Numerical Solution of Optimal Control Prob- lems Governed ... · Numerical Solution of Optimal Control Prob- ... Assuming a suﬃciently smooth state,formal adjoint and gradi-

Numerical Solution of Optimal Control Prob-lems Governed by the Compressible Navier–Stokes Equations

S. Scott Collis, Kaveh Ghayour, Matthias Heinkenschloss,Michael Ulbrich, and Stefan Ulbrich

Abstract. Theoretical and practical issues arising in optimal boundary con-trol of the unsteady two–dimensional compressible Navier–Stokes equationsare discussed. Assuming a sufficiently smooth state, formal adjoint and gradi-ent equations are derived. For a vortex rebound model problem wall normalsuction and blowing is used to minimize cost functionals of interest, here thekinetic energy at the final time.

1. Introduction

Recently, optimal control and optimal design problems governed by fluid flow mod-els have received significant attention in the mathematical and in the engineeringliterature. See, e.g., the collections and reviews [7, 10, 11]. The coupling of ac-curate computational fluid dynamics analyses with optimal control theory holdsthe promise for modifying a wide-range of fluid flows to achieve enhancement ofdesirable flow characteristics. Reduction of skin-friction drag, separation suppres-sion, and increased lift to drag ratios for airfoils are examples of the types ofoptimization that such an approach enables. Moreover, advances in smart materi-als and microelectromechanical systems (MEMS) have increased the possibilitiesto actually implement controllers in physical systems. Optimal control problemshave been studied mathematically and numerically for steady and unsteady in-compressible Navier–Stokes flow. The references [1, 2, 5, 11, 12] present a smallsample of the work in this area. Compressible steady state Euler equations andNavier–Stokes equations have been used in the context of optimal design, see, e.g.,[3, 14, 15, 16, 21].

In this paper we study the optimal control of two–dimensional unsteady com-pressible Navier–Stokes flows. To the best of the authors’ knowledge, this is thefirst attempt to apply optimal control to problems governed by the unsteady com-pressible Navier–Stokes equations. Our research is motivated by the potential to

2 S. S. Collis, K. Ghayour, M. Heinkenschloss, M. Ulbrich, and S. Ulbrich

develop novel and effective flow control strategies for inherently compressible phe-nomena including aeroacoustics and heat transfer by utilizing optimal control the-ory. Specifically, we plan to control the sound arising from Blade-Vortex Interaction(BVI) that can occur for rotorcraft in low speed, descending flight conditions, suchas on approach to landing. When BVI occurs, tip vortices shed by a preceding bladeinteract with subsequent blades resulting in a high amplitude, impulsive noise thatcan dominate other rotorcraft noise sources. Reduction of the noise generated bythis mechanism can alleviate restrictions on civil rotorcraft use near city centersand thereby enhance community acceptance. High frequency loading associatedwith this phenomenon also causes fatigue and hence reductions in BVI can have adirect impact on maintenance costs associated with blade failure in fatigue mode.

We use adjoint based gradient methods to solve the discretized optimal con-trol problem. A critical issue in the numerical solution of optimal control problemsis the accuracy of adjoint and gradient information. For successful optimizationof the discretized problem, it is indispensable that the gradient approximation issufficiently close to the derivative of the discretized objective function. To ensurethat the solution of the discretized optimization problem approximates the in-finite dimensional optimal control, it is also important that adjoint and gradientapproximations used for the discretized problem converge towards their infinite di-mensional counterparts as the discretization is refined. This requires a comprehen-sive view of the problem that integrates well posedness of the infinite dimensionalproblem, existence of adjoint equations and gradient equations, and properties ofthe discretization. Unfortunately, the mathematical foundation for optimal controlproblems governed by the unsteady compressible Navier–Stokes equations is notsufficiently developed to allow a rigorous and comprehensive study of gradient andadjoint accuracy in the previous sense. Even mathematical existence theories forthe unsteady compressible Navier–Stokes equations are less developed than for theincompressible case.

