Stable Poiseuille Flow Transfer for a Navier-Stokes System

Transcription

1 Proceeings of the 26 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 26 WeB2.2 Stable Poiseuille Flow Transfer for a Navier-Stokes System Rafael Vázquez, Emmanuel Trélat an Jean-Michel Coron Abstract We consier the problem of generating an tracking a trajectory between two arbitrary parabolic profiles of a perioic 2D channel flow, which is linearly unstable for high ynols numbers. Also known as the Poisseuille flow, this problem is frequently cite as a paraigm for transition to turbulence. Our approach consists in generating an exact trajectory of the nonlinear system that approaches exponentially the objective profile. A bounary control law guarantees then that the error between the state an the trajectory ecays exponentially in the L 2 norm. The result is first prove for the linearize Stokes equations, then shown to hol for the nonlinear Navier-Stokes system. y Fig. 1. y = 1 U(y) x y = 2D Channel Flow with an equilibrium profile I. INTRODUCTION One of the few situations in which analytic expressions for solutions of the stationary flow fiel are available is the channel flow problem. Also known as the Poiseuille flow, this problem is frequently cite as a paraigm for transition to turbulence. Poiseuille flow requires an impose external pressure graient for being create an sustaine [3. The magnitue of the pressure graient etermines the value of the centerline velocity, which parameterizes the whole flow. It is very well known that this solution goes linearly unstable for ynols numbers greater than the so-calle critical ynols number, CR 5772 [11. The problem of locally stabilizing the equilibrium has been solve by means of optimal control [8, an backstepping [17. Observers have been evelope using ual methos [18. However, all prior works consier a constant pressure graient an a evelope flow which is alreay close to the esire solution. The problem of tracking time varying profiles generate by unsteay pressure graients has, so far, not been consiere from a control point of view. Stability for channel flow riven by unsteay pressure graient has been previously stuie [9. In this paper we consier the problem of moving the state from one Poiseuille equilibrium to another. For example, we may wish to smoothly accelerate flui at rest up to a given ynols number, probably over the critical value, avoiing transition to turbulence. The means at our isposal are the impose pressure graient an bounary control of the velocity fiel (only at one wall). This is a problem of practical interest which, to the best of our knowlege, has not been solve or even been consiere R. Vázquez is with the Department of Mechanical an Aerospace Engineering, University of California San Diego, La Jolla, CA E. Trélat is with the Univ. Paris-Su, Lab. Math., UMR 8628, 9145 Orsay Ceex, France. Jean-Michel Coron is with the Institut Universitaire e France an Univ. Paris-Su, Lab. Math., UMR 8628, 9145 Orsay Ceex, France. so far, since all control laws in the literature are esigne for one given Poiseuille flow (fixe ynols number). A possible solution for the problem woul be to apply quasi-static eformation theory; this woul require to moify the pressure graient very slowly, an simultaneously gain-scheule a fixe ynol number bounary controller like [17 for tracking a (slowly) time varying trajectory, which in general woul not be an exact solution of the system. This iea has been alreay use for moving between equilibria of a nonlinear parabolic equation [4, or a wave equation [5. Other applications inclue the shallow water problem [6 an the Couette-Taylor flow [12. In this paper, however, we follow an alternative approach, fining an exact, fast trajectory of the system which is then stabilize by means of bounary control. The avantage of this