BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU ...
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL OF VORTEX SHEDDING∗ ´§ OLE MORTEN AAMO † , ANDREY SMYSHLYAEV ‡ , AND MIROSLAV KRSTIC Abstract. In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology, for the stabilization of unstable, parabolic partial differential equations. We consider the linearized Ginzburg-Landau equation, which models, for instance, vortex shedding in bluff body flows. Asymptotic stabilization is achieved by means of boundary control via state feedback in the form of an integral operator. The kernel of the operator is shown to be twice continuously differentiable, and a series approximation for its solution is given. Under certain conditions on the parameters of the Ginzburg-Landau equation, compatible with vortex shedding modelling on a semi-infinite domain, the kernel is shown to have compact support, resulting in partial state feedback. Simulations are provided in order to demonstrate the performance of the controller. In summary, the paper extends previous work in two ways: 1) it deals with two coupled partial differential equations, and; 2) under certain circumstances handles equations defined on a semi-infinite domain. Key words. partial differential equations, boundary control, stabilization, flow control AMS subject classifications. 35B37, 35B65, 93D15
1. Introduction. In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology [6], for the stabilization of unstable, parabolic partial differential equations [3, 2, 10, 14]. We consider the linearized Ginzburg-Landau equation given by ∂A (˘ x, t) ∂A (˘ x, t) ∂ 2 A (˘ x, t) + a2 (˘ x) x) A (˘ x, t) = a1 + a3 (˘ ∂t ∂x ˘2 ∂x ˘
(1.1)
for x ˘ ∈ (0, xd ), with boundary conditions A (0, t) = u (t) , A (xd , t) = 0,
(1.2) (1.3)
and where A : [0, xd ] × R+ → C, a2 ∈ C 2 ([0, xd ]; C) , a3 ∈ C 1 ([0, xd ]; C), a1 ∈ C, xd > 0, and u : R+ → C is the control input. a1 is assumed to have strictly positive real part. In order to achieve asymptotic stabilization of the equilibrium at A ≡ 0, backstepping is applied resulting in a boundary control law that essentially cuts the term a3 (˘ x) A (˘ x, t) from equation (1.1). The result extends the work of [10, 14] in two ways: 1) it deals with two coupled partial differential equations, and; 2) under certain circumstances handles equations defined on a semi-infinite domain (xd → ∞). The theory is supplemented with a case study involving control of vortex shedding in bluff body flows. Controllers for this problem have previously been designed for finite dimensional approximations of equation (1.1) [7, 8, 1, 9]. In [12, 13], it was shown numerically that the Ginzburg-Landau model for Reynolds numbers close to Rc can ∗ This work was supported by the National Science Foundation and the Norwegian Research Council. † Department of Engineering Cybernetics, Norwegian University of Science and Technology, N7491 Trondheim, Norway (Tel: +47 73594386, Fax: +47 73594399, E-mail: [email protected]). ‡ Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093-0411, USA. § Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093-0411, USA.
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´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
be stabilized using proportional feedback from a single measurement downstream of the cylinder, to local forcing at the location of the cylinder. In [5], an optimal solution to a boundary control problem formulated for a stationary Ginzburg-Landau model of superconductivity defined on a bounded domain was shown to exist, and the optimality system of equations was solved by employing the finite element method. The paper is organized as follows. In Section 2, equation (1.1) is rewritten in terms of real variables and coefficients, and the problem statement is given. The main result is stated in Section 2. In Section 4 partial differential equations governing the feedback kernel are derived, and in Section 5 they are transformed to corresponding integral equations. We find a unique solution to the integral equations in Section 6.1, and show that the solution also yields the unique feedback kernel. Stability properties of the chosen target system are established in Section 7.1. In Section 9, the results are applied to a model of vortex shedding behind a bluff body immersed in a moving fluid, and it is shown that stabilizing feedback kernels that have compact support can be found even when the domain is semi-infinite. Concluding remarks are offered in Section 10. 2. Problem Statement. We now rewrite equation (1.1) to obtain two coupled partial differential equations in real variables and coefficients by defining ¢ 1¡ ¯ (x, t) , B (x, t) + B 2 ¢ 1 ¡ ¯ (x, t) , ι (x, t) = =(B(x, t)) = B (x, t) − B 2i
ρ (x, t) = 0 is a constant that we will determine later. Clearly, (6.10)—(6.11) hold for n = 0. Noting that Z ξZ η Z Z M Kn ξ η n |Gn (τ, s)| dsdτ ≤ (τ + s) dsdτ n! η 0 η 0 Z ξh i M Kn n+1 = − τ n+1 dτ (τ + η) (n + 1)! η Z ξ M Kn n+1 ≤ (τ + η) dτ (n + 1)! η Z ξ M Kn n+1 ≤ (ξ + η) dτ (n + 1)! η ≤ 2M K n
(ξ + η)n+1 , (n + 1)!
(6.12)
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
9
we obtain from (6.4) that n+1
|Gn+1 (ξ, η)| ≤
1 (ξ + η) M (B + Bc ) K n , 2 (n + 1)!
(6.13)
and from (6.6) that |Gc,n+1 (ξ, η)| satisfies the same bound (6.13). Therefore, setting K = M , we obtain |Gn+1 (ξ, η)| ≤ M K n+1
(ξ + η)n+1 , (n + 1)!
|Gc,n+1 (ξ, η)| ≤ M K n+1
(ξ + η) . (n + 1)!
(6.14)
n+1
(6.15)
Thus, (6.10) and (6.11) are proved by induction, and the series G (ξ, η) =
∞ X
Gn (ξ, η) , and Gc (ξ, η) =
n=0
∞ X
Gc,n (ξ, η) ,
(6.16)
n=0
¡ ¢ converge uniformly in T 1 , and is a solution of (5.1)—(5.2). G and Gc are C 2 T 1 since ¡ ¢ b and bc are C 1 T 1 . The bounds (6.1)—(6.2) follow from (6.10)—(6.11), (6.16) and the fact that K = M . It can be shown by the method of successive approximations that if (G1 , Gc,1 ) and (G2 , Gc,2 ) are two different solutions of (5.1)—(5.2), the resulting homogeneous integral equation for (G, Gc ) = (G1 − G2 , Gc,1 − Gc,2 ) has a unique solution (G, Gc ) = 0, which proves that the solution (6.16) is unique. We can check that (6.16) satisfies (5.10)—(5.15) by direct substitution. Equations (5.10)—(5.15) have a unique solution by Lemma 5.1. Exponential stability of the target system (4.3)—(4.5) in the L2 and H1 norms is proved in the next section. In order to be able to imply stability of the closed loop system (2.4)—(2.7) from that result, we need to establish equivalence of norms of (ρ, ι) and (˜ ρ, ˜ι) in L2 and H1 . This is done by proving that transformation (4.1)—(4.2) is invertible. The inverse transformation has the form Z x ρ (x, t) = ρ˜ (x, t) − [l (x, y) ρ˜ (y, t) + lc (x, y) ˜ι (y, t)] dy, (6.17) Z 0x [−lc (x, y) ρ˜ (y, t) + l (x, y) ˜ι (y, t)] dy. (6.18) ι (x, t) = ˜ι (x, t) − 0
The following result holds for the kernels l (x, y) and lc (x, y) of transformation (6.17)— (6.18). Theorem 6.2. If the pair of kernels, l (x, y) and lc (x, y), satisfy the partial differential equation lxx = lyy − β(y, x)l − βc (y, x)lc ,
(6.19)
lc,xx = lc,yy + βc (y, x)l − β(y, x)lc ,
(6.20)
´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
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with boundary conditions l(x, x) =
1 2
lc (x, x) = − l (x, 0) = 0, lc (x, 0) = 0,
Z
1 2
x
β(γ, γ)dγ,
(6.21)
0
Z
x
βc (γ, γ)dγ,
(6.22)
0
(6.23) (6.24)
and if (˜ ρ, ˜ι) satisfies (4.3)—(4.5), then ¡(ρ,¢ι) satisfies (2.4)—(2.7) with (2.11)—(2.12). System (6.19)—(6.24) has a unique C 2 T solution satisfying |l (x, y)| ≤ M e2Mx ,
|lc (x, y)| ≤ M e
2Mx
(6.25)
,
(6.26)
where M is given in (6.7). Proof. The proof is similar to those of Lemmas 4.1 and 5.1, and Theorem 6.1. 7. Stability Analysis. Theorem 7.1. Suppose c > 0, and select fR (x) and fI (x) such that ¶ µ 1 1 (7.1) sup fR (x) + |fI0 (x)| ≤ − c. 2 2 x∈[0,1] Then the solution (˜ ρ, ˜ι) ≡ (0, 0) of system (4.3)—(4.5) is exponentially stable in the L2 (0, 1) and H1 (0, 1) norms. Corollary 7.2. Suppose c > 0, and set fR (x) = −c and fI (x) ≡ 0. Then the solution (˜ ρ, ˜ι) ≡ (0, 0) of system (4.3)—(4.5) is exponentially stable in the L2 (0, 1) and H1 (0, 1) norms. Proof. Consider the function E (t) =
1 2
Z
1
0
³ ´ ρ˜ (x, t)2 + ˜ι (x, t)2 dx.