In this paper, we discuss some of the theoretical issues arising in the formu-lation and solution of optimal boundary control problems governed by the com-pressible Navier–Stokes equations. Assuming a sufficiently smooth state, we deriveformal adjoint and gradient equations. Finally, we present optimal control resultsfor a vortex rebound test problem.

2. Problem Formulation

Let Ω =x ∈ IR2 : x2 > 0

denote the spatial domain occupied by the fluid and

let Γ denote its spatial boundary. By

u = (ρ, v1, v2, T )T

we denote the primitive flow variables, where ρ(t,x) is the density, vi(t,x) de-notes the velocity in xi-direction, i = 1, 2, v = (v1, v2)T , and T (t,x) denotes the

Optimal Control of Compressible Navier–Stokes Equations 3

temperature. The pressure p and the total energy per unit mass E are given by

p =ρT

γM2 , E =T

γ(γ − 1)M2+

12vT v,

respectively, where γ is the ratio of specific heats and M is the reference Machnumber. We write the conserved variables as functions of the primitive variables,

q(u) = (ρ, ρv1, ρv2, ρE)T

and we define the inviscid flux terms

F1(u) =

ρv1

ρv21 + p

ρv2v1

(ρE + p)v1

, F2(u) =

ρv2

ρv1v2

ρv22 + p

(ρE + p)v2

, (1)

and the viscous flux terms

Gi(u,∇u) =1Re

0τ1i

τ2i

τ1iv1 + τ2iv2 + κPrM2(γ−1)Txi

, (2)

i = 1, 2, where τij are the elements of the stress tensor τ = µ(∇v+∇vT )+λ(∇·v)I.Here µ, λ are first and second coefficients of viscosity, κ is the thermal conductivity,Pr is the reference Prandtl number, and Re is the reference Reynolds number.For the demonstration problems presented here, constant Prandtl number andfluid properties (viscosities and thermal conductivity) are assumed along withStokes hypothesis for the second coefficient of viscosity, λ = −2µ/3. Variable fluidproperties can be easily accommodated and these effects will be included in futurestudies.

The two–dimensional compressible Navier–Stokes equations for the time in-terval [t0, tf ] can now be written as

q(u)t +2∑

i=1

(Fi(u)xi − Gi(u,∇u)xi

)= 0 in (t0, tf ) × Ω, (3)

B(u,∇u,g) = 0 on (t0, tf ) × Γ, (4)

u(t0,x) = u0(x) in Ω. (5)

The function g in the boundary conditions (4) acts as the control, which is takento be suction and blowing in the wall normal direction on Γc ⊂ Γ. This is modeledby

v = b + g on Γc, (6)

where b is a given boundary velocity that satisfies the compatibility conditionv(t0,x) = b(t0,x) for x ∈ Γ. Since Γc ⊂ x : x2 = 0, we have g = (0, g2)T .

To the best of our knowledge, the question of existence and uniqueness ofglobal solutions for the full compressible Navier-Stokes equations (3)–(5) in 2D


and 3D is still open for large initial data. The existence and uniqueness of globalsolutions with (ρ−ρ0,v, T−T0) ∈ C(t0, tf ;H3)∩C1(t0, tf ;H1) for the initial valueproblem (3), (5) is shown in [18] if the initial data u0 are close in H3 to a constantstate (ρ0, 0, 0, T0)T , ρ0, T0 > 0. Local existence in time can be shown also for largedata. An analogous result for the initial-boundary value problem (3)–(5) on thehalf space or on the exterior of any bounded region with smooth boundary is shownin [19] for the boundary conditions v|(t0,tf )×Γ = 0 and either T |(t0,tf )×Γ = T0 or∂T∂n = 0, where n denotes the outward unit normal. Similar results can be found inthe review article [22]. The global existence of weak solutions for the initial valueproblem is shown in [13], if ρ(t0, ·)−ρ0 is small in L2∩L∞, v(t0, ·) is small in L2∩L4,and T (t0, ·)−T0 is small in L2 with constants ρ0, T0 > 0. It is shown that v, T areHolder-continuous in space and time for t > t0, v(t, ·), T (t, ·)−T0 ∈ H1, but merelyρ(t, ·)− ρ0 ∈ L2 ∩L∞, ρ− ρ0 ∈ C(t0, tf ;H−1). The question of uniqueness ist leftopen. Other results, mostly for the barotropic case in which pressure depends onlyon ρ and which decouples the energy equation from the remaining ones, can befound in [17, 22].

