A Closed-Form Feedback Controller for Stabilization of the Linearized 2D Navier-Stokes Poisseuille System

A Closed-Form Feedback Controller for Stabilization of the Linearized 2D Navier-Stokes Poisseuille System Rafael Vazquez and Miroslav Krstic Submitted to IEEE Transactions on Automatic Control September 19, 2005 Abstract We present a formula for a boundary control law which stabilizes the parabolic profile of an infinite channel flow, which is linearly unstable for high Reynolds numbers. Also known as the Poisseuille flow, this problem is frequently cited as a paradigm for transition to turbulence, whose stabilization for arbitrary Reynolds numbers, without using discretization, has so far been an open problem. Our result achieves exponential stability in the L2,H1and H2norms, for the linearized Navier-Stokes equations, guaranteeing local stability for the nonlinear system. Explicit solutions are obtained for the closed loop system. This is the first time explicit formulae are produced for solutions of the Navier-Stokes equations. The result is presented for the 2D case for clarity of exposition. An extension to 3D is available and will be presented in a future publication. This work was supported by NSF grant number CMS-0329662. Department of Mechanical and Aerospace Engineering, University of California at San Diego 2 I. INTRODUCTION We present an explicit boundary control law which stabilizes a benchmark 2D linearized Navier-Stokes system. Despite the deceptive simplicity of the channel flow geometry, there is a number of complex issues underlying this problem [13], making it extremely hard to solve. Controllability and stabilizability results for the Navier-Stokes equations are available for general geometries; for example, see [9], [10], [12] and references therein. However, these results do not provide the means of computing a feedback controller. Many efforts in the design of feedback controllers for the Navier-Stokes system employ indomain actuation, using optimal control methods [7] or model reduction techniques [4]. For boundary feedback control, optimal control theory has also been developed [16], and specialized to specific geometries, like cylinder wake [15]. There are also new techniques arising for specific flow control problems like separation control [3]. Optimal control has so far been the most successful technique for addressing channel flow stabilization [11], in a periodic setting, by using a discretized version of the equations and employing high-dimensional algebraic Riccati equations for computation of gains. The computational complexity of this approach is formidable if a very fine grid is necessary in the discretizations, for example if the Reynolds number is very large. Using a Lyapunov/passivity approach, another control design [1], [5] was developed for stabilization of the (periodic) channel flow; the design was simple and explicit and did not rely on discretization or linearization, but its theory was restricted to low Reynolds numbers though in simulations the approach was successful at high Reynolds numbers, above the linear instability threshold. The approach we present in this paper is the first result that provides an explicit control law (with symbolically computed gains) for stabilization at an arbitrarily high Reynolds number in non-discretized Navier-Stokes equations, and it is applicable to both infinite and periodic September 17, 2005 DRAFT 3 channels with arbitrary periodic box size, and also extends to 3D. Thanks to the explicitness of the controller, we are able to obtain approximate analytical solutions for the Navier-Stokes equations. Exponential stability in the L2,H1and H2norms is proved for the linearized Stokes system around the Pouiseuille profile, therefore local stability is achieved for the nonlinear Navier-Stokes system. We do not prove well-posedness, however, with