Proceedings of the WeC12.444th IEEE Conference on Decision and Control, andthe European Control Conference 2005Seville, Spain, December 12-15, 2005

A Closed-Form Observer for the Channel Flow Navier-Stokes System

Rafael Vazquez and Miroslav Krstic

Abstract- The observer design for non-discretized Navier-Stokes partial differential equations has so far been an openproblem. We present a nonlinear PDE observer that estimatesthe velocity and pressure fields for an infinite channel flow. Alsoknow as the Poisseuille flow, this problem is frequently cited asa paradigm for transition to turbulence, and is a benchmark forflow control and turbulence estimation. Our observer consistsof a copy of the nonlinear Navier-Stokes equations, combinedwith linear injection of output estimation error, with observergains designed in closed form using backstepping. Pressure andskin friction at one of the walls are the only quantities neededfor measurement. For a fully developed channel flow (whetherlaminar or turbulent), with a Reynolds number possibly abovethe critical value, but not too far above it, the observer isguaranteed to be convergent to the actual velocity and pressurefield, provided its initial estimates are not too far from the actualinitial profile. An output feedback result for the linearizedinfinite channel flow is also presented, combining the observer inthis paper with a controller presented in a companion CDC'05paper. Both results are presented for the 2D case for clarity ofexposition, however an extension to 3D is straightforward.

I. INTRODUCTION

The absence of effective state estimators for Navier-Stokesequations modeling turbulent fluid flows is considered one ofthe key obstacles to reliable, model-based weather forecast-ing. We present a nonlinear observer for the channel flowbenchmark problem. The observer measures only pressureand skin friction at one of the walls of the channel andestimates velocities and pressure throughout the channel.Previous observer designs for the channel flow were inthe form of an Extended Kalman Filter for the spatiallydiscretized Navier-Stokes equations and employed high-dimensional algebraic Riccati equations for computation ofobserver gains [2]. The structure of our observer is similarto an Extended Kalman Filter, however it is designed forthe continuum Navier-Stokes model and the output injectiongains are are given explicitly, by a symbolically computableformula.

For notational clarity the result is presented in 2D. Itextends to 3D in a straightforward way due to spatialinvariance in the spanwise direction.We start the paper stating the mathematical model of

the problem, which is the fully nonlinear incompressibleNavier-Stokes PDE (Section II). In Section III we introducethe nonlinear observer for estimation of the velocity andpressure fields, with explicit expressions for the outputinjection gains. Section IV deals with the proof of localconvergence for the observer error system. Fourier transformallows separate analysis for different wave numbers. Forsome wave numbers, a combination of output injection and a

This work was supported by NSF grant number CMS-0329662.Department of Mechanical and Aerospace Engineering, University of

California at San Diego

Fig. 1. 2D Channel Flow and equilibrium profile

stabilizing term found through a backstepping transformationguarantee observer convergence. Finding the kernel of thetransformation is the design task. We derive an integralequation which is used to symbolically compute the observergains. For the very large and the very small wave numbersthe system is known to be exponentially stable, and simplya copy of the plant is used. These two results are combinedto prove observer error convergence for all wave numbers.Section V presents a complete output feedback design com-bining the results of the present paper with control results ina companion paper [7].

II. MODELConsider a 2D incompressible channel flow evolving in a

semi-infinite rectangle (x, y) C (- oc, oc) x [O, 1] like shownin Figure 1. The dimensionless velocity field is governed bythe Navier-Stokes equations

ut =I

(UXX + UYY)-UUx-VUY-PxRvt =

I (Vxx + V11)Reand the continuity equation

(1)

UVx-VVy -Py, (2)

Ux+V =0, (3)where U denotes the streamwise velocity, V the wall-normalvelocity, P the pressure, and Re is the Reynolds number.Instead of using (3) we can derive a Poisson equation thatP verifies, just combining (1), (2) and (3)

Pxx + Pyy =-2(Vy)2 -2VxUy. (4)The boundary conditions verified by (1), (2) and (4) are

U(t, ,0)V(t, ,0)

Py(t, x, 0)

U(t,X,1) = 0,V(t,X,1) = 0,

RIUxy(t, x, O),

(5)(6)(7)

(8)Py (t,x, 1) = 1 uxy(t, : I),where (7)-(8) are deduced from (2) and (3).