(7.2)
Its time derivative along solutions of system (4.3)—(4.5) is E˙ (t) =
Z
1
[˜ ρ (aR ρ˜xx + fR (x) ρ˜ − aI ˜ιxx − fI (x) ˜ι)
0
+˜ι (aI ρ˜xx + fI (x) ρ˜ + aR ˜ιxx + fR (x) ˜ι)] dx Z 1 = (˜ ρ (aR ρ˜xx + fR (x) ρ˜ − aI ˜ιxx ) + ˜ι (aI ρ˜xx + aR ˜ιxx + fR (x) ˜ι)) dx 0
=− ≤
Z
Z 1
0
1
0
¡ ¢ aR ρ˜2x + ˜ι2x dx +
¡ ¢ fR (x) ρ˜2 + ˜ι2 dx.
Z
0
1
¡ ¢ fR (x) ρ˜2 + ˜ι2 dx + aI
Z
0
1
(˜ ρx˜ιx − ˜ιx ρ˜x ) dx (7.3)
So, from (7.1), and the comparison principle, we have E (t) ≤ E (0) e−ct , for t ≥ 0.
(7.4)
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
Set 1 V (t) = 2
Z
¢ ¡ 2 ρ˜x (x, t) + ˜ι2x (x, t) dx.
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(7.5)
The time derivative of V (t) along solutions of system (4.3)—(4.5) is Z 1 (˜ ρx ρ˜xt + ˜ιx˜ιxt ) dx V˙ (t) = 0
=− =−
Z
1
(˜ ρxx ρ˜t + ˜ιxx˜ιt ) dx
0
Z
1
[˜ ρxx (aR ρ˜xx + fR (x) ρ˜ − aI ˜ιxx − fI (x) ˜ι)
0
+˜ιxx (aI ρ˜xx + fI (x) ρ˜ + aR ˜ιxx + fR (x) ˜ι)] dx Z 1 Z 1 ¡ ¡ 2 ¢ ¢ fR (x) ρ˜2x + ˜ι2x dx = −aR ρ˜xx + ˜ι2xx dx + +
Z
0
0
0
1
fI0 (x) (˜ιx ρ˜ − ρ˜x˜ι) dx −
1 2
Z
1
0
¡ ¢ fR00 (x) ρ˜2 + ˜ι2 dx
Z 1 Z ¡ 2 ¡ ¢ ¢ 1 1 00 2 0 fR (x) ρ˜x + ˜ιx dx + fI (x) (˜ιx ρ˜ − ρ˜x ˜ι) dx − fR (x) ρ˜2 + ˜ι2 dx ≤ 2 0 0 0 ¶ Z 1µ Z 1 ¡ ¡ ¢ ¢ 1 1 (|fI0 (x)| − fR00 (x)) ρ˜2 + ˜ι2 dx fR (x) + |fI0 (x)| ρ˜2x + ˜ι2x dx + ≤ 2 2 0 0 ¶ Z 1µ Z 1 ¡ 2 ¢ ¡ 2 ¢ 1 0 1 ≤ fR (x) + |fI (x)| ρ˜x + ˜ι2x dx + c2 ρ˜ + ˜ι2 dx, 2 2 0 0 c −ct ≤ − V (t) + c2 E (0) e (7.6) 2 Z
1
where we have used (7.1) and defined ( c2 , max
sup
x∈[0,1]
(|fI0
(x)| −
fR00
)
(x)) , 0 .
From the comparison principle, we get ³ ´ c c2 c2 V (t) ≤ V (0) + 2 E (0) e− 2 t − 2 E (0) e−ct , c c
(7.7)
(7.8)
so we obtain
V (t) ≤
µ ¶ c 2c2 V (0) + E (0) e− 2 t , for t ≥ 0. c
(7.9)
1 V (t) , 2
(7.10)
Since (Poincaré inequality) E (t) ≤ we get c
V (t) ≤ c3 V (0) e− 2 t , for t ≥ 0, with c3 = 1 + c2 /c.
(7.11)
´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
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8. Proof of Theorem 3.1. From Theorem 7.1, (˜ ρ, ˜ι) = 0 is exponentially stable in the L2 and H1 norms. Since Theorems 6.1 and 6.2 establish equivalence of norms of (ρ, ι) and (˜ ρ, ˜ι) in L2 and H1 , the stability statements of Theorem 7.1 also hold for the solution (ρ, ι) ≡ (0, 0) of system (2.4)—(2.5). From standard results for uniformly parabolic1 equations (see, for instance, [4]), it follows that system (4.3)—(4.4), with Dirichlet boundary conditions (4.5) and initial data ρ˜0 , ˜ι0 ∈ L∞ (0, 1), has a unique classical solution ρ˜, ˜ι ∈ C 2,1 ((0, 1) × (0, ∞)). The smoothness properties of k, kc , l, and lc stated in Theorems 6.1 and 6.2 then provide well posedness of system (2.4)—(2.7) in closed loop with (2.11)—(2.12). 9. Application to a Model of Vortex Shedding. The objective of this section is to provide a numerical demonstration of our results applied to a fluid flow control problem. An interesting feature of the system we study in this example, is that it is defined on an infinite domain (xd → ∞), yet, we obtain feedback gain kernels which have compact support. 9.1. The model. In flows past submerged obstacles, the phenomenon of vortex shedding occurs provided the Reynolds number is sufficiently large. A popular prototype model flow for studying vortex shedding, is the flow past a 2D circular cylinder, as sketched in Figure 9.1. The vortices, which are alternatively shed from the upper and lower sides of the cylinder, induce an undesirable periodic force that acts on the cylinder. The dynamics of the cylinder wake, often referred to as the von Kármán vortex street, is governed by the Navier-Stokes equation, however, in [13], a simplified model was suggested in terms of the Ginzburg-Landau equation ∂A ∂2A ∂A 2 x) x) A + a4 |A| A + δ (˘ x) u, = a1 2 + a2 (˘ + a3 (˘ ∂t ∂x ˘ ∂x ˘
(9.1)
where x ˘ ∈ R, A : R × R+ → C, a1 , a4 ∈ C, and a2 , a3 : R → C. δ denotes the Dirac distribution and u : R+ → C is the control input. Thus, actuation is in the form of local forcing at x ˘ = 0, which is the location of the cylinder. The boundary conditions are A (±∞, t) = 0, that is, homogeneous Dirichlet boundary conditions. A (x, t) may represent any physical variable (velocities (u, v) or pressure p), or derivations thereof, along the centerline y = 0, see Figure 9.1. The choice will have an impact on the performance of the Ginzburg-Landau model, and associating A with the transverse fluctuating velocity v (x, y = 0, t) seems to be a particularly good choice [11]. In order to implement the scheme in practice, transfer functions between A (0) and the physical actuation, and the physical sensing and A (x), would have to be determined, either experimentally or computationally. The physical actuation could for instance be micro/synthetic jet actuators distributed on the cylinder surface. Numerical values for the coefficients in (9.1) were determined from experiments in [13], and are reproduced in Appendix 1. We now simplify this problem to fit into the framework of the previous analysis. We linearize around the zero solution, discard the upstream subsystem by replacing the local forcing at x ˘ = 0 with boundary input at this location, and truncate the downstream subsystem at some xd > 0. The resulting system is of the form (1.1)— (1.3), defined on the interval [0, xd ]. We justify the truncation of the system by noting that the upstream subsystem (the region to the left of the cylinder in Figure 9.1) is approximately uniform flow, whereas the downstream subsystem (the region to the right of the cylinder in Figure 9.1) can be approximated to any desired level 1 System
(4.3)—(4.4) is uniformly parabolic in (0, 1), with module of parabolicity aR .