approach is that it reaches the objective profile much faster. The organization of the paper is as follows. We begin stating the moel in Section II. In Section III, we solve the problem of generating an exact trajectory between two Poiseuille profiles. Section IV presents the bounary control laws an our main results. We follow with Section V, where we present the tools we use to solve the problem. Section VI presents a sketch of proof of the main results. II. CHANNEL FLOW MODEL We consier a 2-D incompressible flui filling a region Ω between two infinite planes separe from each other a istance L, as shown in Fig. 1. Define U c as the maximum centerline velocity, ρ an ν as the ensity an the kinematic viscosity of the flui, respectively, an the ynols number,, as = U c h/ν. Then, using L, L/U c an ρνu c /L as length, time an pressure scales respectively, we can write /6/$2. 26 IEEE 775

2 the nonimensional 2-D Navier-Stokes equations as follows, u t = u p x uu x vu y, (1) v t = v p y uv x vv y, (2) where u is the streamwise velocity, v the wall-normal velocity, an p the pressure, with bounary conitions u(t, x, ) = v(t, x, ) =, (3) u(t, x, 1) = U(t, x), (4) v(t, x, 1) = V (t, x). (5) In (4) an (5), U an V are the actuators locate at the upper wall. Aitionally we consier an incompressible flui, so the velocity fiel must verify in Ω the ivergence-free conition u x + v y =. (6) In this nonimensional coorinates, Ω can be efine as Ω = {(x, y) R 2 : y 1}, (7) with bounary Ω = Ω Ω 1, where Ω an Ω 1 are the lower an upper wall, respectively, an will be referre to as the uncontrolle an controlle bounary. III. TRAJECTORY GENERATION AND CONTROL OBJECTIVE The stationary family of solutions of (1) (5) is the Poiseuille family of parabolic profiles, P δ, which is escribe by a single parameter δ (the maximum centerline velocity) in the following way ( P δ = (u δ, v δ, p δ ) = 4δy(1 y),, 8δ ) x. (8) Velocity actuation at the wall is zero for P δ, since both u δ an v δ are zero at the bounaries. The pressure graient p δ x = 8δ must be externally sustaine [3. Our first task is, given δ an δ 1, generate an unsteay trajectory path Θ(t) = (u(t), v(t), p(t)), where space epenence is omitte for clarity, connecting P δ to P δ1. We assume δ = an δ 1 = 1 for simplicity. Consier the trajectory Θ q (t) efine as Θ q (t) = (u q (t), v q (t), p q (t)) = (g(t, y),, xq(t)), (9) where q is the chosen external pressure graient. Then, by substitution we see that (9) verifies (1) (5) whenever g t = g yy q. (1) Since P, we set Θ q () =, which implies g(, y) = q() =. We impose g(t, ) = g(t, 1) = so no velocity control effort is neee to steer the trajectory. Given these ata, choosing q completely etermines g from (1) an consequently Θ q (t), so q(t) parameterizes Θ q (t). In particular, choosing q(t) as q(t) = 8 ( 1 e ct ), (11) g(t,y) 8 4 y.5 1 Fig. 2. Evolution of g(t, y) for c = 1, = 1. for c >, then q() = an lim t q(t) = 8/. Introucing (11) in (1), we can solve for g analytically. Supposing c π 2 (2m + 1) 2 / for m Z, g is then m= g = 16 [ m= sin ((2m + 1)πy) (2m + 1) 3 π 3 1 e π2 (2m+1) 2 t e ct e π c 1 π 2 (2m+1) 2 t (2m+1) 2 t As time grows, g goes exponentially to its steay state m= lim g(t, y) = 16 t m=.(12) sin ((2m + 1)πy) (2m + 1) 3 π 3 = 4y(1 y). (13) It can be prove as well 1 that g(t, y) is analytic on its omain of efinition an verifies g(t, y) 1, (14) g y (t, y) 4. (15) In Figure 2 we represent g, compute numerically from (1), for c = 1, = 1. Hence Θ q (t) solves trajectory generation problem, since it verifies (1) (5), Θ q () = P an lim t Θ q (t) = P 1. It follows that Θ q (t) connects the chosen Poiseuille profiles 2. Using (9), we efine the error variables as (ũ, ṽ, p) = (u, v, p) Θ q (t) = (u g(t, y), v, p xq(t)). (16) The error variables verify the following equations, ũ t = ũ p x ũũ x ṽũ y g(t, y)ũ x g y (t, y)ṽ, (17) ṽ t = ṽ p y ũṽ x ṽṽ y g(t, y)ṽ x, (18) an the same bounary conitions an ivergence-free conition as before. Our new objetive is to stabilize the equilibrium 1 Using the maximum principle an other heat equation properties. 2 aching P 1 only after an infinitely long time, however by construction through rapily ecaying exponentials, Θ q closely approaches P 1 after a short time, as shown in Fig. 2. In this sense, we consier Θ q a fast trajectory. 776