0-7803-9568-9/05/$20.00 ©2005 IEEE 5959

III. OBSERVERThe observer consists on a copy of (1)-(2), to which we

add output injection of the pressure P and the streamwisevelocity gradient Uy (proportional to friction) at the wall

(it=R(uI x + u)yy fPluu - vuy

-4elc/jQ1 (x y)(Uy (&,O)- (y O))d

+ J Q2(X y) (P(, 0) -P(t, 0)) d, (9)

=R (vxx + vy) -Py -uvx-v

-R~ 08 )J X(k)l(k, y, 0)e27ik(x-4)dk<tRe _, 00

+ e/ Q2(X-(,y) (Uy((, O)-(Uy((,O)) <t

+ J Qi(x - , y) (P( ),O)-P(, 0)) d, (10)

where as usual the observer (estimated) variables are denotedby a hat. The pressure P is governed by a copy of (4)

Pxx + Pyy -2(VY)2 -2VxUY, (1 1)

Qj(x,y)500-

O-

0-5005

0 0~~~~~~.5x y

-5 1

Fig. 2. Q1(x,y)

Re f`c% X(k)l(k, y, O)e2ik(x dk. The purpose of this gainis stabilization of the observer error equation and it is alsoa form of output injection, because by continuity VY(x, 0) isknown to be zero. Therefore, the value of Vy(x. 0), whichshould be zero if the observer has converged to the real value,measures a discrepancy between the observer and the realsystem, which is fed back. For this term,

and the boundary conditions for the observer are

U(t, ,0)V(t, , 0)

Py(t, x, 0)

Py(t, x, 1)

U(t,x, 1) = O,V(t,X,1) = ,

Re Uxy(t, x, 0)1 U

(12)(13)

X(k) { 1, m<Ik <M

where m = 32Re and M(14)

(15) is recursively defined as

The observer employs two output injection kernels defined

Ri (x, y, M)R2(X, y, M)

Ri(x, y, m), (16)R2(X, y, m), (17)

2x 2)2 [y cosh (27ky) sin (2wkx)-x sinh (27ky) cos (27kx)]

2k ~~[y cosh (2wky) cos (2wkx)(x +y2)

10 = -R ik(l -y) ((1 _ y)2 + 3q2 3)3

+4iRey(y -1) sinh (27k(y rj))

+2(1 -y)ie (1 -cosh (27k(y i-)))=k

in in- 1

+x sinh (2wky) sin (2wkx)]sinh (2wky) cos (2wkx)

+ 1r(X2 2 )

R2(X, y, k)

(18)

(2+2 2 [x cosh (27ky) sin (2wkx)+y sinh (27ky) cos (27kx)]

2k

-( 2+y2) [x cosh (2wky) cos (2wkx)-y sinh (2wky) sin (27kx)]cosh (2wky) sin (2wkx)

(X2 2Y)(19)Q, is shown in Fig. 2, and Q2 has a very similarshape. In equation (10) there is also an additional term,

e2 (y n y- yn

Re ik / / (-y + - 2)22

X(-~+ 6jl- k 6d (23)

The terms of this recursion can be computed symbolically asthey only involve integration of polynomials and exponen-tials. In implementation, only a few steps will be sufficient toobtain a highly accurate approximation because the recursionis rapidly convergent [5]. The first order approximation

(24)

1This recursion is proved convergent, smooth, and uniformly boundedover (y, r) [0, 1]2, with a bound continuously dependent on k.

5960

(20)

Qi (X, Y)Q2(X, Y)

1 R, and2'

where

R, (x, y, k)

(21)

(22)

2- (y+n) r8-7 Fd+8Re7ki cosh (7k(t, + 6))

x(r)/0-l)n-1 k,^ 2 ' 2 ( fdr

I (k, y, TI) = lim 1, (k, y, TI)n--.oc

Oc

10 (X, Y, 0) = X(k)lo(k, y. O)e2-Fik (`.)dk,-C)c

defined in (16) and (17) written in Fourier space are

Q1 (k, y)Q2(k, y)

X(k)27k sinh (2wky),X(k)27ki cosh (2wky),

-40 0.5-5 055

0 Nyx 5 0

where we have used the truncating function x(k). Thetruncation for large wave numbers doesn't imply a loss ofsmoothness in physical space, as evident from the smoothexpressions (16)-(19).