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
13
Fig. 9.1. Vortex shedding from a cylinder visualized by passive tracer particles.
Fig. 9.2. bR (x) and bI (x) for xd = 2.5, xd = 5, and xd = 7.5.
of accuracy by selecting xd sufficiently large. We are now in a position to apply our results, and we will do so for different choices of xd . In this numerical example, we set the Reynolds2 number to R = 50, which corresponds to supercritical flow for which vortex shedding will occur in the uncontrolled case. For this choice of Reynolds number, the numerical coefficients of (2.4)—(2.5) derived from the coefficients given in Appendix 1, are aR = 0.156/x2d , aI = 0, and bR (x) and bI (x) are plotted in Figure 9.2 for xd = 2.5, xd = 5, and xd = 7.5. 9.2. Feedback kernels. In terms of the feedback gain kernels, k (1, x) and kc (1, x), the boundary feedback (1.2) is given by µ µ ¶ µ ¶¶ Z xd xd − x xd − x 1 ˘ ˘ k 1, − ikc 1, × u (t) = xd xd xd 0 à ! Z x˘ 1 exp a2 (τ ) dτ A (˘ x, t) d˘ x. (9.2) 2a1 0 Thus, the feedback gain kernel for the original system (1.1)—(1.3) is complex-valued, and given by à ! µ µ ¶ µ ¶¶ Z x˘ ˘ ˘ xd − x xd − x 1 1 k 1, − ikc 1, exp x) = a2 (τ ) dτ . (9.3) ku (˘ xd xd xd 2a1 0 Setting fR (x) = −0.2, and fI (x) = 0, exponential stability is assured by Corollary 7.2, and the stabilizing feedback gain kernel (9.3) can be calculated numerically using 2 The Reynolds number for flow past a circular cylinder is usually defined as R = ρU D/µ, where ∞ U∞ is the free stream velocity, D is the cylinder diameter, and ρ and µ are density and viscosity of the fluid, respectively. Vortex shedding occurs when R > 47.
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´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
formulas (6.3)—(6.6), (6.16), (5.4)—(5.5), and (9.3). Figure 9.3 shows the feedback gain kernel (9.3) for xd = 2.5, xd = 5, and xd = 7.5. It is clear that the feedback gain kernels grow rather rapidly with increasing xd , which is an undesirable feature since we want to make xd large in order to minimize the effect of truncating the downstream subsystem. The increase can be seen in connection with Figure 9.2, which shows that the absolute value of the differences bR (x) − fR (x) and bI (x) − fI (x) increase with increasing xd . In other words, the control effort needed to change the dynamics of system (2.4)—(2.7) into that of (4.3)—(4.5) increases with the degree to which the two systems differ. Therefore, the functions fR (x) and fI (x) must be chosen more intelligently than the simple case of setting them constant. Theorem 7.1 allows some flexibility in choosing fR (x) and fI (x), within the constraints of (7.1). In order to postpone choosing xd , we study fR , fI , bR , and bI as functions of x ˘ rather than x in the following. This is convenient since fR , fI , bR , and bI are invariant of xd when treated as functions of x ˘. Recall that when xd is chosen, the two domains are related by x = (xd − x ˘)/xd . We propose to choose fR (˘ x) and fI (˘ x) as close to bR (˘ x) and bI (˘ x) as possible, without violating the conditions of Theorem 7.1, which we now write ¶ µ 1 1 sup fR (˘ x) + |fI0 (˘ x)| ≤ − c. (9.4) 2 2 x ˘ x) = bR (˘ x) and fI (˘ x) = bI (˘ x), Towards that end, we first set them equal, that is fR (˘ and plot (9.4) along with − 12 c = −0.2. The result is shown in Figure 9.4, for x ˘ ∈ [0, 20]. The figure shows that the conditions for stability are already satisfied, without control, for x ˘ ∈ [xs , 20] (in fact, the stability conditions are satisfied for x ˘ ∈ [xs , ∞)), which means that it suffices to alter fR (˘ x) and fI (˘ x) in [0, xs ) in order to satisfy (9.4). Thus, we set3 ½ 1 x)| , for 0 ≤ x < xs , − 2 c − 12 |b0I (˘ fR (˘ x) = (9.5) bR (˘ x) , for x ˘ ≥ xs , x) = bI (˘ x) , for all x ˘. fI (˘
(9.6)
With these choices of fR (˘ x) and fI (˘ x), we calculate numerically the stabilizing feedback gain kernel (9.3) for xd = 10, xd = 20, and xd = 40. Figure 9.5 shows the result. As expected, the feedback gain kernels look similar, and in particular, they appear to be zero for x ˘ larger than approximately 7.5. In fact, they are identical and have compact support, as stated formally in the next theorem. Theorem 9.1. Given c > 0, suppose there exists xs ∈ (0, ∞) such that x) + bR (˘
1 0 1 x)| ≤ − c, for x |b (˘ ˘ ≥ xs . 2 I 2
(9.7)
x) and fI (˘ x) , satisfying (9.4), can be chosen such that fR (˘ x) = bR (˘ x) and Then fR (˘ fI (˘ x) = bI (˘ x) for x ˘ ∈ [xs , xd ]. The resulting stabilizing feedback gain kernel (9.3) has compact support contained in [0, 2xs ]. Moreover, all choices of xd ≥ 2xs , will produce the same stabilizing feedback gain kernel (9.3) in [0, 2xs ]. Proof. The existence of fR (˘ x) and fI (˘ x) satisfying the criterion for stability (9.4) follows trivially from (9.7). To prove that the kernel has support contained in [0, 2xs ], 3 By the choice of x , f (˘ s R x) is continuous. In this example, we ignore the fact that our choice of fR (˘ x) may not be C 1 , although this can easily be achived by smoothing fR (˘ x) in a small neighborhood of xs .
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
a)
15
b)
c)
Fig. 9.3. Feedback kernel (9.3) for a) xd = 2.5, b) xd = 5, and c) xd = 7, 5. fR (x) = −0.2, and fI (x) = 0.
Fig. 9.4. The stability criterion (9.4) when fR (˘ x) = bR (˘ x), and fI (˘ x) = bI (˘ x), is satisfied for x ˘ ≥ xs .
´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
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a)
b)
c)
Fig. 9.5. Feedback kernels (9.3) for fR (˘ x) and fI (˘ x) as defined in (9.5)—(9.6), and for xd = 10 (a), 20 (b), and 40 (c).
we show that it is identically zero outside this interval. We have that b (ξ, 0) = β(x, x) = 0, bc (ξ, 0) = βc (x, x) = 0,
(9.8) (9.9)
· µ ¶¶ xs ξ ∈ 0, 2 1 − . xd
(9.10)
for
It follows that µ ¶ xs G0 (ξ, η) = 0, Gc,0 (ξ, η) = 0, for (ξ, η) ∈ T1 , ξ ≤ 2 1 − . xd
(9.11)
Now, suppose that ¶ µ xs Gn (ξ, η) = 0, Gc,n (ξ, η) = 0, for (ξ, η) ∈ T1 , ξ ≤ 2 1 − . xd
(9.12)
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
17
From (6.4) and (6.6), we get µ ¶ xs Gn+1 (ξ, η) = 0, Gc,n+1 (ξ, η) = 0, for (ξ, η) ∈ T1 , ξ ≤ 2 1 − . xd
(9.13)
Thus, (9.12) is proved by induction, and k (1, y) = G (1 + y, 1 − y) = 0 kc (1, y) = Gc (1 + y, 1 − y) = 0
¾
µ
xs for 0 ≤ y ≤ 2 1 − xd
¶
− 1.