the high-order Sobolev estimates that we derive it is certainly possible, though lengthy and not trivial. The method we use for solving the stabilization problem is based on the recently developed backstepping technique for parabolic systems [20], which has been successfully applied to flow control problems, for example to the vortex shedding problem [2] and to feedback stabilization of an unstable convection loop [24]. We start the paper by stating, in Section II, the mathematical model, which consists of the linearized Navier-Stokes equations for the velocity fluctuation around the (Pouisseuille) equilibrium profile. In Section III, we introduce the control law that stabilizes the equilibrium profile. Explicit solutions for the closed loop system are then stated in Section IV along with the main results of the paper. Sections V, VI, and VII deal with the proof of, respectively, L2,H1and H2stability of the closed loop system. A Fourier transform approach allows separate analysis for each wave number. For certain wave numbers, a normal velocity controller puts the system into a form where a linear Volterra operator, combined with boundary feedback, can transform the original normal velocity PDE into a stable heat equation. For the rest of wave numbers the system is proved to be open loop exponentially stable, and is left uncontrolled. These two results are combined to prove stability of the closed loop system for all wave numbers and in the physical space. Section VIII is devoted to study and prove some properties of the control laws. In Section IX, we finish the paper with a discussion of the results. September 17, 2005 DRAFT 4 y = 0 y = 1 x y U ( y ) Fig. 1. 2D channel flow and equilibrium profile. Actuation is on the top wall. II. MODEL Consider a 2D incompressible channel flow evolving in a semi-infinite rectangle (x, y)∈ (−∞,∞)×[0,1] as in Figure 1. The dimensionless velocity field is governed by the Navier- Stokes equations Ut=1 Re (Uxx +Uyy)−UUx−V Uy−Px,(1) Vt=1 Re (Vxx +Vyy)−UVx−V Vy−Py,(2) and the continuity equation Ux+Vy= 0,(3) where Udenotes the streamwise velocity, Vthe wall-normal velocity, Pthe pressure, and Re is the Reynolds number. The boundary conditions for the velocity field are the no-penetration, no-slip boundary conditions for the uncontrolled case, i.e., V(x, 0) = V(x, 1) = U(x, 0) = U(x, 1) = 0. Instead of using (3) we derive a Poisson equation that Pverifies, combining (1), (2) and (3) Pxx +Pyy =−2(Vy)2−2VxUy,(4) with boundary conditions Py(x, 0) = (1/Re)Vyy(x, 0) and Py(x, 1) = (1/Re)Vyy(x, 1), which are obtained evaluating (2) at y= 0,1. September 17, 2005 DRAFT 5 The equilibrium solution of (1)–(3) is the parabolic Poisseuille profile Ue= 4y(1 −y),(5) Ve= 0,(6) Pe=P0−8 Rex, (7) shown in Figure 1. This equilibrium is unstable for high Reynolds numbers [19]. Defining the fluctuation variables u=U−Ueand p=P−Pe, and linearizing around the equilibrium profile (5)–(7), the plant equations become the Stokes equations ut=1 Re (uxx +uyy) + 4y(y−1)ux+ 4(2y−1)V−px,(8) Vt=1 Re (Vxx +Vyy) + 4y(y−1)Vx−py,(9) pxx +pyy = 8(2y−1)Vx,(10) with boundary conditions u(x, 0) = 0,(11) u(x, 1) = Uc(x),(12) V(x, 0) = 0,(13) V(x, 1) = Vc(x),(14) py(x, 0) = Vyy(x, 0) Re ,(15) py(x, 1) = Vyy(x, 1) + (Vc)xx(x) Re −(Vc)t(x).(16) The continuity equation is still verified ux+Vy= 0.