Substracting the observer equations from the plant equa-tions we obtain the error equations, with states U = U-U,V V V-,P = P- P. Note that from continuity,VY (x, O) = 0 and therefore VY (x, 0) Vy (x, 0).

First we eliminate the pressure error, whose equation inFourier space is

Fig. 3. lo(x,y,0)

is plotted in Fig. 3. Comparing this gain with the previouslydefined Qi and Q2 gains, it can be seen that the outputinjections of P(x, 0) and Uy(x, 0) are important away fromthe measurements and near the y = 1 boundary, while theterm that accounts for the discrepancy in Vy(x, 0) is biggernear the y = 0 boundary.Remark 1: The output injection operators in the observer

are spatially invariant in the x direction in the sense of [1].2This is due to the fact that the plant itself is invariant in thex direction. This is very significant since it allows an easierdesign by exploiting the structure of the plant. It also makespossible a straightforward extension to a 3D result, for thecase of a channel infinite in two directions, with the newdirection z being also of spatial invariance.Remark 2: It should be noted that the observer estimated

"velocity field" (U, V) is not required to verify the incom-pressibility condition (3). In fact, it can be shown fromequations (9)-(15) that as the estimated velocity field evolvesin time, its divergence will also verify an evolution equationwhich will make it change, even if the initial condition forthe observer is a divergence-free velocity field. This is notreally a problem, since the proof shows that the observererror converges to zero and therefore the observer convergesto the real, divergence-free velocity field. The transient valueof the estimated field is allowed to have nonzero divergenceas an extra degree of freedom.

IV. OBSERVER CONVERGENCE PROOF

As common for infinite channels, we use a Fourier trans-form in x. The transform pair (direct and inverse transform)has the following definition

00

f (k,y) = {f(x,y)} / f(x,y)e- 27ikxdx, (25)

f (x y) = 1 {f (k y)} f (k, y)e27ikx dx. (26)-00

Note that we use the same symbol f for both the originalf(x, y) and the image f(k, y). In hydrodynamics k is re-ferred to as the "wave number." The output injection kernels

2The feedback operator commutes with the translation in the x direction.

-4w2k P + Pyy

Py(k, 0)

Py(k, 1)

2¶{ (Vv)2 (V)2

+VxUy VXUy },

27kiyRe

2wki Cy( )Re

The solution to this equation isY sinh (27k(y -))¶{(V)2

+VfxyUy-VxUy} (k, r)dr1

cosh (27ky) f1 cosh (27k(1 -

sinh (27k) Jo7k

2(VY)2 + vxuy

(29)

(30)

(31)

_ (Vy)2

l) F {(,8)2

VU}Uy (k, j)dr

iLcosh 2k)Uy(kl) + (sinh(27ky)Re sinh (2wk)

(32)-cosh (27ky) iosh(2 k)) Uy(k:0)],and noting that

P(k, 0) = sinh (2lFk) ( k

-2(Vy)2 + VxUy-VxUy} (k, Tj)dTj

iUv(k, 1)-cosh (27k) Uy(k,O ), (33)

we can write the pressure error as

Y sinh (27F (y -)){(V)2

+VxUy- V Uy (k, r)dri+ cosh (2wky) P(k, 0)- sinh (27ky) Uy (k, 0).Re

(Vy)2

(34)

We define the fluctuation variable

u(t, x, y) = U(t, x, y) - U(y), (35)

where U(y) = 4y( -y) is the parabolic equilibrium profileof the channel flow, shown in Fig. 1.

5961

I0(XY,O)40-

(27)(28)

Tl)) ,F (,.Y)2

Using (34) for eliminating the pressure, and the Fouriertransforms for the output injection kernels (27)-(28), thevelocity error equations expressed in Fourier space are

1 /ut 4Re(-472k2u + uCY) + 8iwky(y -1)U

+4(2y- l)VY

+16wk j sinh (2wk(y r-)) (2r1- 1)fV(k, r)dr1

-(1 -X(k))2iwk cosh (27ky) P(k, 0)-(1 -X(k)) Re sinh (2wky) Uy (k, 0)+Ar4u f, (I ,UV , (36)

=R (-4w2k2V+V) + 8iwky(y -1)V-16wki j cosh (2wk(y r-)) (2r -1)V(k, j)drj

-(1 -X(k))2wk sinh (27ky) P(k, 0)+(1 -X(k)) R cosh (2wky) Uy(k, 0)Re

-X(k) Re (k, y, 0)VyV(k 0)