(9.14)
x) = 0, for Therefore, ku (˘ µ ¶ xs xd − x ˘ ≤2 1− − 1, 0≤ xd xd
(9.15)
˘ ≥ 2xs . xd ≥ x
(9.16)
which yields
In order to prove the last part of the theorem, we need to show that for any xd,1 , xd,2 ∈ [2xs , ∞), ³ ´ ³ ´ xd,1 −˘ x xd,2 −˘ x 1 1 k k 1, = 1, x x d,1 d,2 xd,1 xd,1 ´ xd,2 ³ ³ xd,2 ´ (9.17) for x ˘ ∈ [0, 2xs ], x −˘ x x −˘ x 1 kc,x 1, d,1 = 1 kc,x 1, d,2 xd,1
d,1
xd,1
xd,2
d,2
xd,2
where the additional subscripts, xd,1 and xd,2 , on variables identify the domain of the problem from which they stem. We have ¶ µ ¶ µ 1 xd − x 1 ˘ ˘, x ˘ . (9.18) = Gxd 2 − x kxd 1, xd xd xd From (9.8)—(9.10) it follows that ¶ ¶ µ µ Z 1 1 2xs 1 1 xd G0,xd 2 − x ˘, x ˘ =− bxd 2 − τ, 0 dτ, xd xd 4 x˘ xd
for xd ≥ 2xs . From the definition of b, we have that ¶ ¶ µ µ 1 1 τ, 0 = x2d,1 bxd,2 2 − τ, 0 , x2d,2 bxd,1 2 − xd,1 xd,2 for τ ∈ [0, 2xs ]. It follows from (9.19)—(9.20) that ¶ ¶ µ µ 1 1 1 1 x ˘, x ˘ = xd,1 G0,xd,2 2 − x ˘, x ˘ . xd,2 G0,xd,1 2 − xd,1 xd,1 xd,2 xd,2 Similar arguments for Gc,0 , Gn and Gc,n yield ¶ ¶ µ µ 1 1 1 1 x ˘, x ˘ = xd,1 Gxd,2 2 − x ˘, x ˘ , xd,2 Gxd,1 2 − xd,1 xd,1 xd,2 xd,2
(9.19)
(9.20)
(9.21)
(9.22)
which in turn gives (9.17). The significance of Theorem 9.1 is that it guarantees stabilization of the system evolving on an infinite domain by solving the stabilization problem on a finite domain. The procedure for verifying the conditions of the theorem was demonstrated above, but for clarity we repeat it in the following remark.
18
´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
a) Re(A) - Uncontrolled
b) Im(A) - Uncontrolled
c) Re(A) - Controlled
d) Im(A) - Controlled
Fig. 9.6. Comparison of the uncontrolled and controlled cases. For clarity, only a part of the computational domain is shown.
Remark 2. The key to applying Theorem 9.1 is being able to find an xs that satisfies (9.7). This is most easily done by inspecting a graph as the one shown in Figure 9.4. Once xs is found, a possible choice of fR (˘ x) and fI (˘ x) is given in (9.5)—(9.6). Other choices are possible, and in particular, care should be taken to ensure necessary smoothness properties of fR (˘ x). Also, note that the estimate for the support of (9.3) is not tight, as suggested by Figures 9.4 and 9.5, which indicate that ku (˘ x) is supported on approximately [0, 7.5] while 2xs = 11.2. Theorem 9.1 states that [0, 7.5] ⊆ [0, 2xs ], which is true. 9.3. Numerical simulations. For completeness, we include numerical simulations of the controlled and uncontrolled systems. The simulations have been performed by discretizing (1.1) on the domain x ˘ ∈ [0, 15], using finite differences on a grid of 200 nodes. To make the simulation study more interesting, the nonlinear term in (9.1) is accounted for in the simulations. Figures 9.6a and 9.6b show the real and imaginary parts of A (˘ x, t) for the uncontrolled case. The system is in a periodic state reminiscent of vortex shedding. Figures 9.6c and 9.6d show the real and imaginary parts of A (˘ x, t) for the controlled case. The figures show that A (˘ x, t) is effectively driven to zero by the control. Figure 9.7 shows the control effort. 10. Conclusions. This paper extends previous work in two ways: 1) it deals with two coupled partial differential equations, and; 2) under certain circumstances handles equations defined on a semi-infinite domain. For the linearized Ginzburg-
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
19
Fig. 9.7. Control effort, u(t).
Landau equation, asymptotic stabilization is achieved by means of boundary control via state feedback in the form of an integral operator. The kernel of the operator is shown to be twice continuously differentiable, and a series approximation for its solution is given. Under certain conditions (given in (9.7)) on the parameters of the Ginzburg-Landau equation, compatible with vortex shedding modelling on a semiinfinite domain, the kernel is shown to have compact support, resulting in partial state feedback. Simulations are provided in order to demonstrate the performance of the controller. Acknowledgements. We thank Professor Peter Monkewitz for helpful explanations on relationships between Navier-Stokes and Ginzburg-Landau models of vortex shedding and on implementability of GL-based controllers on NS simulations or experiments. REFERENCES [1] O.M. Aamo and M. Krsti´c, ”Global stabilization of a nonlinear Ginzburg-Landau model of vortex shedding,” European Journal of Control, vol. 10, no. 2, 2004. [2] A. Balogh and M. Krsti´c, ”Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability,” European Journal of Control, vol. 8, no. 2, pp. 165—176, 2002. [3] D.M. Boskovi´c, M. Krsti´c, and W.J. Liu, ”Boundary control of an unstable heat equation via measurement of domain-averaged temperature,” IEEE Transactions on Automatic Control, vol. 46, no. 12, pp. 2022—2028, 2001. [4] A. Friedman, Partial Differential Equations of Parabolic Type, Robert E. Krieger Publishing Company, 1983. [5] M.D. Gunzburger, L.S. Hou, and S.S. Ravindran, ”Analysis and approximation of optimal control problems for a simplified Ginzburg-Landau model of superconductivity,” Numerische Mathematik, 77, pp. 243—268, 1997. [6] M. Krsti´c, I. Kanellakopoulos, and P. Kokotovi´c, Nonlinear and Adaptive Control Design, Wiley, New York, 1995. [7] E. Lauga and T.R. Bewley, ”H∞ control of linear global instability in models of non-parallel wakes,” Proceedings of the Second International Symposium on Turbulence and Shear Flow Phenomena, Stockholm, Sweden, 2001.
´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
20
[8] E. Lauga and T.R. Bewley, ”The decay of stabilizability with Reynolds number in a linear model of spatially developing flows,” Proceedings of the Royal Society of London, 459, pp. 2077—2095, 2003. [9] E. Lauga and T.R. Bewley, ”Performance of a linear robust control strategy on a nonlinear model of spatially-developing flows,” to appear in Journal of Fluid Mechanics. [10] W.J. Liu, ”Rapid boundary feedback stabilization of an unstable heat equation,” SIAM Journal on Control and Optimization, vol. 42, pp. 1033—1043, 2003. [11] P.A. Monkewitz, private communication. [12] D.S. Park, D.M. Ladd, and E.W. Hendricks, ”Feedback control of a global mode in spatially developing flows,” Physics Letters A, 182, pp. 244—248, 1993. [13] K. Roussopoulos and P.A. Monkewitz, ”Nonlinear modelling of vortex shedding control in cylinder wakes,” Physica D, 97, pp. 264—273, 1996. [14] A. Smyshlyaev and M. Krsti´c, ”Regularity of hyperbolic PDE’s governing backstepping gain kernels for parabolic PDE’s”, submitted. Downloadable from http://mae.ucsd.edu/research/krstic/papers/smykrs.pdf.
1. Coefficients for the Ginzburg-Landau Equation. The numerical coefficients below are taken from [13, Appendix A]. Rc xt ω0t k0t t ωkk t ωxx kxt
= 47 = 1.183 − 0.031i = 0.690 + 0.080i + (−0.00159 + 0.00447i)(R − Rc ) = 1.452 − 0.844i + (0.00341 + 0.011i)(R − Rc ) = −0.292i = 0.108 − 0.057i = 0.164 − 0.006i ¢2 1 t ¡ x) = ω0t + ωxx ω0 (˘ x ˘ − xt 2 ¡ ¢ k0 (˘ x) = k0t + kxt x ˘ − xt 1 t a1 = iωkk 2 t a2 (˘ x) = ωkk k0 (˘ x) µ ¶ 1 t 2 a3 (˘ x) = − ω0 (˘ x) + ωkk k0 (˘ x) i 2 a4 = −0.0225 + 0.0671i.