(17) We have added in (12) and (14) the actuation variables Uc(x)and Vc(x), respectively for streamwise and normal velocity boundary control. The actuators are placed along the top wall, September 17, 2005 DRAFT 6 y= 1, and we assume they can be independently actuated for all x∈R. No actuation is done inside the channel or at the bottom wall. Taking Laplacian in equation (9) and using (10), we get an autonomous equation for the normal velocity, the well-known Orr-Sommerfeld equation, △Vt=1 Re△2V+ 4y(y−1)△Vx−8Vx,(18) with boundary conditions (13)–(14), as well as Vy(x, 0) = 0,Vy(x, 1) = −(Uc)x, derived from (11)–(12) and (17). This equation is numerically studied in hydrodynamic theory to determine stability of the channel flow [17]. Defining Y=−Vy, it is possible to partially solve (18) and obtain an evolution equation for Y, Yt=1 Re (Yxx +Yyy) + 4y(y−1)Yx+Zy 0Z∞ −∞ Y(ξ, η)Z∞ −∞ 16πke2πik(x−ξ) ×[πk(2y−1) −2 sinh (2πk(y−η)) 2πk(2η−1) cosh (2πk(y−η))] dkdξdη +Z1 0Z∞ −∞ Y(ξ, η)Z∞ −∞ 32πke2πik(x−ξ)cosh (2πky) sinh (2πk)[cosh (2πk(1 −η)) +πk(2η−1) sinh (2πk(1 −η))] dkdξdη +Z∞ −∞ Z∞ −∞ Yy(ξ, 1) −(Vc)xx(ξ) Re + (Vc)t(ξ)2πke2πik(x−ξ)cosh (2πky) sinh (2πk)dkdξ −Z∞ −∞ Z∞ −∞ Yy(ξ, 0) Re 2πke2πik(x−ξ)cosh (2πk(1 −y)) sinh (2πk)dkdξ, (19) with boundary conditions Yy(x, 0) = 0 and Y(x, 1) = (Uc)x. Equation (19) governs the channel flow, since from Yand using (17), we recover both components of the velocity field: V(x, y) = −Zy 0 Y(x, η)dη, (20) u(x, y) = Zx −∞ Y(ξ, y)dξ. (21) Equation (19) displays the full complexity of the Navier-Stokes dynamics, which the PDE system (8)–(10) conceals through the presence of the pressure equation (10), and the Orr- September 17, 2005 DRAFT 7 Sommerfeld equation (18) conceals through the use of fourth order derivatives. Besides being unstable (for high Reynolds numbers), the Ysystem incorporates (on its right-hand side) the components of Y(x, y)from everywhere in the domain. This is the main source of difficulty for both controlling and solving the Navier-Stokes equations. A perturbation somewhere in the flow is instantaneously felt everywhere—a consequence of the incompressible nature of the flow. Our approach to overcoming this obstacle is to use one of the two control variables (normal velocity Vc(x), which is incorporated explicitly inside the equation) to prevent perturbations from propagating in the direction from the controlled boundary towards the uncontrolled boundary. This is a sort of “spatial causality” on y, which in the nonlinear control literature is referred to as the ‘strict-feedback structure’ [14]. III. CONTROLLER The explicit control law consists of two parts—the normal velocity controller Vc(x)and the streamwise velocity controller Uc(x).Vc(x)makes the integral operator in the third to fifth lines of (19) spatially causal in y,1which is a necessary structure for the application of a “backstepping” boundary controller for stabilization of spatially causal partial integro-differential equations [20]. Uc(x)is a backstepping controller which stabilizes the spatially causal structure imposed by Vc(x). The expressions for the control laws are Uc(t, x) = Z1 0Z∞ −∞ Qu(x−ξ, η)u(t, ξ, η)dξdη, (22) Vc(t, x) = h(t, x),(23) where hverifies the equation ht=hxx +g(t, x),(24) 1The first, second and sixth lines are already spatially causal in y. September 17, 2005 DRAFT 8 where g=Z1 0Z∞ −∞ QV(x−ξ, η)V(t, ξ, η)dξdη +Z∞ −∞ Q0(x−ξ) (uy(t, ξ, 0) −uy(t, ξ, 1)) dξ, (25) and the kernels Qu,QVand Q0are defined as Qu=Z∞ −∞ χ(k)K(k, 1, η)e2πik(x−ξ)dk, (26) QV=Z∞ −∞ χ(k)16πki(2η−1) cosh (2πk(1 −η)) e2πik(x−ξ)dk, (27) Q0=Z∞ −∞ χ(k)2πki Re e2πik(x−ξ)dk. (28) In expressions (26)–(28), χ(k)is a truncating function in the wave number space whose definition is χ(k) =    1, m < |k|< M 0,otherwise (29) where mand Mare respectively the low and high cut-off wave numbers, two design parameters which can be conservatively chosen as m≤1 32πRe and M≥1 πqRe 2. The function K(k, y, η) appearing in (26) is a (complex valued) gain kernel defined as K(k, y, η) = lim n→∞ Kn(k, y, η),(30) September 17, 2005 DRAFT 9 where Knis recursively defined as 2 K0=−2πkcosh (2πk(1 −y+η)) −cosh (2πk(y−η)) sinh (2πk)+ 4iReη(η−1) sinh (2πk(y−η)) −Re 3πikη 21y2−6y(3 + 4η) + η(12 + 7η)−6ηiRe πk (1 −cosh (2πk(y−η))) (31) Kn=Kn−1−4πkiRe Zy+η y−ηZy−η 0Zδ −δsinh (πk(ξ+δ)) πk −(2ξ−1) +2(γ−δ−1) cosh (πk(ξ+δ)) Kn−1k, γ+δ 2,γ+ξ 2dξdδdγ +Re 2πik Zy+η y−ηZy−η 0 (γ−δ)(γ−δ−2)Kn−1k, γ+δ 2,γ−δ 2dδdγ +2πk Zy−η 0 