+./v f,U,cu,VU ,where

_inh(Fkjetcosh (27k(1

x (2r -1)V(k,r1)dr1+JVP (f,,U, ,V)

with boundary conditionsU(t, ,0)V(t, ,0)

and higher order terms

'1))

U(t,x, 1) = O,V(t,X,I) = 0,

FiUUx-Uux- uX+VUyVUy -VUy

+2ij sinh(2k(1rT)) F{2Vfy(x,rT)Vy(x,T1)

-(Vy)2( X, rT) + Vx(X, rT)uy(X, T1)

+Vx(x, T1)Uy(x, TI) - x(x, rq) vy(x, q) }dr1 (41)

which are quadratic in V, U, u, V.When the observer state (U, V) is close to the actual state

(U, V), and when the fluctuation (u, V) around the equilib-rium state is small, then JVu(U, V, , V), JVv(U, V, a, V)and JVP (U, V, u, V) are small and dominated by the linearterms in the equations. The linearization of the subsystem(with J\Vu, JVv, and J\Vp neglected) is uncoupled for eachwave number, and therefore can be studied separately form < k < M, which we will refer as the observed wavenumber range, and k > M, k < m, the unobserved wavenumber range, where m and M are parameters that will bedefined to ensure stability for the unobserved wave numberequations.

A. Observed wave number analysisFor m < Ikl < M, the output injection kernels eliminate

the presence of P(k, 0) and Uy(k, 0). The linearized plant is

(it = ((-472kU + uCY) + 8iwky(y -1)U

+(2y- 1)V + 16wk / sinh (2wk(y- q))x (2rj -1)V(k,rT)drT,

(37) V = 14 (-4wr2k2V + V ) + 8iwky(y-16wki j cosh (2wk(y r-)) (2r-

1

(44)1)V

1)V(k, j)drj

(45)

with boundary conditions (39)-(40). Equation (44) is expo-(38) nentially stable when V = 0. Thus for U(t) to be locally

convergent, it is enough to ensure that V goes to zero.Therefore we have to stabilize the V subsystem. For this

(39) we use the gain l (k, y, 0), which is designed using the(40) backstepping observer design technique [6] by transforming

V equation for m < k < M into the family of heatequations (parametrized by k)

at =Re (-4w2k2a+a0)a(k, 0)a(k, 1)

0

0,

(46)

(47)(48)

where

A/v I{UV - UV, - uVf +VVy - VVy - VVy

+2 cosh (2wk(1rj)) F {2Vl(x, jT)1v(x, T)

-(Vy)2( X, rT) + Vx(X, rT)uy(X, T1)

+Vx(x, T1)Uy(x, TI) - x(x, rq) vy(x, q), dr1, (42)1

.VP = / cosh (27k(1r-))7k sinh (27k)x F {2Vy (x, r1)V (x, rj) (V )2 (x, rj)

+Vx(x, rI)uy(X, ,j) + Vx(X, r1)Uy(x, r)

-VG (x, q) Uy(x, rq) dr (43)

Jy~~~~~V= a- II(k, y, q1)a(t, k, r1)dr10

(49)

is the backstepping transformation as defined in [6]. Thekernel I is found to verify the following equation

Re ' RIYY + 87iky(y -1)1-16wki(2r1 -1) cosh (2wk(y r-))+16wki (2 -1) cosh (2wk(y -))

(50)

a hyperbolic partial integro-differential equation in the region7T {(y,rT): 0 < y < 1,0 < r1 < y} with boundary

5962

P(k, 0)

J\vU =

x l(k, (, T1)<(:

conditions2Re 2l(y, y) = - ~7wiky2(2y -3), (51)

1(I, T) =0. (52)

This equation can be transformed into an integral equationfrom which the explicit solution given in (22)-(23) is ob-tained, following the steps outlined in [6]. Estimating thesolution of the integral equation one gets the following result.

Theorem 1: The equation (50) with boundary conditions(51)-(52) has a unique C2(7-) solution analytic on k, ex-pressed by series (22)-(23).

The proof uses the same argument as in [5], slightlymodified to account for a complex-valued kernel and usingthe fact that the terms are analytic in k. See [7] for anextended explanation.Once the kernel I is found, the gain that appears in (45) is

explicitly known and guarantees exponential stability of V,and therefore of U, for all m < k < M.