1. Introduction. In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology [6], for the stabilization of unstable, parabolic partial differential equations [3, 2, 10, 14]. We consider the linearized Ginzburg-Landau equation given by ∂A (˘ x, t) ∂A (˘ x, t) ∂ 2 A (˘ x, t) + a2 (˘ x) x) A (˘ x, t) = a1 + a3 (˘ ∂t ∂x ˘2 ∂x ˘
(1.1)
for x ˘ ∈ (0, xd ), with boundary conditions A (0, t) = u (t) , A (xd , t) = 0,
(1.2) (1.3)
and where A : [0, xd ] × R+ → C, a2 ∈ C 2 ([0, xd ]; C) , a3 ∈ C 1 ([0, xd ]; C), a1 ∈ C, xd > 0, and u : R+ → C is the control input. a1 is assumed to have strictly positive real part. In order to achieve asymptotic stabilization of the equilibrium at A ≡ 0, backstepping is applied resulting in a boundary control law that essentially cuts the term a3 (˘ x) A (˘ x, t) from equation (1.1). The result extends the work of [10, 14] in two ways: 1) it deals with two coupled partial differential equations, and; 2) under certain circumstances handles equations defined on a semi-infinite domain (xd → ∞). The theory is supplemented with a case study involving control of vortex shedding in bluff body flows. Controllers for this problem have previously been designed for finite dimensional approximations of equation (1.1) [7, 8, 1, 9]. In [12, 13], it was shown numerically that the Ginzburg-Landau model for Reynolds numbers close to Rc can ∗ This work was supported by the National Science Foundation and the Norwegian Research Council. † Department of Engineering Cybernetics, Norwegian University of Science and Technology, N7491 Trondheim, Norway (Tel: +47 73594386, Fax: +47 73594399, E-mail: [email protected]). ‡ Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093-0411, USA. § Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093-0411, USA.
1
2
´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
be stabilized using proportional feedback from a single measurement downstream of the cylinder, to local forcing at the location of the cylinder. In [5], an optimal solution to a boundary control problem formulated for a stationary Ginzburg-Landau model of superconductivity defined on a bounded domain was shown to exist, and the optimality system of equations was solved by employing the finite element method. The paper is organized as follows. In Section 2, equation (1.1) is rewritten in terms of real variables and coefficients, and the problem statement is given. The main result is stated in Section 2. In Section 4 partial differential equations governing the feedback kernel are derived, and in Section 5 they are transformed to corresponding integral equations. We find a unique solution to the integral equations in Section 6.1, and show that the solution also yields the unique feedback kernel. Stability properties of the chosen target system are established in Section 7.1. In Section 9, the results are applied to a model of vortex shedding behind a bluff body immersed in a moving fluid, and it is shown that stabilizing feedback kernels that have compact support can be found even when the domain is semi-infinite. Concluding remarks are offered in Section 10. 2. Problem Statement. We now rewrite equation (1.1) to obtain two coupled partial differential equations in real variables and coefficients by defining ¢ 1¡ ¯ (x, t) , B (x, t) + B 2 ¢ 1 ¡ ¯ (x, t) , ι (x, t) = =(B(x, t)) = B (x, t) − B 2i
ρ (x, t) = 0 is a constant that we will determine later. Clearly, (6.10)—(6.11) hold for n = 0. Noting that Z ξZ η Z Z M Kn ξ η n |Gn (τ, s)| dsdτ ≤ (τ + s) dsdτ n! η 0 η 0 Z ξh i M Kn n+1 = − τ n+1 dτ (τ + η) (n + 1)! η Z ξ M Kn n+1 ≤ (τ + η) dτ (n + 1)! η Z ξ M Kn n+1 ≤ (ξ + η) dτ (n + 1)! η ≤ 2M K n
(ξ + η)n+1 , (n + 1)!
(6.12)
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
9
we obtain from (6.4) that n+1
|Gn+1 (ξ, η)| ≤
1 (ξ + η) M (B + Bc ) K n , 2 (n + 1)!
(6.13)
and from (6.6) that |Gc,n+1 (ξ, η)| satisfies the same bound (6.13). Therefore, setting K = M , we obtain |Gn+1 (ξ, η)| ≤ M K n+1
(ξ + η)n+1 , (n + 1)!
|Gc,n+1 (ξ, η)| ≤ M K n+1
(ξ + η) . (n + 1)!
(6.14)
n+1
(6.15)
Thus, (6.10) and (6.11) are proved by induction, and the series G (ξ, η) =
∞ X
Gn (ξ, η) , and Gc (ξ, η) =
n=0
∞ X
Gc,n (ξ, η) ,
(6.16)
n=0
¡ ¢ converge uniformly in T 1 , and is a solution of (5.1)—(5.2). G and Gc are C 2 T 1 since ¡ ¢ b and bc are C 1 T 1 . The bounds (6.1)—(6.2) follow from (6.10)—(6.11), (6.16) and the fact that K = M . It can be shown by the method of successive approximations that if (G1 , Gc,1 ) and (G2 , Gc,2 ) are two different solutions of (5.1)—(5.2), the resulting homogeneous integral equation for (G, Gc ) = (G1 − G2 , Gc,1 − Gc,2 ) has a unique solution (G, Gc ) = 0, which proves that the solution (6.16) is unique. We can check that (6.16) satisfies (5.10)—(5.15) by direct substitution. Equations (5.10)—(5.15) have a unique solution by Lemma 5.1. Exponential stability of the target system (4.3)—(4.5) in the L2 and H1 norms is proved in the next section. In order to be able to imply stability of the closed loop system (2.4)—(2.7) from that result, we need to establish equivalence of norms of (ρ, ι) and (˜ ρ, ˜ι) in L2 and H1 . This is done by proving that transformation (4.1)—(4.2) is invertible. The inverse transformation has the form Z x ρ (x, t) = ρ˜ (x, t) − [l (x, y) ρ˜ (y, t) + lc (x, y) ˜ι (y, t)] dy, (6.17) Z 0x [−lc (x, y) ρ˜ (y, t) + l (x, y) ˜ι (y, t)] dy. (6.18) ι (x, t) = ˜ι (x, t) − 0
The following result holds for the kernels l (x, y) and lc (x, y) of transformation (6.17)— (6.18). Theorem 6.2. If the pair of kernels, l (x, y) and lc (x, y), satisfy the partial differential equation lxx = lyy − β(y, x)l − βc (y, x)lc ,
(6.19)
lc,xx = lc,yy + βc (y, x)l − β(y, x)lc ,
(6.20)
´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
10
with boundary conditions l(x, x) =
1 2
lc (x, x) = − l (x, 0) = 0, lc (x, 0) = 0,
Z
1 2
x
β(γ, γ)dγ,
(6.21)
0
Z
x
βc (γ, γ)dγ,
(6.22)
0
(6.23) (6.24)
and if (˜ ρ, ˜ι) satisfies (4.3)—(4.5), then ¡(ρ,¢ι) satisfies (2.4)—(2.7) with (2.11)—(2.12). System (6.19)—(6.24) has a unique C 2 T solution satisfying |l (x, y)| ≤ M e2Mx ,
|lc (x, y)| ≤ M e
2Mx
(6.25)
,
(6.26)
where M is given in (6.7). Proof. The proof is similar to those of Lemmas 4.1 and 5.1, and Theorem 6.1. 7. Stability Analysis. Theorem 7.1. Suppose c > 0, and select fR (x) and fI (x) such that ¶ µ 1 1 (7.1) sup fR (x) + |fI0 (x)| ≤ − c. 2 2 x∈[0,1] Then the solution (˜ ρ, ˜ι) ≡ (0, 0) of system (4.3)—(4.5) is exponentially stable in the L2 (0, 1) and H1 (0, 1) norms. Corollary 7.2. Suppose c > 0, and set fR (x) = −c and fI (x) ≡ 0. Then the solution (˜ ρ, ˜ι) ≡ (0, 0) of system (4.3)—(4.5) is exponentially stable in the L2 (0, 1) and H1 (0, 1) norms. Proof. Consider the function E (t) =
1 2
Z
1
0
³ ´ ρ˜ (x, t)2 + ˜ι (x, t)2 dx.