3 at the origin in (17) (18) by means of U an V. This woul imply that the trajectory Θ q is stabilize. Linearizing (17) (18) aroun Θ q, an ropping tiles, we obtain the unsteay Stokes equations u t = u p x g(t, y)u x g y (t, y)v, (19) v t = v p y g(t, y)v x, (2) with bounary conitions u(t, x, ) = v(t, x, ) =, (21) u(t, x, 1) = U(t, x), (22) v(t, x, 1) = V (t, x). (23) We aim to stabilize the origin of (19) (2), hence achieving local stabilization 3 for the original nonlinear system. IV. MAIN RESULTS Consier the following control laws. The controller V (t, x) is a ynamic controller, foun as the unique solution of the following force parabolic equation V t = V xx [ 2i < n <M h e iγn(ξ x) g y (t, ) cosh (γ n (1 )) V (τ, ξ, ) ξ, (24) i u y(t, ξ, ) u y (t, ξ, 1) initialize at zero, whereas the control law U is given by U = < n <M h e iγn(ξ x) K n (t, 1, )u(t, ξ, )ξ (25) where M = 2h π an γ n = πn/h. K n in (25) is the solution, for each n, of the following (well-pose)kernel equation K nt = 1 (K nyy K n ) λ n (t, )K n + f(y, ) f(ξ, )K n (t, y, ξ)ξ, (26) a linear partial integro-ifferential equation in the region Γ = (t, y, ) (, ) T, where T = {(y, ) R 2 : y 1}, with bounary conitions: ( K n (t, y, y) = λ(y) y ) 2 + µ n(), (27) [ K n (t, y, ) = µ n (σ)k n (t, y, σ)σ µ n (y),(28) 3 Local stabilization suffices, since we assume the initial ata are zero, i.e. the velocity fiel starts at the origin itself. an where the functions that appear in (26) (28) are λ n (t, y) = iγ n g(t, y), (29) [ f n (t, y, ) = iγ n g y (t, y) + 2γ n g y (t, σ) sinh (γ n (y σ)) σ, (3) µ n (y) = γ n cosh (γ n (1 y)) cosh (γ n y)).(31) sinh γ n We state now our results. Theorem 4.1: For any ynols number, the equilibrium u v of Stokes system (19) (23) with control laws (24) (25) is exponentially stable in the L 2 norm, i.e., if w = (u, v), there exist numbers C 1 (), C 2 () > such that for t, w(t) C 1 e C2t w(). (32) The result above is vali for any initial conition. If we consier the nonlinear terms, local stability follows. Theorem 4.2: For any ynols number, the equilibrium u v of the Navier-Stokes system (17) (18) with bounary conitions (21) (23) an control laws (24) (25) is locally exponentially stable in the L 2 norm, i.e., if w = (u, v), there exist numbers ɛ(), C 1 (), C 2 () > such that if w() < ɛ, an for t, w(t) C 1 e C2t w(). (33) From Section III an Theorem 4.2, the final result follows. Theorem 4.3: For any ynols number, Θ q (t) efine by (9) (12) is a solution of system (1) (5), with impose pressure graient (11), an control laws (24) (31) expresse in the error variables (16). Moreover, this solution is locally exponentially stable in the L 2 norm. In particular, if the state is initialize close enough to rest, it closely follows Θ q (t) an approaches the steay equilibrium P 1 exponentially fast. mark 1: Even though the controller (24) (31) looks rather involve, it is not har to implement. A finite set of linear PIDE equations has to be solve for computing the controllers, which can be one fast an efficiently [13. mark 2: This result can be extene in a number of ways. An output feeback esign is possible applying a ual backstepping observer methoology [15, [18, only requiring bounary measurements of pressure an skin friction. A 3D channel flow, perioic in two irections, is also tractable, aing some refinements which inclue actuation of the spanwise velocity at the wall. Stability in the H 1, H 2 norms can be obtaine as well. We skip the etails ue to page limit. mark 3: From (24) it follows that the mean of V is zero, hence verifying the zero net flux conition. In the next sections we introuce a mathematical framework an prove some of the results, skipping some proofs ue to space restriction. Full etails will be provie in a future publication. 777