The optimal control problem treated in this paper is the minimization ofkinetic energy in Ω0 ⊂ Ω at final time, more precisely,

ming∈G

J(g) def=12

∫Ω0

ρ(tf ,x)‖v(tf ,x)‖22dx

+∫ tf

t0

∫Γc

(α1

2‖gt‖2

2 +α2

2‖∇g‖2

2 +α3

2‖g‖2

2

)dxdt.

(7)

where α1, α2, α3 > 0 and where the control space is chosen to be

G =g : g ∈ L2(t0, tf ;H1

0 (Γc)), gt ∈ L2(t0, tf ;L2(Γc)), g(t0,x) = 0 in Γc

.

Here ∇g is the gradient of g on the boundary, in our case ∇g = (0, (g2)x1)T .The second part of J is a regularization term which, together with G, must bechosen so that (7) is well–posed. In particular the regularity requirements on thecontrol must be compatible with the regularity of the trace of v on Γc. For theincompressible Navier–Stokes equations such trace regularity estimates have beenprovided recently in [8, 9] and our choice of the control space and of the regular-ization term follows [12]. There is no theoretical justification yet that this choiceis suitable for the compressible case. Significantly stronger regularity requirementson the controls seem necessary in connection with the theory in [18, 19]. Our regu-larity requirements are closer aligned with what one would expect from the theoryin [13]. Application of general existence results such as those in [8] to (7) are notyet known. However, our numerical results indicate that (7) is well–posed for theflows we have considered. A relaxation of the regularity requirements, i.e., settingα1 = 0 or even α1 = α2 = 0 leads to highly oscillatory controls. More detailedgrid convergence studies are under way. We remark that the choice of the controlspace and regularization term does not affect the adjoint equations computed in


Sections 3.2, 3.3, it only effects how the gradient is computed given the adjoint(see (8)).

3. Adjoint Equation

3.1. Adjoint Equation and Gradient for an Abstract Problem

We first consider the gradient computation for an abstract functional whose eval-uation involves implicitly defined functions. Let G be a Hilbert space with innerproduct 〈·, ·〉G and let U , C be Banach spaces. We consider an equation C(u,g) = 0,where C : U × G → C. Suppose that for every g ∈ G the equation C(u,g) = 0 hasa unique solution u(g). We consider the abstract problem

ming∈G

J(g) = J(u(g),g).

We assume the existence of neighborhoods G, U of g and u = u(g), respectively,such that C is Frechet–differentiable. Further, we assume that u(g) is differentiableon G. This holds, e.g., if C is continuously Frechet–differentiable on U×G and thepartial derivative Cu(u, g) is continuously invertible. However, in our context theselatter requirements seem to be too restrictive, since the results in [18, Prop. 4.1]indicate that the solution of the linearized state equation is less regular than thestate u about which the linearization is done. Now suppose that J is Frechet–differentiable on U × G. Then the Frechet–derivative Jg(g) ∈ G∗ and the gradient∇J(g) ∈ G of J can be computed from Jg(g) = Ju(u, g) ug(g) + Jg(u, g) and〈Jg(g),g′〉G∗×G = 〈∇J(g),g′〉G for all g′ ∈ G, respectively, where G∗ denotes thetopological dual of G and 〈·, ·〉G∗×G denotes the duality pairing between G∗ and G.