(7.2)
Its time derivative along solutions of system (4.3)—(4.5) is E˙ (t) =
Z
1
[˜ ρ (aR ρ˜xx + fR (x) ρ˜ − aI ˜ιxx − fI (x) ˜ι)
0
+˜ι (aI ρ˜xx + fI (x) ρ˜ + aR ˜ιxx + fR (x) ˜ι)] dx Z 1 = (˜ ρ (aR ρ˜xx + fR (x) ρ˜ − aI ˜ιxx ) + ˜ι (aI ρ˜xx + aR ˜ιxx + fR (x) ˜ι)) dx 0
=− ≤
Z
Z 1
0
1
0
¡ ¢ aR ρ˜2x + ˜ι2x dx +
¡ ¢ fR (x) ρ˜2 + ˜ι2 dx.
Z
0
1
¡ ¢ fR (x) ρ˜2 + ˜ι2 dx + aI
Z
0
1
(˜ ρx˜ιx − ˜ιx ρ˜x ) dx (7.3)
So, from (7.1), and the comparison principle, we have E (t) ≤ E (0) e−ct , for t ≥ 0.
(7.4)
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
Set 1 V (t) = 2
Z
¢ ¡ 2 ρ˜x (x, t) + ˜ι2x (x, t) dx.
11
(7.5)
The time derivative of V (t) along solutions of system (4.3)—(4.5) is Z 1 (˜ ρx ρ˜xt + ˜ιx˜ιxt ) dx V˙ (t) = 0
=− =−
Z
1
(˜ ρxx ρ˜t + ˜ιxx˜ιt ) dx
0
Z
1
[˜ ρxx (aR ρ˜xx + fR (x) ρ˜ − aI ˜ιxx − fI (x) ˜ι)
0
+˜ιxx (aI ρ˜xx + fI (x) ρ˜ + aR ˜ιxx + fR (x) ˜ι)] dx Z 1 Z 1 ¡ ¡ 2 ¢ ¢ fR (x) ρ˜2x + ˜ι2x dx = −aR ρ˜xx + ˜ι2xx dx + +
Z
0
0
0
1
fI0 (x) (˜ιx ρ˜ − ρ˜x˜ι) dx −
1 2
Z
1
0
¡ ¢ fR00 (x) ρ˜2 + ˜ι2 dx
Z 1 Z ¡ 2 ¡ ¢ ¢ 1 1 00 2 0 fR (x) ρ˜x + ˜ιx dx + fI (x) (˜ιx ρ˜ − ρ˜x ˜ι) dx − fR (x) ρ˜2 + ˜ι2 dx ≤ 2 0 0 0 ¶ Z 1µ Z 1 ¡ ¡ ¢ ¢ 1 1 (|fI0 (x)| − fR00 (x)) ρ˜2 + ˜ι2 dx fR (x) + |fI0 (x)| ρ˜2x + ˜ι2x dx + ≤ 2 2 0 0 ¶ Z 1µ Z 1 ¡ 2 ¢ ¡ 2 ¢ 1 0 1 ≤ fR (x) + |fI (x)| ρ˜x + ˜ι2x dx + c2 ρ˜ + ˜ι2 dx, 2 2 0 0 c −ct ≤ − V (t) + c2 E (0) e (7.6) 2 Z
1
where we have used (7.1) and defined ( c2 , max
sup
x∈[0,1]
(|fI0
(x)| −
fR00
)
(x)) , 0 .
From the comparison principle, we get ³ ´ c c2 c2 V (t) ≤ V (0) + 2 E (0) e− 2 t − 2 E (0) e−ct , c c
(7.7)
(7.8)
so we obtain
V (t) ≤
µ ¶ c 2c2 V (0) + E (0) e− 2 t , for t ≥ 0. c
(7.9)
1 V (t) , 2
(7.10)
Since (Poincaré inequality) E (t) ≤ we get c
V (t) ≤ c3 V (0) e− 2 t , for t ≥ 0, with c3 = 1 + c2 /c.
(7.11)
´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
12
8. Proof of Theorem 3.1. From Theorem 7.1, (˜ ρ, ˜ι) = 0 is exponentially stable in the L2 and H1 norms. Since Theorems 6.1 and 6.2 establish equivalence of norms of (ρ, ι) and (˜ ρ, ˜ι) in L2 and H1 , the stability statements of Theorem 7.1 also hold for the solution (ρ, ι) ≡ (0, 0) of system (2.4)—(2.5). From standard results for uniformly parabolic1 equations (see, for instance, [4]), it follows that system (4.3)—(4.4), with Dirichlet boundary conditions (4.5) and initial data ρ˜0 , ˜ι0 ∈ L∞ (0, 1), has a unique classical solution ρ˜, ˜ι ∈ C 2,1 ((0, 1) × (0, ∞)). The smoothness properties of k, kc , l, and lc stated in Theorems 6.1 and 6.2 then provide well posedness of system (2.4)—(2.7) in closed loop with (2.11)—(2.12). 9. Application to a Model of Vortex Shedding. The objective of this section is to provide a numerical demonstration of our results applied to a fluid flow control problem. An interesting feature of the system we study in this example, is that it is defined on an infinite domain (xd → ∞), yet, we obtain feedback gain kernels which have compact support. 9.1. The model. In flows past submerged obstacles, the phenomenon of vortex shedding occurs provided the Reynolds number is sufficiently large. A popular prototype model flow for studying vortex shedding, is the flow past a 2D circular cylinder, as sketched in Figure 9.1. The vortices, which are alternatively shed from the upper and lower sides of the cylinder, induce an undesirable periodic force that acts on the cylinder. The dynamics of the cylinder wake, often referred to as the von Kármán vortex street, is governed by the Navier-Stokes equation, however, in [13], a simplified model was suggested in terms of the Ginzburg-Landau equation ∂A ∂2A ∂A 2 x) x) A + a4 |A| A + δ (˘ x) u, = a1 2 + a2 (˘ + a3 (˘ ∂t ∂x ˘ ∂x ˘
(9.1)
where x ˘ ∈ R, A : R × R+ → C, a1 , a4 ∈ C, and a2 , a3 : R → C. δ denotes the Dirac distribution and u : R+ → C is the control input. Thus, actuation is in the form of local forcing at x ˘ = 0, which is the location of the cylinder. The boundary conditions are A (±∞, t) = 0, that is, homogeneous Dirichlet boundary conditions. A (x, t) may represent any physical variable (velocities (u, v) or pressure p), or derivations thereof, along the centerline y = 0, see Figure 9.1. The choice will have an impact on the performance of the Ginzburg-Landau model, and associating A with the transverse fluctuating velocity v (x, y = 0, t) seems to be a particularly good choice [11]. In order to implement the scheme in practice, transfer functions between A (0) and the physical actuation, and the physical sensing and A (x), would have to be determined, either experimentally or computationally. The physical actuation could for instance be micro/synthetic jet actuators distributed on the cylinder surface. Numerical values for the coefficients in (9.1) were determined from experiments in [13], and are reproduced in Appendix 1. We now simplify this problem to fit into the framework of the previous analysis. We linearize around the zero solution, discard the upstream subsystem by replacing the local forcing at x ˘ = 0 with boundary input at this location, and truncate the downstream subsystem at some xd > 0. The resulting system is of the form (1.1)— (1.3), defined on the interval [0, xd ]. We justify the truncation of the system by noting that the upstream subsystem (the region to the left of the cylinder in Figure 9.1) is approximately uniform flow, whereas the downstream subsystem (the region to the right of the cylinder in Figure 9.1) can be approximated to any desired level 1 System
(4.3)—(4.4) is uniformly parabolic in (0, 1), with module of parabolicity aR .