4 V. MATHEMATICAL PRELIMINARIES A. Perioic function spaces We assume that the velocity fiel (u, v) an the pressure p are perioic in x with some perio 2h > [16. This requires for consistency that U an V are perioic with the same perio; a property alreay verifie by expressions (24) (25). Ω an its bounary are ientifie with Ω h = {(x, y) Ω : x h}, (34) Ω hi = {(x, y) Ω i : x h}. (35) Let L 2 (Ω h ) be the usual Lebesgue space of square-integrable functions, enowe with the scalar prouct (φ, ψ) L2 (Ω h ) = h Define then L 2 h (Ω) = L2 (Ω h ), where now φ(x, y)ψ(x, y)yx. (36) (φ, ψ) L 2 h (Ω) = (φ Ωh, ψ Ωh ) L 2 (Ω h ). (37) B. Fourier series expansion Given a function φ we efine the sequence of its complex Fourier coefficients (φ n (y)) n Z as φ n (y) = 1 2h h φ(x, y)e inπ h x x, n Z. (38) We will simply write φ n in the sequel. It can be shown that if φ L 2 (Ω h ), then (38) is well efine an φ n is in the (complex value) l 2 L 2 (, 1) space, i.e., n Z φ n (y) 2 y <. (39) One can recover φ by writting its Fourier series, φ(x, y) = n Z φ n (t, y)e inπ h x. (4) Equation (4) yiels a L 2 (Ω h ) function if φ n l 2 L 2 (, 1). One important result is Parseval s formula (φ, ψ) L 2 (Ω h ) = (φ n, ψ n ) l 2 L 2 (,1) (41) where the l 2 L 2 (, 1) scalar prouct is (φ n, ψ n ) l2 L 2 (,1) = n Z φ n (y) ψ n (y)y, (42) an where the bar enotes the complex conjugate. In the sequel we frequently omit the subinexes when clear from the context. Using (41), an given ψ in L 2 (Ω h ), we can compute its norm by computing its Fourier coefficients ψ n. Then, where ψ 2 L 2 (Ω h ) = ψ 2 l 2 L 2 (,1) = ψ n 2 L 2 (,1), (43) ψ n 2 L 2 (,1) = ψ n (y) 2 y. (44) C. H 1 spaces We efine the space Hh 1 (Ω) as H 1 h(ω) = {f Ωh H 1 (Ω h ), f x= = f x=h a.e.}. (45) The H 1 norm is efine as φ 2 Hh 1(Ω) = φ 2 L 2 h (Ω) + φ y 2 L 2 h (Ω) + φ x 2 L 2 h (Ω), (46) or in terms of the Fourier coefficients φ 2 Hh 1(Ω) = [(1 + 4π 2 n 2 ) φ n 2 L 2 (,1) + φ ny 2 L 2 (,1). (47) We state the following lemma, whose proof we skip. Lemma 5.1: Suppose that φ Hh 1(Ω) such that φ Ω, an ψ L 2 h (Ω). Then: (φ 2, ψ 2 ) L 2 h (Ω) φ y 2 L 2 h (Ω) ψ 2 L 2 h (Ω). (48) D. Spaces for the velocity fiel Calling w = (u, v), we efine H h (Ω) = {w [L 2 h(ω) 2 : w =, w Ω = }(49) H 1 h(ω) = H h (Ω) [H 1 h(ω) 2, (5) enowe with the scalar prouct of, respectively, [L 2 h (Ω)2 an [H 1 h (Ω)2. See [16 for the precise meaning of ivergence an trace in this space. E. Transformations of L 2 functions Our approach uses the backstepping metho [13. The metho is base on fining a invertible mapping of the state variables into others with esire stability properties. We stuy the kin of transformations that appear in the metho. Definition 5.1: Given complex value functions f L 2 (, 1) an K L (T ), we efine the transforme variable g = (I K)f, where the operator Kf is efine as Kf = K(y, )f(), (51) i.e. a Volterra operator. We call I K the irect transformation with kernel K. If there exists a function L L (T ) such that f = (I + L)g, then we say that the transformation is invertible, an we call I + L the inverse transformation, an L the inverse kernel (or the inverse of K). We state some important results [7. Proposition 5.1: For K L (T ), the transformation I K is always invertible. Moreover, L is relate to K by L(y, ) = K(y, ) + = K(y, ) + K(y, σ)l(σ, )σ L(y, σ)k(σ, )σ. (52) Proposition 5.2: If f L 2 (, 1) then g = (I K)f is in L 2 (, 1). Similarly, if g L 2 (, 1) then f = (I + L)g is in L 2 (, 1). Moreover, g 2 L 2 (,1) (1 + K ) 2 f 2 L 2 (,1), (53) f 2 L 2 (,1) (1 + L ) 2 g 2 L 2 (,1). (54) 778

