1

Γ4 Γ42 λ2 Γ41 λ2 Γ42

...........................................................................

1 2 1 1 2 2

Γ

2 p−1 Γ2 p−1 λ p Γ2 p−1 + Γ2 p λ p Γ2 p−1 + Γ2 p

Γ 1 Γ22p λ p Γ21p λ p Γ22p 2p λ1N −2 (λ1 Γ11 + (N − 1)Γ21 )

.. . .. . .. . .. .

.. . .. .

λ1N −2 (λ1 Γ12 + (N − 1)Γ22 )



N −1 2 N −1 1 λ1 Γ 2 λ1 Γ 2

N −2 N −2 1 1 2 2

λ2 (λ2 Γ3 + (N − 1)Γ4 ) λ2 (λ2 Γ3 + (N − 2)Γ4 )

N −2 2 N −1 1

λ2 Γ 4 λ2 Γ 4

............................................................................................

λ Np −2 (λ p Γ21p−1 + (N − 1)Γ21p ) λ Np −2 (λ p Γ22p−1 + (N − 1)Γ22p )

λ N −1 Γ 1 λ N −1 Γ 2 p

p

2p

2p

and the determinant

.. N −2

2

Γ 1

λ1 Γ11 +Γ21 . λ1 (λ1 Γ11 +(N −1)Γ21 ) λ1N −2 (λ1 Γ12 +(N −1)Γ22 )



.

Γ 2

.. λ N −1 Γ 1 λ1 Γ21 λ1N −2 Γ22

2

1 2



det ..................................................................................................................

. N −2 .

Γ 2

1 1 1 1 2 2 N −2

2 p−1 λ p Γ2 p−1 +Γ2 p . λ1 (λ p Γ2 p−1 +(N −1)Γ2 p ) λ p (λ p Γ2 p−1 +(N −1)Γ2 p )



.. N −1 1

Γ 2

1 2 N −1 λ p Γ2 p . λ p Γ2 p λ p Γ2 p 2p

(3.20)

can be made 6= 0 for an appropriate choice of µ. We may argue by induction with respect to p. Indeed, for p = 1 2 2 Γ Γ λ1 Γ11 + Γ21 Γ11 1 1 det 2 det 2 − Γ21 Γ22 6= 0 = λ1 Γ2 λ1 Γ21 Γ2 Γ21 if µ is suitable chosen. Assume now that µ is chosen such that conditions (2.6) hold and all 2( p − 1) order determinants Di of constructed by elements of lines 3 ≤ j ≤ 2 p are 6= 0. Since (3.20) is a linear combination of Di and second-order determinants arise in the first two lines, we conclude that there is µ such that determinant (3.20) is 6= 0 as claimed. ek (t)} = {ϕk (t), ψk (t)}, for 0 ≤ t ≤ T , {e ek (t)} = 0 for Next, we choose the controller {e ϕk (t), ψ ϕk (t), ψ t ≥ T , and insert it into system (3.9). Since the homogeneous system (3.13) is exponentially stable (because σ ((I − PN )A N ) ⊂ {λ; Re λ < 0} and (I − PN )A N generates an analytic semigroup on H01 (0, 1) we infer that the solution Vk to (3.9) is exponentially decreasing to zero). More precisely, we have   kVk (t)k H 1 (0,1) ≤ Ce−γ t kVk0 k H 1 (0,1) + 1 , ∀t > 0, 0

0

as claimed. Next, we come back to system (2.27) and take the stabilizing controller {ϕk , ψk } found above for |k| ≤ M, and ϕk , ψk ≡ 0,

for |k| > M.

V. Barbu / Nonlinear Analysis 67 (2007) 2573–2588

2585

By (3.1) (or (3.2)), it follows that such a controller exponentially stabilizes system (2.27). More precisely, inserting this controller in (2.27) yields (Vk )t = Ak (Vk ) − µ1 ϕk0 (t) − µ2 ψk0 (t) + ϕk (t)(νµ001 − (νk 2 + ikU )µ1 − G 1k ) + ψk (t)(νµ002 − (νk 2 + ikU )µ2 − G 2k ), if |k| ≤ M, t ≥ 0 (Vk )t = Ak (Vk ), if |k| > M.

(3.21)



We have therefore (see (3.2) and (3.8)).