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
13
Fig. 9.1. Vortex shedding from a cylinder visualized by passive tracer particles.
Fig. 9.2. bR (x) and bI (x) for xd = 2.5, xd = 5, and xd = 7.5.
of accuracy by selecting xd sufficiently large. We are now in a position to apply our results, and we will do so for different choices of xd . In this numerical example, we set the Reynolds2 number to R = 50, which corresponds to supercritical flow for which vortex shedding will occur in the uncontrolled case. For this choice of Reynolds number, the numerical coefficients of (2.4)—(2.5) derived from the coefficients given in Appendix 1, are aR = 0.156/x2d , aI = 0, and bR (x) and bI (x) are plotted in Figure 9.2 for xd = 2.5, xd = 5, and xd = 7.5. 9.2. Feedback kernels. In terms of the feedback gain kernels, k (1, x) and kc (1, x), the boundary feedback (1.2) is given by µ µ ¶ µ ¶¶ Z xd xd − x xd − x 1 ˘ ˘ k 1, − ikc 1, × u (t) = xd xd xd 0 à ! Z x˘ 1 exp a2 (τ ) dτ A (˘ x, t) d˘ x. (9.2) 2a1 0 Thus, the feedback gain kernel for the original system (1.1)—(1.3) is complex-valued, and given by à ! µ µ ¶ µ ¶¶ Z x˘ ˘ ˘ xd − x xd − x 1 1 k 1, − ikc 1, exp x) = a2 (τ ) dτ . (9.3) ku (˘ xd xd xd 2a1 0 Setting fR (x) = −0.2, and fI (x) = 0, exponential stability is assured by Corollary 7.2, and the stabilizing feedback gain kernel (9.3) can be calculated numerically using 2 The Reynolds number for flow past a circular cylinder is usually defined as R = ρU D/µ, where ∞ U∞ is the free stream velocity, D is the cylinder diameter, and ρ and µ are density and viscosity of the fluid, respectively. Vortex shedding occurs when R > 47.
14
´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
formulas (6.3)—(6.6), (6.16), (5.4)—(5.5), and (9.3). Figure 9.3 shows the feedback gain kernel (9.3) for xd = 2.5, xd = 5, and xd = 7.5. It is clear that the feedback gain kernels grow rather rapidly with increasing xd , which is an undesirable feature since we want to make xd large in order to minimize the effect of truncating the downstream subsystem. The increase can be seen in connection with Figure 9.2, which shows that the absolute value of the differences bR (x) − fR (x) and bI (x) − fI (x) increase with increasing xd . In other words, the control effort needed to change the dynamics of system (2.4)—(2.7) into that of (4.3)—(4.5) increases with the degree to which the two systems differ. Therefore, the functions fR (x) and fI (x) must be chosen more intelligently than the simple case of setting them constant. Theorem 7.1 allows some flexibility in choosing fR (x) and fI (x), within the constraints of (7.1). In order to postpone choosing xd , we study fR , fI , bR , and bI as functions of x ˘ rather than x in the following. This is convenient since fR , fI , bR , and bI are invariant of xd when treated as functions of x ˘. Recall that when xd is chosen, the two domains are related by x = (xd − x ˘)/xd . We propose to choose fR (˘ x) and fI (˘ x) as close to bR (˘ x) and bI (˘ x) as possible, without violating the conditions of Theorem 7.1, which we now write ¶ µ 1 1 sup fR (˘ x) + |fI0 (˘ x)| ≤ − c. (9.4) 2 2 x ˘ x) = bR (˘ x) and fI (˘ x) = bI (˘ x), Towards that end, we first set them equal, that is fR (˘ and plot (9.4) along with − 12 c = −0.2. The result is shown in Figure 9.4, for x ˘ ∈ [0, 20]. The figure shows that the conditions for stability are already satisfied, without control, for x ˘ ∈ [xs , 20] (in fact, the stability conditions are satisfied for x ˘ ∈ [xs , ∞)), which means that it suffices to alter fR (˘ x) and fI (˘ x) in [0, xs ) in order to satisfy (9.4). Thus, we set3 ½ 1 x)| , for 0 ≤ x < xs , − 2 c − 12 |b0I (˘ fR (˘ x) = (9.5) bR (˘ x) , for x ˘ ≥ xs , x) = bI (˘ x) , for all x ˘. fI (˘
(9.6)
With these choices of fR (˘ x) and fI (˘ x), we calculate numerically the stabilizing feedback gain kernel (9.3) for xd = 10, xd = 20, and xd = 40. Figure 9.5 shows the result. As expected, the feedback gain kernels look similar, and in particular, they appear to be zero for x ˘ larger than approximately 7.5. In fact, they are identical and have compact support, as stated formally in the next theorem. Theorem 9.1. Given c > 0, suppose there exists xs ∈ (0, ∞) such that x) + bR (˘
1 0 1 x)| ≤ − c, for x |b (˘ ˘ ≥ xs . 2 I 2
(9.7)
x) and fI (˘ x) , satisfying (9.4), can be chosen such that fR (˘ x) = bR (˘ x) and Then fR (˘ fI (˘ x) = bI (˘ x) for x ˘ ∈ [xs , xd ]. The resulting stabilizing feedback gain kernel (9.3) has compact support contained in [0, 2xs ]. Moreover, all choices of xd ≥ 2xs , will produce the same stabilizing feedback gain kernel (9.3) in [0, 2xs ]. Proof. The existence of fR (˘ x) and fI (˘ x) satisfying the criterion for stability (9.4) follows trivially from (9.7). To prove that the kernel has support contained in [0, 2xs ], 3 By the choice of x , f (˘ s R x) is continuous. In this example, we ignore the fact that our choice of fR (˘ x) may not be C 1 , although this can easily be achived by smoothing fR (˘ x) in a small neighborhood of xs .
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
a)
15
b)
c)
Fig. 9.3. Feedback kernel (9.3) for a) xd = 2.5, b) xd = 5, and c) xd = 7, 5. fR (x) = −0.2, and fI (x) = 0.
Fig. 9.4. The stability criterion (9.4) when fR (˘ x) = bR (˘ x), and fI (˘ x) = bI (˘ x), is satisfied for x ˘ ≥ xs .
´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
16
a)
b)
c)
Fig. 9.5. Feedback kernels (9.3) for fR (˘ x) and fI (˘ x) as defined in (9.5)—(9.6), and for xd = 10 (a), 20 (b), and 40 (c).
we show that it is identically zero outside this interval. We have that b (ξ, 0) = β(x, x) = 0, bc (ξ, 0) = βc (x, x) = 0,
(9.8) (9.9)
· µ ¶¶ xs ξ ∈ 0, 2 1 − . xd
(9.10)
for
It follows that µ ¶ xs G0 (ξ, η) = 0, Gc,0 (ξ, η) = 0, for (ξ, η) ∈ T1 , ξ ≤ 2 1 − . xd
(9.11)
Now, suppose that ¶ µ xs Gn (ξ, η) = 0, Gc,n (ξ, η) = 0, for (ξ, η) ∈ T1 , ξ ≤ 2 1 − . xd
(9.12)
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
17
From (6.4) and (6.6), we get µ ¶ xs Gn+1 (ξ, η) = 0, Gc,n+1 (ξ, η) = 0, for (ξ, η) ∈ T1 , ξ ≤ 2 1 − . xd
(9.13)
Thus, (9.12) is proved by induction, and k (1, y) = G (1 + y, 1 − y) = 0 kc (1, y) = Gc (1 + y, 1 − y) = 0
¾
µ
xs for 0 ≤ y ≤ 2 1 − xd
¶
− 1.