cosh (2πk(1 −δ)) −cosh (2πkδ) sinh (2πk)Kn−1(k, y −η, δ)dδ. (32) The terms of this series can be computed symbolically as they only involve integration of polynomials and exponentials. In implementation, a few terms are sufficient to obtain a highly accurate approximation because the series is rapidly convergent [20]. Remark 1: (23) is a dynamic controller whose magnitude is determined by the variable h(t, x), which evolves according to (24). We use an initial condition h(0, x)≡0. The stabilization result remains valid for h(0, x)6= 0, however it involve additional routine effort to account for the exponentially stable effect of the compensator internal dynamics (which are of heat equation type). Remark 2: Control kernels (27) and (28) can be explicitly expressed as QV(ξ, η) = 8(2η−1)RV(ξ, η, M)−RV(ξ, η, m) ξ2+ (1 −η)2(33) Q0(ξ, η) = R0(ξ, η, M)−R0(ξ, η, m) Re ξ ,(34) 2This infinite sequence is convergent, smooth, and uniformly bounded over (y, η)∈[0,1]2, and analytic in k. September 17, 2005 DRAFT 16 Note that the relation between Yin (19) and uin (64) is that Y(k, y) = 2πkiu(k, y). Now, we design the controller in two steps. First, we set Vcso that (64) has a strict-feedback form in the sense previously defined: (Vc)t=2πki (uy(k, 0) −uy(k, 1)) −4π2k2Vc Re −16πki Z1 0 (2η−1)V(k, η) cosh (2πk(1 −η)) dη. (67) This can be integrated and explicitly stated as a dynamic controller in the Laplace domain: Vc=2πki s+4π2k2 Re uy(s, k, 0) −uy(s, k, 1) Re × −8Z1 0 (2η−1)V(s, k, η) cosh (2πk(1 −η)) dη.(68) Control law (67) can be expressed in the time domain and physical space as (23)–(25) and (27), (28), by use of the convolution theorem of the Fourier transform. Introducing Vcin (64) yields ut=1 Re −4π2k2u+uyy+ 8πkiy(y−1)u +8iZy 0 {πk(2y−1) −2 sinh (2πk(y−η)) −2πk(2η−1) cosh (2πk(y−η))}u(k, η)dη −2πkcosh (2πky)−cosh (2πk(1 −y)) sinh (2πk) uy(k, 0) Re .(69) Equation (69) can be stabilized using the backstepping technique for parabolic partial integrodifferential equations [20]. Using backstepping, we map u, for each wave number m < |k|< M, into the family of heat equations αt=1 Re −4π2k2α+αyy,(70) α(k, 0) = 0 ,(71) α(k, 1) = 0 ,(72) September 17, 2005 DRAFT 17 where α=u−Zy 0 K(k, y, η)u(t, k, η)dη , (73) u=α+Zy 0 L(k, y, η)α(t, k, η)dη , (74) are respectively the direct and inverse transformation. The kernel Kis found to verify the following equation 1 ReKyy =1 ReKηη + 8πikη(η−1)K−8i{πk(2y−1) −sinh (2πk(y−η)) −2πk(2η−1) cosh (2πk(y−η))}+ 8iZy η {πk(2ξ−1) −2 sinh (2πk(ξ−η)) −2πk(2η−1) cosh (2πk(ξ−η))}K(k, y, ξ)dξ, (75) a hyperbolic partial integro-differential equation (PIDE) in the region T={(y, η) : 0 ≤η≤ y≤1}with boundary conditions: K(y, y) = −2Re 3πiky2(2y−3) −2πkcosh (2πk)−1 sinh (2πk),(76) K(y, 0) = 2πk sinh (2πk)cosh (2πky)−cosh (2πk(1 −y)) +Zy 0 K(k, y, ξ) [cosh (2πk(1 −ξ)) −cosh (2πkξ)] dξ.(77) The equation can be transformed into an integral equation for calculating the kernel symbolically. To do this, we transform the PIDE into an integral equation and solve it explicitly via a successive approximation series. The series definition of Kis (30)–(32). We skip the details, since we follow [20] exactly, with the only difference that the kernel is complex valued, which does not change the proof. In addition, using the estimates of the proof and the fact that the terms in the series definition (31)–(32) of Kare analytic in k, it can be shown that the kernel itself is also analytic as a complex function of k, for any bounded k[18], so in particular, it will be analytic in the annulus m < |k|< M. September 17, 2005 DRAFT 18 From the transformation (73) and the boundary condition (65) the control law is Uc=Z1 0 K(k, 1, η)u(t, k, η)dη. (78) Using the convolution theorem of the Fourier transform we write the control law (78) back in physical space. The resulting expressions is (22). The equation for the inverse kernel Lin (74) is similar to the one of Kand enjoys similar properties 1 ReLyy =1 ReLηη −8πiky(y−1)L−8i{πk(2y−1) −2 sinh (2πk(y−η)) −2πk(2η−1) cosh (2πk(y−η))} − 8iZy η {πk(2y−1) −sinh (2πk(y−ξ)) +2πk(2ξ−1) cosh (2πk(y−ξ))}L(k, ξ, η)dξ, (79) again a hyperbolic partial integro-differential equation in the region Twith boundary conditions L(y, y) = −2Re 3πiky2(2y−3) −2πkcosh (2πk)−1 sinh (2πk),(80) L(y, 0) = 2πk sinh (2πk)cosh (2πky)−cosh (2πk(1 −y)) .