5 Proposition 5.2 allows to efine an L 2 equivalent norm, f 2 KL 2 (,1) = (I K)f 2 L 2 (,1) = g 2 L 2 (,1). (55) For C 1 (T ) kernels K an L, one has an equivalent version of Proposition 5.1 an Proposition 5.2, allowing to efine a KH 1 (, 1) norm, which is equivalent to the H 1 (, 1) norm. F. Transformations of the velocity fiel Definition 5.2: Consier A = {a 1,..., a j } Z an K = (K n (y, )) n A a family of L (T ) kernels. Then, for w = (u, v) H h (Ω), one efines the transforme variable ω = (α, β) = (I K)w, through its Fourier components, { ((I Kn )u ω n = n, ) n A, (56) w n, otherwise. The inverse transformation, w = (I + L)ω, is efine as { ((I + Ln )α w = n, ˆL n α n ) n A, (57) ω n, otherwise, where the new operator ˆL n is efine as: ˆL n f = πi n ( ) f() + L(, σ)f(σ)σ. (58) h Using the ivergence-free conition in Fourier space, πi n h u n+v ny =, an the bounary conition v n () =, it is straightforwar to show that the inverse is correctly efine. Even though v n is apparently lost, it can be recovere since the transformation is invertible. Using a similar argument as in Proposition 5.2, ω 2 H h (Ω) (1 + K ) 2 w 2 H h (Ω), (59) w 2 H h (Ω) (1 + N 2 )(1 + L ) 2 ω 2 H h (Ω), (6) where N = max n A {π n h }, an K = max n }, n A (61) L = max n }. n A (62) This allows the efinition of an H h (Ω) equivalent norm, as in (55), that we call KH h (Ω). VI. PROOF OF THEOREM 4.1 Equations (19) (2) in Fourier space are u nt = nu n iγ n (p n + g(t, y)u n ) g y (t, y)v n (63) v nt = nv n p ny iγ n g(t, y)v n, (64) where n = yy γn. 2 The bounary conitions are u n (t, ) = v n (t, ) =, (65) u n (t, 1) = U n (t), (66) v n (t, 1) = V n (t), (67) an the ivergence-free conition becomes iγ n u n + v ny =. (68) From (63) (64) an equation for the pressure can be erive, p nyy γ 2 np n = 2iγ n g y (t, y)v n, (69) with bounary conitions obtaine from evaluating (64) at the bounaries an using (66) (67), u ny (, t) p ny (, t) = iγ n, (7) p ny (1, t) = iγ n u ny (1, t) V n γn 2 V n. (71) Equations for ifferent n s are uncouple, allowing separate consieration for each moe. Most moes, which we refer to as uncontrolle, are naturally stable an thus left without control. A finite set of moes, calle controlle, are unstable an require control. A. Uncontrolle moes 1) n = (mean velocity fiel): From (68), v. u verifies u t = u yy, (72) with u () = u (1) =. The following estimate hols: t u (t) 2 L 2 (,1) e 2 t u () 2 L 2 (,1). (73) 2) Moes for large n : If w n = (u n, v n ), then, consiering no control (V n = U n = ): t w n 2 = 2 w ny 2 2γ 2 w n 2 n (g y u n, v n ) (g y v n, u n ) (u n, iγ n p n ) (iγ n p n, u n ) (v n, p ny ) (p ny, v n ). (74) Consier the pressure terms like those in the last two lines of (74). Using (68) an integration by parts, (u n, iγ n p n ) = (v ny, p n ) = (v n, p ny ). (75) Then, using Young s inequality with the remaining terms, t w n 2 2 w ny 2 2γ 2 w n 2 n + g y L w n 2. (76) Since g y (t, y) 4, choosing γ n 2, i.e., yiels n M = 2h, (77) π t w n 2 2 w ny 2 γ 2 w n 2 n 2 w n 2, (78) by Poincare s inequality, therefore achieving L 2 exponential stability for large moes ( n M). B. Controlle moes. Construction of control laws The moes < n < M are open-loop unstable an must be controlle. We esign the control in several steps. 779