(9.14)
x) = 0, for Therefore, ku (˘ µ ¶ xs xd − x ˘ ≤2 1− − 1, 0≤ xd xd
(9.15)
˘ ≥ 2xs . xd ≥ x
(9.16)
which yields
In order to prove the last part of the theorem, we need to show that for any xd,1 , xd,2 ∈ [2xs , ∞), ³ ´ ³ ´ xd,1 −˘ x xd,2 −˘ x 1 1 k k 1, = 1, x x d,1 d,2 xd,1 xd,1 ´ xd,2 ³ ³ xd,2 ´ (9.17) for x ˘ ∈ [0, 2xs ], x −˘ x x −˘ x 1 kc,x 1, d,1 = 1 kc,x 1, d,2 xd,1
d,1
xd,1
xd,2
d,2
xd,2
where the additional subscripts, xd,1 and xd,2 , on variables identify the domain of the problem from which they stem. We have ¶ µ ¶ µ 1 xd − x 1 ˘ ˘, x ˘ . (9.18) = Gxd 2 − x kxd 1, xd xd xd From (9.8)—(9.10) it follows that ¶ ¶ µ µ Z 1 1 2xs 1 1 xd G0,xd 2 − x ˘, x ˘ =− bxd 2 − τ, 0 dτ, xd xd 4 x˘ xd
for xd ≥ 2xs . From the definition of b, we have that ¶ ¶ µ µ 1 1 τ, 0 = x2d,1 bxd,2 2 − τ, 0 , x2d,2 bxd,1 2 − xd,1 xd,2 for τ ∈ [0, 2xs ]. It follows from (9.19)—(9.20) that ¶ ¶ µ µ 1 1 1 1 x ˘, x ˘ = xd,1 G0,xd,2 2 − x ˘, x ˘ . xd,2 G0,xd,1 2 − xd,1 xd,1 xd,2 xd,2 Similar arguments for Gc,0 , Gn and Gc,n yield ¶ ¶ µ µ 1 1 1 1 x ˘, x ˘ = xd,1 Gxd,2 2 − x ˘, x ˘ , xd,2 Gxd,1 2 − xd,1 xd,1 xd,2 xd,2
(9.19)
(9.20)
(9.21)
(9.22)
which in turn gives (9.17). The significance of Theorem 9.1 is that it guarantees stabilization of the system evolving on an infinite domain by solving the stabilization problem on a finite domain. The procedure for verifying the conditions of the theorem was demonstrated above, but for clarity we repeat it in the following remark.
18
´ O.M. AAMO, A. SMYSHLYAEV, AND M. KRSTIC
a) Re(A) - Uncontrolled
b) Im(A) - Uncontrolled
c) Re(A) - Controlled
d) Im(A) - Controlled
Fig. 9.6. Comparison of the uncontrolled and controlled cases. For clarity, only a part of the computational domain is shown.
Remark 2. The key to applying Theorem 9.1 is being able to find an xs that satisfies (9.7). This is most easily done by inspecting a graph as the one shown in Figure 9.4. Once xs is found, a possible choice of fR (˘ x) and fI (˘ x) is given in (9.5)—(9.6). Other choices are possible, and in particular, care should be taken to ensure necessary smoothness properties of fR (˘ x). Also, note that the estimate for the support of (9.3) is not tight, as suggested by Figures 9.4 and 9.5, which indicate that ku (˘ x) is supported on approximately [0, 7.5] while 2xs = 11.2. Theorem 9.1 states that [0, 7.5] ⊆ [0, 2xs ], which is true. 9.3. Numerical simulations. For completeness, we include numerical simulations of the controlled and uncontrolled systems. The simulations have been performed by discretizing (1.1) on the domain x ˘ ∈ [0, 15], using finite differences on a grid of 200 nodes. To make the simulation study more interesting, the nonlinear term in (9.1) is accounted for in the simulations. Figures 9.6a and 9.6b show the real and imaginary parts of A (˘ x, t) for the uncontrolled case. The system is in a periodic state reminiscent of vortex shedding. Figures 9.6c and 9.6d show the real and imaginary parts of A (˘ x, t) for the controlled case. The figures show that A (˘ x, t) is effectively driven to zero by the control. Figure 9.7 shows the control effort. 10. Conclusions. This paper extends previous work in two ways: 1) it deals with two coupled partial differential equations, and; 2) under certain circumstances handles equations defined on a semi-infinite domain. For the linearized Ginzburg-
BOUNDARY CONTROL OF THE LINEARIZED GINZBURG-LANDAU MODEL
19
Fig. 9.7. Control effort, u(t).
Landau equation, asymptotic stabilization is achieved by means of boundary control via state feedback in the form of an integral operator. The kernel of the operator is shown to be twice continuously differentiable, and a series approximation for its solution is given. Under certain conditions (given in (9.7)) on the parameters of the Ginzburg-Landau equation, compatible with vortex shedding modelling on a semiinfinite domain, the kernel is shown to have compact support, resulting in partial state feedback. Simulations are provided in order to demonstrate the performance of the controller. Acknowledgements. We thank Professor Peter Monkewitz for helpful explanations on relationships between Navier-Stokes and Ginzburg-Landau models of vortex shedding and on implementability of GL-based controllers on NS simulations or experiments. REFERENCES [1] O.M. Aamo and M. Krsti´c, ”Global stabilization of a nonlinear Ginzburg-Landau model of vortex shedding,” European Journal of Control, vol. 10, no. 2, 2004. [2] A. Balogh and M. Krsti´c, ”Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability,” European Journal of Control, vol. 8, no. 2, pp. 165—176, 2002. [3] D.M. Boskovi´c, M. Krsti´c, and W.J. Liu, ”Boundary control of an unstable heat equation via measurement of domain-averaged temperature,” IEEE Transactions on Automatic Control, vol. 46, no. 12, pp. 2022—2028, 2001. [4] A. Friedman, Partial Differential Equations of Parabolic Type, Robert E. Krieger Publishing Company, 1983. [5] M.D. Gunzburger, L.S. Hou, and S.S. Ravindran, ”Analysis and approximation of optimal control problems for a simplified Ginzburg-Landau model of superconductivity,” Numerische Mathematik, 77, pp. 243—268, 1997. [6] M. Krsti´c, I. Kanellakopoulos, and P. Kokotovi´c, Nonlinear and Adaptive Control Design, Wiley, New York, 1995. [7] E. Lauga and T.R. Bewley, ”H∞ control of linear global instability in models of non-parallel wakes,” Proceedings of the Second International Symposium on Turbulence and Shear Flow Phenomena, Stockholm, Sweden, 2001.
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[8] E. Lauga and T.R. Bewley, ”The decay of stabilizability with Reynolds number in a linear model of spatially developing flows,” Proceedings of the Royal Society of London, 459, pp. 2077—2095, 2003. [9] E. Lauga and T.R. Bewley, ”Performance of a linear robust control strategy on a nonlinear model of spatially-developing flows,” to appear in Journal of Fluid Mechanics. [10] W.J. Liu, ”Rapid boundary feedback stabilization of an unstable heat equation,” SIAM Journal on Control and Optimization, vol. 42, pp. 1033—1043, 2003. [11] P.A. Monkewitz, private communication. [12] D.S. Park, D.M. Ladd, and E.W. Hendricks, ”Feedback control of a global mode in spatially developing flows,” Physics Letters A, 182, pp. 244—248, 1993. [13] K. Roussopoulos and P.A. Monkewitz, ”Nonlinear modelling of vortex shedding control in cylinder wakes,” Physica D, 97, pp. 264—273, 1996. [14] A. Smyshlyaev and M. Krsti´c, ”Regularity of hyperbolic PDE’s governing backstepping gain kernels for parabolic PDE’s”, submitted. Downloadable from http://mae.ucsd.edu/research/krstic/papers/smykrs.pdf.
1. Coefficients for the Ginzburg-Landau Equation. The numerical coefficients below are taken from [13, Appendix A]. Rc xt ω0t k0t t ωkk t ωxx kxt
= 47 = 1.183 − 0.031i = 0.690 + 0.080i + (−0.00159 + 0.00447i)(R − Rc ) = 1.452 − 0.844i + (0.00341 + 0.011i)(R − Rc ) = −0.292i = 0.108 − 0.057i = 0.164 − 0.006i ¢2 1 t ¡ x) = ω0t + ωxx ω0 (˘ x ˘ − xt 2 ¡ ¢ k0 (˘ x) = k0t + kxt x ˘ − xt 1 t a1 = iωkk 2 t a2 (˘ x) = ωkk k0 (˘ x) µ ¶ 1 t 2 a3 (˘ x) = − ω0 (˘ x) + ωkk k0 (˘ x) i 2 a4 = −0.0225 + 0.0671i.