Proposition 3.3. There are the functions µ1 , µ2 satisfying (2.6) and a controller {ϕk , ψk }|k|≤M with compact support on [0, T ] such that the corresponding system (2.27), i.e., (3.9) is exponentially stable. More precisely, one has
kVk (t)k1,k ≤ Ce−δt kVk (0)k1,k + 1 , ∀k 6= 0, t > 0 (3.22) where δ > 0 and C are independent of k. The stabilizing controller found above is an open loop controller but {ϕk , ψk }|k|≤M can be found in feedback form. Indeed, since for each |k| ≤ M system (3.10) is controllable, it follows by (3.17) that the matrix Z 1 k k ∗ WN = e−A N t (B N1 − AkN B N2 )(B N1 − AkN B N2 )∗ e−(A N ) t dt (3.23) 0

is positive definite and a little calculation reveals that (−AkN + (B N1 − AkN B N2 )(B N1 − AkN B N2 )∗ W N−1 )W N + W N (−(AkN )∗ + (B N1 − AkN B N )(B N1 − AkN B N2 )∗ W N−1 ) = e−A N (B N1 − A N B N2 )(B N1 − AkN B N2 )∗ e−(A N ) + W N (B N1 − A2N B N2 )(B 1 N − AkN B N2 )∗ W N−1 . k

k ∗

Since in virtue of the Kalman rank condition (3.17) the latter is a positive definite matrix, we infer by Lyapunov’s theorem that the matrix AkN − (B N1 − AkN B N2 )(B N1 − AkN B N2 )∗ W N−1 is asymptotically stable (Hurwitzian). This implies that the feedback controller   e ϕk (t) 1 k 2 ∗ −1 ek (t) = −(B N − A N B N ) W N z N , |k| ≤ M ψ asymptotically stabilizes the system z 0N

=

AkN z N

+ (B N1

−

AkN B N2 )



 e ϕk ek , ψ

|k| ≤ M

or equivalently, system (3.10). We conclude therefore that the feedback controller   ϕk (t) = −(B N1 − AkN B N2 )∗ W N−1 PN Vk (t), |k| ≤ M ψk (t)   ϕk (t) = 0, |k| ≤ M, ψk (t)

(3.24)

exponentially stabilizes system (3.9) and consequently (2.27). We have proved therefore Proposition 3.4. There are µ1 , µ2 satisfying (2.6) and a feedback controller of the form (3.24) which, inserted into (2.27), stabilizes the system, i.e., kVk (t)k1,k ≤ Ce−δt kVk (0)k1,k , where C and δ are independent of k.

∀t > 0,

(3.25)

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4. Boundary stabilization of system (1.3) Theorem 4.1 below is the main result of this paper. M Theorem 4.1. There is a controller ϕ(t) = {ϕ j } M j=1 , ψ(t) = {ψ j } j=1 , given in feedback form   ϕk (t) = −(B N1 − AkN B N2 )∗ W N−1 PN ((v(t), e−ikx ) L 2 (Q) ψk (t) − µ1 (y)ϕk (t) − µ2 (y)ψk (t)), for t ≥ 0, |k| ≤ M   ϕk (t) ≡ 0 for |k| > M ψk (t)

(4.1)

such that the solution (u, v) to system (1.3) with u 0 , v0 ∈ V and boundary value conditions u(t, x, 0) = u(t, x, 1) = 0, v(t, x, 0) =

M X

ei j x ψ j (t),

v(t, x, 1) =

| j|=1

M X

(4.2)

ei j x ϕ j (t)

| j|=1

H 1 -norm,

is exponentially decreasing to zero in i.e. Z ∞ (ku(t)k2H 1 (Q) + kv(t)k2H 1 (Q) )dt ≤ C;

(4.3)

0

ku(t)k L 2 (Q) + kv(t)k H 1 (Q) ≤ C(ku(0)k L 2 (Q) + kv(0)k H 1 (Q) )e−γ t ,

∀t > 0

for some γ > 0. Here Q = (0, 2π ) × (0, 1). Broadly, this means that the linearized system around parabolic flow profile (U, 0) is stabilizable using a finite number of scalar controllers {ϕ j (t), ψ j (t)} multiplying the periodic modes ei j x . The dimension M of the controller is exactly the number defined by (3.1); i.e. h a i M= + 1. (4.4) 2ν 2 Proof of Theorem 4.1. We recall that the solution (u, v) to (1.3) is given by X X u= (Uk + ξk )eikx , v= (Vk + ηk )eikx , k

k6=0

where Vk is the solution to system (3.9), and ξk , ηk are given by (2.7). By Proposition 3.4 we know that Vk satisfies estimate (3.28), while ηk = (µ1 ϕk + µ2 ψk ), ηk = 0,

ξk = 0

ξk = − for |k| > M.

1 0 (µ ϕk + µ02 ψk ) ik 1

for |k| ≤ M, k 6= 0;

This yields kv(t)k2H 1 (Q) =

X k

≤C

(|Vk0 + ηk0 |2L 2 (0,1) + k 2 |Vk + ηk |2L 2 (0,1) )

X k6=0

2

e−γ0 |k| t + (|Vk0 (0, y)|2L 2 (0,1) + k 2 |Vk (0)|2L 2 (0,1) )

≤ Ce−γ0 t kv(0)k2H 1 (Q)

∀t ≥ 0.