B. Unobserved wave number analysisWhen k > M or k K< m, the observer error linearization

verifies the following equations

1Y /Re (\4w2k2u +uYY + 8iwky(y l)UI

+166k j sinh (2uk(y ri)) (2r,- 1)fV(k, r,1)dr+(2y -1)V -2iwk cosh (2wky) P(k, 0)

2- kRe~ sinh (2wky) Uly(k, 0), (53)

t = 12Re2(4k +Vyy) + 8iwky(y -1)VRe

-16wki j cosh (2wk(y rj)) (2r -1)V(k, j)drj

-2wk sinh (27ky) P(k, 0)

+ R cosh (2wky) Uy(k, 0), (54)Rewhere

P(k 0) sinh(2kj (2lk(1 ]))x (2r -1)V(k,r)drj (55)

with boundary conditions (39)-(40). These are just the lin-earized Navier-Stokes equations. Then, following exactly thesame arguments as in [7] for the uncontrolled Navier-Stokesequations, choosing m = 1 and M 1 R stability32orRe r 2 itof the unobserved wave number range is guaranteed.

C2 such that, if the L2 norms of the initial conditions for Uand V are less than C1, i.e.

j1 J/ U2(0 z y) +V2(0,x,y)) dxdy < Cl, (56)

and if the turbulent kinetic energy of U and V (defined as theL2 norm of the fluctuation with respect to the Pouisseuilleequilibrium profile) is less than C2 for all time, i.e. Vt > 0,

r1 C)jJX (u2(t,x y) + V2(t,X,y))dxdy<C2, (57)o 00

then the L2 norms of U, V converge to zero:1 00c

lim u(2(t, x, y) +v2(t,x,y)) dxdy = 0. (58)The meaning of this theorem is that, for a fully developed

channel flow (whether laminar or turbulent), with a Reynoldsnumber possibly above the critical value (the boundary oflinear stability) but not too far above it, the observer isguaranteed to be convergent to the real velocity and pressurefield, provided its initial estimates are not too far from theactual initial profile.

V. AN OUTPUT FEEDBACK DESIGNIn a companion paper [7] a state feedback control law is

presented for the Navier-Stokes 2D channel flow linearizedaround the Poisseuille equilibrium profile

1Ut = Re (Uxx + U,,) + 4Y(Y

+4(2y - 1)V - px,vt =

I (Vxx + Vyy) + 4y(yRewith boundary conditions

u(t, x 0)V(t, ,0)u(t, x 1)V(t, , 1)

0O0,

Uc(t, x),Vc(t, x),

1)ux

(59)1 )V -p,, (60)

(61)(62)(63)(64)

where Uc, V, are the actuation variables for tangential andnormal velocity. In this section we complete the resultpresenting an output feedback law, which employs the sameactuators but assume that the state is unknown except atthe uncontrolled wall, where measurements of VY(x, 0) andP(x, 0) are available.The dynamic controller that stabilizes plant (59)-(64) is

Uc(t, x)

Vc(t, x)Qul (x - :, TI) ii(t, (, Tj)<dTr, (65)

h(t, x), (66)where h verifies the equation

ht= hxx +g(t: X):C. Analysis for the entire wave number rangeThe analysis sketched in the previous section can be

combined for all wave numbers, to prove the following result.Theorem 2: Consider the system (1), (2) and (4), with

boundary conditions (5)-(8), and the system (9)-(15), andsuppose that both have classical solutions. Consider now theobserver error system. There exists positive constants C1 and

(67)where

1/01g = Q/ Q(x-(, 1)V(t, (, r1) dqdr

+ / Qo(x - : (uy(t, (, O) -iiy(t, (, 1)) d<,(68)-00

5963

and where V and i are obtained from

itI

Re(uxx + uy) -Px+4y(y - l)iix + 4(2y - 1)V

I Oc

i4eI00Ql(x y)(Uy(& O) i(&O)r00

+ Q2(X y) (p(, 0) -p(&, 0)) d,

1 i \ rRe 'x'y -Py+4y(y-1)Vx14eJ0 (v( o)J X(k)l(k,y O)e27i(Re _c Oo

+ 4eI00Q2(X( Y) (Uy(&,O) -uiy(&CO+ QQi(x- , y) (p(,4O) -p(, O))d&

the pressure p is governed by

pxx +Pyy 8(2y -)Vf,and the boundary conditions for the observer are

(t,x 0) = V(t,x, 0) = 0,(t, x, 1) = uc,

V(t, , 1) = Vc,

Py(t,x,0)