It can be shown that the gradient ∇J(g) can be computed from

〈∇J(g),g′〉G = 〈Cg(u, g)∗λ,g′〉G∗×G + 〈Jg(u, g),g′〉G∗×G (8)

for all g′ ∈ G, if there exists an adjoint state λ ∈ C∗ satisfying the adjoint equationCu(u, g)∗λ = −Ju(u, g) in U∗, i.e., if λ satisfies

〈Cu(u, g)∗λ,u′〉U∗×U = 〈−Ju(u, g),u′〉U∗×U (9)

for all u′ ∈ U . The existence of λ is ensured if Cu(u, g)∗ is onto. Note, however,that it is sufficient that the adjoint equation is solvable for the particular righthand side −Ju(u, g).

3.2. Adjoint Equation for the Compressible Navier–Stokes Equations

We carry out the formal derivation of the adjoint equation for the general situationthat Ω is a domain with C2-boundary Γ. According to (3)–(5) we define

C(u,g) =

q(u)t +∑2

i=1

(Fi(u) − Gi(u,∇u)

)xi

B(u,∇u,g)u − u0

. (10)


To write the linearization of the Navier–Stokes equation it is useful to define M =qu(u),Ai = Fi

u(u), i = 1, 2, and to write

Gi(u,∇u) = Ki1(u)ux1 + Ki

2(u)ux2 , i = 1, 2, (11)

where

K11(u) =

1Re

0 0 0 00 µ 0 00 0 µ 00 µv1 µv2

κPrM2(γ−1)

, K12(u) =

1Re

0 0 0 00 0 λ 00 µ 0 00 µv2 λv1 0

,

K21(u) =

1Re

0 0 0 00 0 µ 00 λ 0 00 λv2 µv1 0

, K22(u) =

1Re

0 0 0 00 µ 0 00 0 µ 00 µv1 µv2

κPrM2(γ−1)

with µ = 2µ + λ. From the representation (11) we obtain Gi

uxj(u,∇u) = Ki

j(u),i, j = 1, 2, and the definition (2) of the viscous terms implies

Giu(u,∇u) = Di(∇u) =

1Re

0 0 0 00 0 0 00 0 0 00 τ1i τ2i 0

, i = 1, 2.

In the following we simply write Kij instead of Ki

j(u), i, j = 1, 2, and Di instead ofDi(∇u), i = 1, 2. With this notation, the linearized state equation can be writtenas

Cu(u,g)u′ =

(Mu′)t +∑

i

(Aiu′ − Diu′ −∑j Ki

ju′xj

)xi

Bu(u,∇u,g)u′ +∑

j Buxj(u,∇u,g)u′

xj

u′

. (12)

The adjoint variables λ are partitioned according to the partition of C in(10) and are denoted by λ = (λd,λb,λ0).

We assume that

〈DuJ(u,g),u′〉U∗×U =∫ tf

t0

∫Ω

(u′)T r +∫

Ω

(u′|t=tf)T rtf

+∫ tf

t0

∫Γ

((u′)T rΓ + (∇u′n)T rΓ,n + (∇u′s)T rΓ,s

),

(13)

where n = (n1, n2)T is the unit outward normal, s = (s1, s2)T = (−n2, n1)T isthe unit tangential vector, and ∇u′ is the Jacobian of u′. This is true for theobjective function in (7), but also for many more general objective functions thatinvolve distributed observations or observations of normal derivatives ∇u n or oftangential derivatives ∇u s of the state.


To derive the adjoint equations, we multiply Cu(u,g)u′ by λ and integratethe resulting terms over (t0, tf ) × Ω, (t0, tf ) × Γ, and Ω, respectively. Integrationby parts leads to (9), which in this case is given by

−∫ tf

t0

∫Ω

(u′)T r −∫

Ω

(u′|t=tf)T rtf

−∫ tf

t0

∫Γ

((u′)T rΓ + (∇u′n)T rΓ,n + (∇u′s)T rΓ,s

)=∫ tf

t0

∫Ω

(u′)T

(−MT λd

t −∑

i

((Ai − Di)T λd

xi+∑

j

((Ki

j)T λdxi

)xj

))

+∫ tf

t0

∫Γ

(u′)T

(∑i

(ni(Ai − Di)T λd +

∑j

nj(Kij)T λd

xi

)+ BT

uλb

)

+∫ tf

t0

∫Γ

∑j

(u′xj

)T

(BT

uxjλb −

∑i

ni(Kij)T λd

)+∫

Ω

(u′)T MT λd|t=tf+∫

Ω

(u′)T (λ0 − MT λd)∣∣∣t=t0

∀u′.