6 1) Pressure shaping: Solving (69) (71), p n = 2i +2i cosh (γ ny) sinh γ n g y (t, ) sinh (γ n (y )) v n (t, ) g y (t, ) cosh (γ n (1 )) v n (t, ) + i cosh (γ n(1 y)) u ny (, t) sinh γ n ( cosh (γ ny)) i u ny(1, t) + V ) n V n + γ n.(79) sinh γ n γ n We set V n to enforce in (79) a strict-feeback structure [1 in y. This property, require by backstepping [13, [14, is a sort of spatial causality. We choose V n as V n γ n V n = γ n iu ny(, t) u ny (1, t) 2i g y (t, ) cosh (γ n (1 )) v n (t, ).(8) 2) Control of velocity fiel: By (68), v n can be compute from u n. Then, only u n is neee. Using (68) to eliminate v n an introucing (8) an (79) into (63), yiels u nt = nu n + λ n (t, y)u n + f n (t, y, )u n (t, ) +µ n (y)u ny (, t), (81) with λ n, f n an µ n as in (29) (31), an bounary conitions u n (t, ) =, (82) u n (t, 1) = U n (t). (83) Following [14 we map u n, for each moe < n < M, into the family of heat equations: where α nt = 1 ( ) γ 2 n α n + α nyy (84) α n (k, ) = α n (k, 1) =, (85) α n = (I K n )u n (86) u n = (I + L n )α n, (87) are respectively the irect an inverse transformations. The kernel K n is foun to verify equations (26) (28), which can be prove well-pose, an L n can be erive from K n. The control law is, from (86), (85) an (83) U n = K n (t, 1, )u n (t, k, ), (88) Stability follows from (84) (85) an (86) (87). We obtain t u n 2 K nl 2 (,1) e 2 t u n () 2 K nl 2 (,1). (89) C. Stability for the whole system If we call A = {n Z : < n < M}, an K = K n (t, y, ) n A, an apply the control laws (88) (8) in physical space, which yiel (25) (24), then it follows that w 2 KH h (Ω) = n/ A w n 2 L 2 (,1) 2 + n A e 2 t [ n/ A w n () 2 L 2 (,1) 2 + u n () 2 K nl 2 (,1) n A u n 2 K nl 2 (,1) e 2 t w() 2 KH h (Ω), (9) an by norm equivalency, this proves Theorem 4.1. ACKNOWLEDGEMENTS We acknowlege the support of a EU Marie Curie Fellowship, in the framework of the CTS, contract number HPMT- CT We thank T. Bewley, M. Krstic an A. Smyshlyaev for valuable suggestions for helpful iscussions. REFERENCES [1 O. M. Aamo an M. Krstic, Flow Control by Feeback: Stabilization an Mixing, Springer, 22. [2 A. Balogh, W.-J. Liu an M. Krstic, Stability enhancement by bounary control in 2D channel flow, IEEE Transactions on Automatic Control, vol. 46, pp , 21. [3 G. K. Batchelor, An Introuction to Flui Mechanics, Cambrige University Press, Lonon, [4 J.-M. Coron an E. Trélat, Global steay-state controllability of 1D semilinear heat equations, SIAM J. Contr. an Opt., vol. 43, pp , 24. [5 J.-M. Coron an E. Trelat, Global steay-state stabilization an controllability of 1D semilinear wave equations, to appear, Commun. Contemp. Math., 26. [6 J.-M. Coron, Local controllability of a 1D tank containing a flui moele by the shallow water equations, ESAIM: Contr. Optim. Calc. Variat., vol. 8, pp , 22. [7 H. Hochstat, Integral Equations, Wiley, [8 M. Hogberg, T. R. Bewley an D. S. Henningson, Linear feeback control an estimation of transition in plane channel flow, Journal of Flui Mechanics, vol. 481, pp , 23. [9 C. H. von Kerczek, The instability of oscillatory plane Poiseuille flow, Journal of Flui Mechanics, vol. 116, pp , [1 M. Krstic, I. Kanellakopoulos an P. V. Kokotovic, Nonlinear an Aaptive Control Design, Wiley, [11 P. J. Schmi an D. S. Henningson. Stability an Transition in Shear Flows, Springer, 21. [12 M. Schmit, E. Trélat, Controllability of Couette flows, Comm. Pure Applie Analysis 5, 1 (26), [13 A. Smyshlyaev an M. Krstic, Close form bounary state feebacks for a class of partial integro-ifferential equations, IEEE Transactions on Automatic Control, vol. 49, pp , 24. [14 A. Smyshlyaev an M. Krstic, On control esign for PDEs with space-epenent iffusivity or time-epenent reactivity, Automatica, vol. 41, pp , 25. [15 A. Smyshlyaev an M. Krstic, Backstepping observers for parabolic PDEs, Systems an Control Letters, vol. 54, pp , 25. [16 R. Temam, Navier-Stokes Equations, Theory an Numerical Analysis, North-Hollan Publishing Co., Amsteram, [17 R. Vázquez an M. Krstic, A close-form feeback controller for stabilization of linearize Navier-Stokes equations: the 2D Poisseuille flow, Procs. of the 44th CDC, Sevilla, 25. [18 R. Vázquez an M. Krstic, A close-form observer for the channel flow Navier-Stokes system, Procs. of the 44th CDC, Sevilla,