Py(t, x, 1)

-1 ^Re Uxy(t, ,0),

Re (UXY(t, , 1)-(VC)t (t, X)

and

Kn Kn-1-Y±r/ Y/ / sinh (7k(t + d))-4wkiRe]

-(2( -1) + 2( -d- 1) cosh (7k(t + ))}

xKn-1 (k, a 2a)dd6d(69)

Re py±r Y-+ 2 wik f/ - 6)(j-}xKn-1 (k '2 )d6da

(`)dkdk2)

fY 17 cosh (2wk(1- ()) -cosh (2wk()+2wk1 sinh (2yk)(70)

The closed loop system verifies the following result.Theorem 3: Suppose the system (59)-(64), (65)-(82),

(71) with gains 1, Qi and Q2 defined by (16)-(23) has a classicalsolution. Then, the equilibrium w(x, y) _ V(x, y) _ 0 isexponentially stable.

(72) This Theorem follows from [7] where the state feedbackresult is presented, combined with the method for observer

(73) design explained in the previous sections.(74)

The kernels Q, Qv and Qo are defined as

Qu= (k)K(k, 1, 1)e2,ik(,-()dk: (77)

Qv = x(k)l67ki(2T -1) cosh (2uk(1 -r))

xe2wik(x`)dk (78)

Qo = / X(k)e2<e ik(x -) dk (79)

The function K(k, y, rq) appearing in (77) is a (complexvalued) gain kernel defined as

K(k,y,ry)= lim Kn(k,Y,rT), (80)

where Kn is recursively defined as 3

Re 2Ko = - w7ikrT (21y2 -6y(3 + 4rT) + TI(12 + 7TI))cosh (2k(1 -y + rj)) -cosh (2wk(y r-))

sinh (2wk)+4iRerq(r -1) sinh (27k(y rj))

-6rji'R (1 -cosh (27k(y -r))), (81)

3This infinite sequence is convergent, smooth, and uniformly boundedover (y, r) c [0, 1]2, and analytic in k.

VI. DISCUSSIONThe result was presented in 2D for ease of notation.

Since 3D channels are spatially invariant in both streamwiseand spanwise direction, it is straightforward to extend thedesign to 3D, just by applying the Fourier transform in bothinvariant directions. For the nonlinear observer, additionalmeasurements of wy(x, 0) are required. The output feedbackresult can be extended to a 3D setting as well, with additionalactuation in the spanwise direction. The result can be alsobe extended to periodic channel flow, 2D or 3D, substitutingthe Fourier transform with Fourier series, with all otherexpressions still holding.

ACKNOWLEDGEMENTWe thank Tom Bewley for numerous helpful discussions

and expert advice.

REFERENCES[1] B. Bamieh, F. Paganini and M.A. Dahleh, "Distributed control of

spatially-invariant systems," IEEE Trans. Automatic Control, vol. 45,pp. 1091-1107, 2000.

[2] J. Hoepffner, M. Chevalier,T.R. Bewley and D.S. Henningson, "Stateestimation in wall-bounded flow systems. Part 1. Perturbed laminarflows," Journal of Fluid Mechanics, to appear.

[3] P.J. Schmid and D.S. Henningson. Stability and Transition in ShearFlows, Springer, 2001.

[4] S. C. Reddy, P.J. Schmid, and D.S. Henningson, "Pseudospectra ofthe Orr-Sommerfeld operator," SIAM J. Appl. Math., vol. 53, no. 1,pp. 15-47, 1993.

[5] A. Smyshlyaev and M. Krstic, "Closed form boundary state feedbacksfor a class of partial integro-differential equations," IEEE Transactionson Automatic Control, vol. 49, pp. 2185-2202, 2004.

[6] A. Smyshlyaev and M. Krstic, "Backstepping observers for parabolicPDEs," Systems and Control Letters, vol. 54, pp. 1953-1971, 2005.

[7] R. Vazquez and M. Krstic, "A closed-form feedback controller forstabilization of linearized Navier-Stokes equations: the 2D Poisseuilleflow," 2005 Conference on Decision and Control, Sevilla.

5964

(VC) XX (t . X)