(14)

If we choose test functions u′ ∈ C∞0 ((t0, tf ) × Ω), then (14) implies

MT λdt +

∑i

((Ai − Di)T λd

xi+∑

j

((Ki

j)T λdxi

)xj

)= r (15)

in (t0, tf ) × Ω. If we choose test functions u′ such that u′ = 0 on t0 × Ω, u′ = 0and ∇u′ = 0 on (t0, tf ) × Γ, then (14) implies (MT λd)|t=tf

= −rtfin Ω or,

equivalently,

λd = −M−T rtfin tf × Ω. (16)

Similarly, if we choose test functions u′ such that u′ = 0 on tf × Ω, u′ = 0 and∇u′ = 0 on (t0, tf )×Γ, then (14) implies λ0 − (MT λd)|t=t0 = 0 in Ω. This meansthat λ0 is determined by λd|t=t0 .

Next we choose test functions u′ such that u′ = 0 on 0, tf × Ω and on(t0, tf)×Γ. The tangential derivatives (∇u′)s of these test functions is zero. Using∇u′ = ∇u′nnT + ∇u′ssT = ∇u′nnT , u′

xj= ∇u′ej = ∇u′nnj and the previous

identities in (14), we obtain

∑j

nj

(BT

uxjλb −

∑i

ni(Kij)T λd

)= −rΓ,n on (t0, tf ) × Γ. (17)


With (15)–(17) the identity (14) reduces to

−∫ tf

t0

∫Γ

((u′)T rΓ + (∇u′s)T rΓ,s

)=∫ tf

t0

∫Γ

(u′)T

(∑i


∑j

nj(Kij)T λd

xi

)+ BT

uλb

)

+∫ tf

t0

∫Γ

(∇u′s)T∑

j

sj

(BT

uxjλb −

∑i

ni(Kij)T λd

)∀u′.

One can employ integration by parts over Γ on the integrals involving (∇u′s) toarrive at the more convenient identity∑

i


∑j

nj(Kij)T λd

xi

)+ BT

uλb

− ∂

∂s

(∑j

sj

(BT

uxjλb −

∑i

ni(Kij)T λd

)+ rΓ,s

)= −rΓ.

(18)

3.3. Adjoint Equation for the Boundary Control of Final–Time Kinetic Energy

In our model problem (7) we assume adiabatic boundary conditions for the tem-perature on the bottom wall Γ =

x ∈ IR2 : x2 = 0

. The velocities on Γ are

prescribed and are equal to b on (t0, tf )× (Γ \Γc) and they are equal to b + g on(t0, tf) × Γc. The boundary condition operator B in (4) is

B(u,∇u,g) =(v − g − b

−Tx2

), B(u,∇u,g) =

(v − b−Tx2

)on (t0, tf )×Γc and on (t0, tf )×(Γ\Γc), respectively. The partial Frechet–derivativeof the objective function J in (7) is given by (13) with r = rΓ,n = rΓ,s = 0 and

rtf(x) = 1Ω0(x)

(12‖v(x, tf )‖2

2, ρ(x, tf )v(x, tf ), 0)T

. (19)

Since Ω =x ∈ IR2 : x2 > 0

, n = (0,−1)T on Γ, the boundary condition (17)

reads

BTux2

λb + (K22)T λd = 0, on (t0, tf ) × Γ,

which, using the definition of B and K22, is equivalent to

1Re

µ 0 µv1

0 2µ + λ (2µ + λ)v2

0 0 κPrM2(γ−1)

λd2

λd3

λd4

=

00λb

3

on (t0, tf ) × Γ.