(4.5)

In order to estimate u, we note that by (1.3) we have (u(t), v(t)) ∈ H for t ≥ T . Then, multiplying first equation by u, second by v and integrating, we get on (T, ∞) that Z 1 d 2 2 2 2 (|u(t)| L 2 (Q) + kv(t)k L 2 (Q) ) + ν(ku(t)kV + kv(t)kV ) + v(t)u(t)U 0 (y)dx dy = 0. 2 dt Q

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V. Barbu / Nonlinear Analysis 67 (2007) 2573–2588

Then, using estimate (4.5), after some calculation, gives Z ∞  ku(t)k2H 1 (Q) + kv(t)k2H 1 (Q) dt ≤ C, 0

ku(t)k2L 2 (Q) ≤ Ce−γ t (|u(0)|2L 2 (Q) + kv(0)k2H 1 (Q) ), as claimed. We introduce the operators Fk = {Fk1 , Fk2 } : H01 (0, 1) → C N × C N Fk (V ) = −(B N1 − AkN B N2 )∗ W N−1 PN V, Fk (V ) = 0,

(4.6)

for |k| ≤ M

for |k| ≤ M.

As seen earlier in Proposition 3.4, the controller ϕk (t) = Fk1 (Vk (t)),

ψk (t) = Fk2 (Vk (t))

stabilizes system (2.17). Equivalently ϕk (t) = Fk1 ((v(t), e−ikx ) L 2 (Q) − µ1 ϕk (t) − µ2 ψk (t)),

(4.7)

ψk (t) = Fk2 ((v(t), e−ikx ) L 2 (Q) − µ1 ϕk (t) − µ2 ψk (t)). This is precisely the feedback controller (4.1) and the proof is complete.



Remark 4.1. The feedback controller (4.1) (or (4.7)) is given in implicit form but we see by (2.6) and (4.6) that one might find (µ1 , µ2 ) such that kD N k < 1 where D N = k(B N1 − AkN B N2 )∗ W N−1 PN µ1 , (B N1 − AkN B N2 )∗ W N−1 PN µ2 k and so 

 ϕk (t) = −(I − D N )−1 (B N1 − AkN B N2 )∗ W N−1 PN (v(t), e−ikx ) L 2 (Q) ψk (t)

for |k| ≤ M.

(4.8)

To insert this controller into nonlinear system (1.1), one might argue, locally stabilizes the steady-state solution (U (y), 0) of (1.1) as in [4]. The details are omitted. Remark 4.2. By Proposition 3.2 it follows that system (1.3) can be stabilized as well by an open-loop controller {ϕk , ψk }|k|≤M with compact support in an arbitrary interval [0, T ]. Acknowledgments This work was completed during the author’s visit to the Department of Mathematics of the University of Virginia, Spring 2005. Fruitful discussion with R. Triggiani on the subject of this paper are acknowledged. Also, the author is indebted to an anonymous referee for several suggestions for improving the presentation and for a simple device to prove Lemma 3.3. References [1] O.M. Aamo, M. Krstic, T.R. Bewley, Control of mixing by boundary feedback in 2D-channel, Automatica 39 (2003) 1597–1606. [2] A. Balogh, W.-J. Liu, M. Krstiˇc, Stability enhancement by boundary control in 2D channel flow, IEEE Transactions on Automatic Control 46 (11) (2001) 1696–1711. [3] V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilizations of Navier–Stokes equations, Memoires of the American Mathematical Society 181 (852) (2006). [4] V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high and low gain feedback controllers, Nonlinear Analysis, Theory, Methods and Applications 64 (12) (2006) 2704–2746. [5] T.R. Bewley, Flow control: New challenges for a new renaissance, Progress in Aerospace Science 37 (2001) 21–58. [6] P. Constantin, C. Foias, Navier–Stokes Equations, University of Chicago Press, Chicago, London, 1989.

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[7] M. Krstiˇc, 2004 (preprint). [8] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, NY, 1966. [9] A.V. Fursikov, Real processes to the 3D Navier–Stokes equations and its feedback stabilization from the boundary, in: M.S. Agranovic et al. (Eds.), Partial Differential Equations, in: Amer. Math. Soc. Translations, vol. 206, 2002, pp. 95–123. [10] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, 1983. [11] J.P. Raymond, Feedback stabilization of the 2-D Navier–Stokes equations, SIAM Journal of Control and Optimization 45 (3) (2006) 790–828. [12] R. Triggiani, Normal stabilization of 2-D channel flow (in press). [13] R. Vasquez, M. Krstic, A closed form feedback controller for stabilization of linearized Navier–Stokes equations: The 2D Poisseuille flow (in press). [14] R. Vasquez, M. Krstic, High order stability properties of a 2D Navier–Stokes systems with an explicit boundary controller (in press).