(81) The equation can be transformed into an integral equation and calculated via the successive approximation series (45)–(46). By using (63) and (73)–(74), Vcan also be expressed in terms of α α=iVy−Ry 0K(k, y, η)Vy(t, k, η)dη 2πk (82) V=−2πki Zy 01 + Zy η L(k, η, σ)dσα(t, k, η)dη . (83) Since we can solve the heat equation (70)–(72) explicitly, the inverse transformations (74) and (83) yield the explicit solutions u∗(t, k, y)and V∗(t, k, y), respectively. Moreover, since (73)–(74) map (69) into (70), stability properties of the velocity field follows from those of the αsystem. September 17, 2005 DRAFT 19 Proposition 1: For any kin the range m < |k|< M, the equilibrium u(t, k, y)≡V(t, k, y)≡ 0of the system (52)–(60) with feedback control laws (67), (78) is exponentially stable in the L2sense, i.e., ||V(t, k)||2 ˆ L2+||u(t, k)||2 ˆ L2≤D0e −1 2Re t||V(0, k)||2 ˆ L2+||u(0, k)||2 ˆ L2,(84) where D0is defined as: D0= (1 + 4π2M2) max m<|k|<M{(1 + ||L||∞)2(1 + ||K||∞)2}.(85) Proof: First, from the αequation (70) it is possible to get an L2estimate ||α(t, k)||2 ˆ L2≤e−1 2Re t||α(0, k)||2 ˆ L2,(86) then employing the direct and inverse transformations (73)–(74) and (83) we get (84)–(85). Now, if we apply the feedback laws (67), (78) for all wave numbers m < |k|< M, then the control laws in physical space are given by expressions (22)–(28), where the inverse transform integrals are truncated at k=m, M in (26)–(28). If we define V∗(t, x, y) = Z∞ −∞ χ(k)V(t, k, y)e2πikxdk, (87) u∗(t, x, y) = Z∞ −∞ χ(k)u(t, k, y)e2πikxdk, (88) which are variables that contain all velocity field information for wave numbers m < |k|< M, the following result holds. Proposition 2: Consider equations (8)–(16) with control laws (22)–(23). Then the variables u∗(t, x, y)and V∗(t, x, y)defined in (87)–(88) decay exponentially: ||V∗(t)||2 L2+||u∗(t)||2 L2≤D0e −1 2Re t||V∗(0)||2 L2+||u∗(0)||2 L2.(89) Proof: The Fourier transform of the star variables is, by definition, the same as the Fourier transform of the original variables for m < |k|< M, and zero otherwise. Therefore, applying September 17, 2005 DRAFT 20 Parseval’s formula and Proposition 1, ||V∗(t)||2 L2+||u∗(t)||2 L2=Z∞ −∞ ||V∗(t, k)||2 ˆ L2+||u∗(t, k)||2 ˆ L2dk =Z∞ −∞ χ(k)||V(t, k)||2 ˆ L2+||u(t, k)||2 ˆ L2dk ≤D0e −1 2Re tZ∞ −∞ χ(k)||V(0, k)||2 ˆ L2+||u(0, k)||2 ˆ L2dk =D0e−1 2Re t||V∗(0)||2 L2+||u∗(0)||2 L2,(90) proving (89). B. Uncontrolled wave number analysis For the uncontrolled system (52)–(53), we define, for each k, the Lyapunov functional Λ(k, t) = 1 2||V(t, k)||2 ˆ L2+||u(t, k)||2 ˆ L2(91) The time derivative of Λis ˙ Λ = −8π2k2 Re Λ−1 Re ||uy(k)||2 ˆ L2+||Vy(k)||2 ˆ L2+ 4 Z1 0 (2y−1)u¯ V+ ¯uV 2dy, (92) where the bar denotes the complex conjugate, and the pressure term has disappeared using integration by parts and the continuity equation (61). The second term in the first line of (92) can also be bounded using the Poincare inequality, thanks to the Dirichlet boundary condition at y= 0: −||uy(k)||2 ˆ L2− ||Vy(k)||2 ˆ L2≤ −Λ 2.(93) Consider now separately the two cases |k| ≤ mand |k| ≥ M. In the first case, we can bound the second line of (92) as ˙ Λ≤ −8π2k2 Re Λ−1 2ReΛ + 4Λ,(94) so, if |k| ≥ 1 πqRe 2, then ˙ Λ≤ − 1 2ReΛ.(95) September 17, 2005 DRAFT 21 Now, consider the case of small wave numbers. We bound the second line of (92) using the continuity equation (61) ˙ Λ≤ −8π2k2 Re Λ−1 2ReΛ + 8π|k|Λ,(96) so, if |k| ≤ 1 32πRe , then ˙ Λ≤ − 1 4ReΛ.