Hence, we obtain

λb3 =

κ

RePrM2(γ − 1)λd

4 on (t0, tf ) × Γ, (20)


and the boundary conditions

λd2 = −v1λ

d4, λd

3 = −v2λd4 on (t0, tf ) × Γ. (21)

These boundary conditions imply (K21)T λd = 0. Hence, the boundary condition

(18) on (t0, tf ) × Γ reduces to

(A2 − D2)T λd + (K12)T λd

x1+ (K2

2)T λdx2

= BTuλb. (22)

By definition, A2 = F2u(u), where F2(u) is one of the inviscid fluxes in (1). Hence,

A2 =

v2 0 ρ 0

v1v2 ρv2 ρv1 0v22 + T

γM2 0 2ρv2ρ

γM2

v2

(T

(γ−1)M2 + 12v

T v)

ρv1v2 ρ(

T(γ−1)M2 + 1

2vT v)

+ ρv22

ρv2(γ−1)M2

.

Moreover, using the definition of K12 and (21) we obtain

(K12)T λd

x1=

1Re

0

µ(λd3)x1 + µv2(λd

4)x1

λ(λd2)x1 + λv1(λd

4)x1

0

=1Re

0

−µ(v2)x1

−λ(v1)x1

0

λd4.

If we insert the previous two equations and (21) into (22) we arrive at the conditionv2 v2

(T

γ(γ−1)M2 − 12v

T v)

0 − 1Re (µ(v2)x1 + τ12)

ρ ρ(

T(γ−1)M2 − 1

2vT v)− 1

Re (λ(v1)x1 + τ22)0 ρv2

(λd

1

λd4

)

+1Re

0 0 0µ 0 µv1

0 2µ + λ (2µ + λ)v2

0 0 κ(γ−1)M2Pr

λd

2

λd3

λd4

x2

=

0λb

1

λb2

0

(23)

on (t0, tf ) × Γ. Equation (23) yields the boundary conditions

v2λd1 = v2

(− T

γ(γ − 1)M2+

12vT v

)λd

4,

ρv2λd4 = − κ

(γ − 1)M2PrRe(λd

4)x2

(24)

on (t0, tf ) × Γ. Thus, the adjoint boundary conditions for λd are given by (21),(24). If desired, the adjoint variables λb for the boundary data can be computedfrom λd using (20) and the second and third equation in (23).

We remark, that the general formulation of the adjoint equations (15)–(18)is also useful, when non–reflecting boundary conditions are introduced on theboundary ∂Ωc \ Γ of the computational domain Ωc ⊂ Ω.


3.4. Gradient Computation

Given the adjoint λ, the gradient can be computed from (8). As in [9, 12] this leadsto an elliptic problem on the time–space boundary (t0, tf )×Γ for ∇J . Due to spacerestrictions the details are omitted here. For the numerical solution of the optimalcontrol problem the solution of this PDE can be avoided with a reformulation ofthe optimal control problem and working with a different, yet appropriate innerproduct. This is described in [4] for the semi–discrete case.

4. Results

We present results for a model problem consisting of two counter-rotating viscousvortices above an infinite wall which, due to the self-induced velocity field, prop-agate downward and interact with the wall. Our non–dimensionalization is basedon initial vortex core radius and the maximum azimuthal velocities at the edgeof the viscous cores. For the computations reported here, the Mach, Reynolds,and Prandtl numbers are M = 0.5,Re = 25,Pr = 1, respectively. Our compu-tational domain is [−15, 15] × [0, 15] in non–dimensional units. The compressibleNavier–Stokes equations are discretized in space using fourth-order accurate cen-tral differences on a 128 × 128 uniform grid. Time integration is performed usingthe classical fourth–order Runge Kutta method with a fixed time step ∆t = 0.05.To compute the initial conditions, two compressible Oseen vortices [6], located at(±2, 7.5), are superimposed at time t = 0. From this superposed field, which is nota solution to the equations, we advance 100 time steps until time t0 = 5, and takethe resulting flow field at time t0 as the initial condition to our problem. The sideand top boundaries are assumed to be located far enough from the main flow re-gion to justify the imposition of a characteristics based inviscid far–field boundarycondition. For details on the model problem and its discretization see [4].