(97) We have just proved the following result: Proposition 3: If m=1 32πRe and M=1 πqRe 2, then for both |k| ≤ mand |k| ≥ Mthe equilibrium u(t, k, y)≡V(t, k, y)≡0of the uncontrolled system (52)–(60) is exponentially stable in the L2sense: ||V(t, k)||2 ˆ L2+||u(t, k)||2 ˆ L2≤e −1 4Re t||V(0, k)||2 ˆ L2+||u(0, k)||2 ˆ L2.(98) Since the decay rate in (98) is independent of k, that allows us to claim the following result for all uncontrolled wave numbers. Proposition 4: The variables ǫu(t, x, y)and ǫV(t, x, y)defined as ǫu(t, x, y) = Z∞ −∞ (1 −χ(k)) u(t, k, y)e2πikxdk, (99) ǫV(t, x, y) = Z∞ −∞ (1 −χ(k)) V(t, k, y)e2πikxdk, (100) decay exponentially as ||ǫV(t)||2 L2+||ǫu(t)||2 L2≤e −1 4Re t||ǫV(0)||2 L2+||ǫu(0)||2 L2.(101) Proof: As in Proposition 2. September 17, 2005 DRAFT 22 C. Analysis for the entire wave number range Using (37)–(38), ||V(t)||2 L2=Z∞ −∞ ||V(t, k)||2 ˆ L2dk =Z1 0Z∞ −∞ (V∗(t, k, y) + ǫV(t, k, y))2dkdy =Z1 0Z∞ −∞ (V∗)2+ǫ2 V+ 2V∗ǫVdkdy =||V∗(t)||2 L2+||ǫV(t)||2 L2,(102) where we have used the fact that V∗(t, k, y)ǫV(t, k, y) = χ(k)(1 −χ(k))V(t, k, y)and χ(k)(1 − χ(k)) is zero for all kby its definition (29). This shows that the L2norm of Vis the sum of the L2norms of V∗(t, k, y)and ǫV(t, k, y). The same holds for u. Therefore, Theorem 1 follows from Propositions 2 and 4. Noting that D0 as defined in (85) is greater than unity, we obtain the following estimate of the decay: ||V(t)||2 L2+||u(t)||2 L2≤D0e −1 4Re t||V(0)||2 L2+||u(0)||2 L2.(103) The explicit solutions are (37)–(38), obtained by solving explicitly (70), using (74) and (83), and applying the inverse Fourier transform, whereas the error bounds are obtained from Proposition 4. VI. H1STABILITY We define the H1norm of f(x, y)as ||f||2 H1=||f||2 L2+||fx||2 L2+||fy||2 L2.(104) We also define the H1norm of f(k, y)with respect to y as ||f(k)||2 ˆ H1= (1 + 4π2k2)||f(k)||2 ˆ L2+||fy(k)||2 ˆ L2.(105) September 17, 2005 DRAFT 23 The ˆ H1norm as a function of kis related to the H1norm as ||f||2 H1=Z∞ −∞ ||f(k)||2 ˆ H1dk. (106) A. H1stability for controlled wave numbers For each k, one has that ||f(k)||2 ˆ H1≤(5 + 16π2M2)||fy(k)||2 ˆ H1,(107) where we have used (105) and Poincare’s inequality. This proves the equivalence, for any k, of the ˆ H1norm of f(k, y)and the ˆ L2norm of just fy(k, y). Therefore, we only have to show exponential decay for uyand Vy. Due to the backstepping transformations (73), (74) and (82) (83), αy=uy−K(k, y, y)u−Zy 0 Ky(k, y, η)u(t, k, η)dη , (108) uy=αy+L(k, y, y)α+Zy 0 Ly(k, y, η)α(t, k, η)dη , (109) α=−1 2πkiVy−Zy 0 K(k, y, η)Vy(t, k, η)dη,(110) Vy=−2πkiα+Zy 0 L(k, y, η)α(t, k, η)dη,(111) and then it is possible to write the following estimates, which are derived from simple estimates on αand αyfrom (70) ||uy(t, k)||2 ˆ L2≤D1e−2 5Re t||uy(0, k)||2 ˆ L2,(112) ||Vy(t, k)||2 ˆ L2≤D0e−1 2Re t||Vy(0, k)||2 ˆ L2,(113) where D1= 5 max m<|k|<M (1 + 4||L||∞+ 4||Ly||∞)2(1 + 4||K||∞+ 4||Ky||∞)2.(114) Using these estimates the following proposition can be stated regarding the velocity field at each kin the controlled range. September 17, 2005 DRAFT 24 Proposition 5: For any kin the range m < |k|< M, the equilibrium u(t, k, y)≡V(t, k, y)≡ 0of the system (52)–(60) with feedback control laws (67), (78) is exponentially stable in the H1sense ||V(t, k)||2 ˆ H1+||u(t, k)||2 ˆ H1≤D2e −2 5Re t||V(0, k)||2 ˆ H1+||u(0, k)||2 ˆ H1,(115) where D2is defined as: D2= (5 + 16π2M2) max{D0, D1}.(116) Thanks to the same argument as in Proposition 2, for all wave numbers m < |k|< M, the following result holds. Proposition 6: Consider equations (8)–(16) with control laws (22)–(23). Then the variables u∗(t, x, y)and V∗(t, x, y)defined in (87)–(88) decay exponentially in the H1norm: ||u∗(t)||2 H1+||V∗(t)||2 H1≤D2e −2 5Re t||u∗(0)||2 H1+||V∗(0)||2 H1.(117) B. H1stability for uncontrolled wave numbers Following the same argument as in (91)–(97), a slightly different bound can be derived that keeps some of the ˆ H1norm in (96) ˙ Λ≤ − Λ 8Re −ΛH 2Re,(118) where ΛH(k, t) = 1 2||uy(t, k)||2 ˆ L2+||Vy(t, k)||2 ˆ L2.(119) September 17, 2005 DRAFT 25 The time derivative of ΛHcan be bounded as dΛH dt =Z1 0 uy¯uyt + ¯uyuyt +¯ VyVyt +Vy¯ Vyt 2dy =−Z1 0 uyy ¯ut+ ¯uyyut+¯ VyyVt+Vyy ¯ Vt 2dy =−1 Re ||uyy||2 ˆ L2+||Vyy||2 ˆ L2+ 4k2π2Z1 0 uyy ¯u+ ¯uyyu+¯ VyyV+Vyy ¯ V 2Re dy +2πki Z1 0 uyy ¯p−¯uyyp 2dy −Z1 0 ¯ Vyypy+Vyy ¯py 2dy −4Z1 0 (2y−1)uyy ¯ V+ ¯uyyV 2dy +8πki Z1 0 y(y−1)uyy ¯u−¯uyyu−¯ VyyV+Vyy ¯ V 2dy, (120) where we have used integration by parts and the Dirichlet boundary conditions of the uncontrolled wave number range. Doing further integration by parts and using the divergence free condition, we can simplify a little the previous expression: dΛH dt =−1 Re ||uyy||2 ˆ L2+||Vyy||2 ˆ L2−8k2π2 Re Λh−16π2k2Z1 0 (2y−1) ¯uV −u¯ V 2dy −¯ Vyyp+Vyy ¯p 2 1 0 .(121) Only the last term remains to be estimated. Using (59)–(60) with Vcbeing zero for uncontrolled wave number, the last term in (121) can be expresssed as ¯ Vyyp+Vyy ¯p 2 1 0 =Re ¯pyp+py¯p 2 1 0 .(122) This quantity can be estimated using the following lemma. Lemma 1: If the pressure pverifies the Poisson equation (54) with boundary conditions (59)– (60), then −¯pyp+py¯p 2 1 0 ≤16||V(t, k)||2 ˆ L2.(123) Proof: Multiplying equation (54) by ¯pand integrating from zero to one, one gets: −4π2k2||p(t, k)||2 ˆ L2+Z1 0 ¯ppyydy =Z1 0 16πki(2y−1)¯pV dy, (124) September 17, 2005 DRAFT 32 Points i and iv are deduced trivially from the fact that (22) and (25) are defined as convolutions, and properties of the heat equation (24). Point ii is verified if Z∞ −∞ Vc(t, x)dx = 0.(163) From the definition of the Fourier transform of Vc, Vc(t, k = 0) = Z∞ −∞ Vc(t, x)dx. (164) Therefore, as k= 0 lies on the uncontrolled wave number range −m < k < m, then Vc(t, k = 0) = 0 and the property is verified. Point iii gives bounds on the decay rate of the kernels (26)–(28). All the kernel definitions are of the form Q(x−ξ, y) = Z∞ −∞ χ(k)f(k, y)e2πik(x−ξ)dk, (165) for some fanalytic in kand smooth in y. Then, integrating by parts, we find that |Q(x−ξ, y)| ≤ (M−m) π|x−ξ|max m<|k|<M  df dk(k, y) +2 π|x−ξ|max m<|k|<M |f(k, y)| =C |x−ξ|,(166) showing that the kernels decay at least like 1/|x−ξ|. This bound is made explicit in Remark 2 for QVand Q0. From the definition of the inverse Fourier transform (48), it is straightforward to show that if the real part of f(k, y)is even and the imaginary part of f(k, y)is odd, then the resulting f(x, y) will always be real. Then, Point v can be proved showing that the functions under the integrals in (26)–(28), which are inverse Fourier transforms, have this property. This is immediate for (27) and (28). For (26), the property must be shown for the kernel K, defined by the sequence (31)–(32). Since Kis the limit of the sequence, it will have the property if all Knshare the property. This can be proved by induction. For K0, the property is evident from its definition September 17, 2005 DRAFT 33 (31) and can be immediately verified. For Kn, if the property is assumed for Kn−1, then from expression (32) and taking into account that the product of even functions is again even, the product of odd functions is also even, and the product of an even function times an odd function is odd, then it can be seen that Knalso shares the property. Therefore, the limit Kwill have a real inverse transform, and kernel Quwill be real. Point vi is deduced from the definition of the kernels (26)–(28) as truncated Fourier inverse integrals, which makes the kernels smooth in x−ξ. Smoothness in ηis deduced from smoothness of the functions under the integrals. For Point vii, consider expression (22) and (26). Then, ||Uc||2 L2=Z∞ −∞ Uc(t, x)2dx =Z∞ −∞ |Uc|(t, k)2dk =Z∞ −∞ χ(k)Z1 0 K(k, 1, η)u(t, y, k)dη 2 dk ≤2(M−m) max m≤|k|≤M{||K||∞}||u(t)||2 L2,(167) and the result follows from Theorem 1. On the other hand, for