We control the flow in the time window t0 = 5, tf = 40. Our control g is thewall normal velocity which for our geometry is given by g = (0, g2). The followingcontrol plots show g2. Positive g2 represents injection (blowing) of fluid into thedomain while negative g2 corresponds to suction of fluid out of the domain.

Our numerical results are produced using a nonlinear conjugate gradient(ncg) algorithm [20] for the solution of the discretized problem. The inner productsused in the ncg method are discretizations of the G inner product (see [4]) tominimize the mesh–dependent behavior of the cg method and to avoid artificialill–conditioning due to discretization. All computations are performed in parallelon an SGI Origin 2000. Using four processors, one optimization run takes about10hours.

We performed three runs, two include a regularization of the time derivativeof the control, the third does not. The coefficients αj in the regularization term,the value of the objective functional in (7) at the initial iterate, i.e., for zero control(J0), at the final control iterate (Jfinal) and the terminal kinetic energy (the firstintegral in (7)) at the final control iterate (TKEfinal) are shown in Table 1. We see


Figure 1. Optimal wall–normal velocity distributions

-0.4

-0.2

0

-10

0

10X

10

20

30

40

time

Control g

Optimal control run I

-0.4

-0.2

0

-10

0

10X

10

20

30

40

time

Control g

Optimal control run III

that because of the large regularization parameter α1, the terminal kinetic energyreduction in run I is less than that for run III. However, a smaller α1 > 0 will givea smaller TKEfinal, while maintaining temporal smoothness in the controls.

Table 1.

Run α1 α2 α3 J0 Jfinal TKEfinal

I 0.5 0.005 0.005 12.43 0.48 0.42II 0.05 0.005 0.005 12.43 0.37 0.32III 0 0.005 0.005 12.43 0.24 0.20

In all cases the optimization is started with zero control. The optimal wall–normal velocity distributions g2 for are plotted in Figure 1. The optimal controlsfor runs I and II are very similar, the amplitudes in the optimal control for runs IIare slightly higher than those for run I, but no additional oscillations arise whenα1 is reduced to 0.05. The plots clearly show the effect of the regularization term∫

α12 ‖gt‖. Without it, the control starts to oscillate in time in the second half of

[t0, tf ] and it exhibits a large jump at t0. If we even set α1 = 0 and α2 = 0, thencontrols are produced that exhibit strong spatial and temporal oscillations whichfrequently led to a failure in the compressible Navier–Stokes solver.

Figure 2 shows the contours of kinetic energy for the uncontrolled flow andthe controlled flow, run I.

5. Acknowledgements

The work of S. Collis, K. Ghayour and M. Heinkenschloss was supported by TexasATP grant 003604–0001, 1999. M. Ulbrich and S. Ulbrich were supported by DFGgrants UL157/3-1 and UL158/2-1, respectively, and by CRPC grant CCR-9120008.


Figure 2. Kinetic energy contours for the uncontrolled flow (toprow) and the controlled flow, run I (bottom row).

X

Y

-15 -10 -5 0 5 10 150

5

10

15

t = t0

X

Y

-15 -10 -5 0 5 10 150

5

10

15

t = t0 + 15X

Y

-15 -10 -5 0 5 10 150

5

10

15

t = tf

X

Y

-15 -10 -5 0 5 10 150

5

10

15

t = t0

X

Y

-15 -10 -5 0 5 10 150

5

10

15

t = t0 + 15X

Y

-15 -10 -5 0 5 10 150

5

10

15

t = tf

Computations were performed on an SGI Origin 2000 which was purchased withthe aid of NSF SCREMS grant 98–72009.

References

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Departments of Computational and Applied Mathematics andof Mechanical Engineering and Materials Science,Rice University,Houston, TX 77005-1892, USAE-mail address: [email protected], [email protected], [email protected]

Lehrstuhl fur Angewandte Mathematik und Mathematische Statistik,Technische Universitat Munchen,D–80290 Munchen, GermanyE-mail address: [email protected], [email protected]

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