Vcone has to use its dynamic equation (24)–(25), and a Lyapunov functional consisting in half its L2norm. One then has, using Young’s inequality d dt |Vc(k)|2 2≤−π2k2 Re |Vc(k)|2+|uy|2(t, k, 0) + |uy|2(t, k, 1) Re +64 cosh (2πM)||V(t, k)||2 ˆ L2,(168) and supposing the control law is initialized at zero (see Remark 1), and using the H2norm to bound the second line of (168) one gets |Vc(t, k)|2≤Zt 0 e−π2m2 Re (t−τ)"10||u(τ, k)||2 ˆ H2 Re + 64 cosh (2πM)||V(τ, k)||2 ˆ L2dτ. (169) September 17, 2005 DRAFT 34 Integrating in k ||Vc(t)||2 L2≤Zt 0 e−π2m2 Re (t−τ)10||u(τ)||2 H2 Re + 64 cosh (2πM)||V(τ)||2 L2dτ, (170) and then the result follows from Theorem 1. For Point viii, consider the jth spatial derivative of Ucand calculate its L2spatial norm  dj dxjUc 2 L2 =Z∞ −∞ dj dxjUc(t, x)2 dx =Z∞ −∞ |2πk|2j|Uc|(t, k)2dk ≤(2πM)2j||Uc||2 L2,(171) so the result for Ucfollows from Point vii. We proceed similarly for Vc, thus proving Point viii. IX. DISCUSSION The result was presented in 2D for ease of notation. Since 3D channels are spatially invariant in both streamwise and spanwise direction, it is possible to extend the design to 3D, by applying the Fourier transform in both invariant directions and following the same steps, with some refinements which include actuation of the spanwise velocity at the wall. The result also trivially extends to periodic channel flow of arbitrary periodic box size, 2D or 3D; it only requires substitution of the Fourier transform by Fourier series, with all other expressions still holding. Our control laws are presented with full state feedback. However, for parabolic PDEs, in [21] we developed an observer design methodology, which is dual to the backstepping control methodology in [20], which we extended to Navier-Stokes equations here to solve the state feedback stabilization problem for the channel flow. Extending the observer concepts in [21] to the Navier- Stokes PDEs has allowed us to also develop an observer for the channel flow, which is presented in the conference paper [23]. While the observer is of interest in its own right (one can use it to estimate turbulent flows without controlling/relaminarizing them), the state feedback controller in September 17, 2005 DRAFT 35 the present paper and the observer in [23] can be combined into an output feedback compensator, which uses measurements of P(x, 0) and uy(x, 0) only, and the actuation of V(x, 1),u(x, 1). Our controller requires actuation of both velocity components at the wall. An assumption made throughout the flow control literature is that the boundary values of velocity are actuated through micro-jet actuators that perform “zero-mean” blowing and suction. Effective actuation of wall velocity at angles as low as 5◦relative to the wall has been demonstrated experimentally using differentially actuated pairs of jets. Unlike in our earlier publications [1], [5] where we included DNS simulation results that demonstrated relaminarization with our controllers, we do not present simulation results in this paper. In another publication, to be submitted to a fluid mechanics journal, we will present an extension to 3D, without the H1, H2stability estimates and without the explicit closed-loop Navier-Stokes solutions (these two issues extend in a rather straightforward manner to 3D because we deal with linearized Navier-Stokes equations), but with simulations results included. The 3D controller will include actuation in the spanwise direction. The numerical tests will focus on turbulence-critical issues like the behavior of the controller at kx= 0 for moderate-to-large kz and other issues which come up only in 3D. ACKNOWLEDGEMENT We thank Tom Bewley for numerous helpful discussions and expert advice, and Jennie Cochran for reviewing the paper and correcting some analytical expressions. REFERENCES [1] O. M. 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