The POD Dirichlet Boundary Control of the
Navier-Stokes Equations: A Low-dimensional
Approach to Optimal Control with High
Smoothness
Ying Wang1, Fredi Tr¨oltzsch2, and G¨unter B¨arwolff2
1TU Braunschweig, Institute for computational modeling in civil engineering,
D-38106 Braunschweig, M¨uhlenpfordtstraße 4-5, Germany
2TU Berlin, Institute of Mathematics, D-10623 Berlin, Str. d. 17. Juni 135,
Germany
Summary. The proper orthogonal decomposition(POD) is an approach to capture
a reduced order basis functions for a dynamical system. Utilizing the order reduc-
tion property of POD basis for minimizing computational cost to unsteady fluid
flow control problem, we present a POD-based framework of the unsteady Dirichlet
boundary control problem for Navier-Stokes equations. An extra basis function can
be therefor constructed and appended into the general POD subspace, which as a
key step enables the POD approach to the Dirichlet boundary control and results
in the control problem merely in time scale. In the paper the excellent quality and
flexibility of the POD approach to Dirichlet boundary flow control are confirmed
numerically in several flow matching control examples.
Key words: Dirichlet boundary control, Galerkin POD method, reduced
order models, Navier-Stokes equations
1 Introduction
Numerical flow simulation is computationally expensive for the purpose of
unsteady fluid flow control. Recent development in computational methods
for the control problems is therefore the design of model reduction to carry
out the flow control problems with less effort. The proper orthogonal decom-
position is one of the most widely applied methods for this purpose.
In the paper, we study the Dirichlet boundary flow control by using
Galerkin POD, for which the fluid motion can be controlled by injection along
a piece of boundary. The cost functional for flow matching is expressed as a
measure of the distance between the controlled velocity and a given target
flow. The control problem is subject to the unsteady Navier-Stokes equations
2 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
for viscous, incompressible fluid. This subject has been studied by many au-
thors, e.g., [2–4,7,8].
The POD was first proposed by Lumley [11] in 1967, as a mathemati-
cal technique to extract a typical structure from turbulence flows. Numerous
literature in the past two decades signifies the great progress in theoreti-
cal, numerical analysis and computational aspects especially for optimization,
e.g. [1,10,13,14,18–20].
The Galerkin proper orthogonal decomposition provides possibility for de-
riving reduced order models of dynamical systems, which is in general de-
scribed by PDEs. It is based on projecting the governing PDEs onto a proper
subspace of snapshots ensemble, which is composed of the solutions for this
dynamical system at pre-specified time distances or even experimental mea-
surements. However, the snapshots are not suitable as the basis for the en-
semble spanned by themselves due to the possible linear dependence. The
global optimal orthogonal basis for the ensemble can be identified by solving
an eigenvalue problem, and these basis will be denoted as POD basis. The
number of these POD basis can be very small in comparison with the number
of snapshots, which nevertheless carry the most dynamic energy of the system.
The POD basis spanned subspace is the subspace onto which the dynamical
system will be projected.
With Galerkin POD method we aim at not only reducing the computa-
tional cost of solving the nonlinear flow dynamics, but also achieving optimal
Dirichlet boundary control for the Navier-Stokes equations. An access to the
subject is firstly to make a new ansatz for the solution of the Navier-Stokes
equations containing an extra basis function which is extracted from the spa-
tial boundary behavior along the time, subsequently summarize the control
action merely in a time dependent function. To guarantee its smoothness in
the Dirichlet boundary condition, the explicit control parameter coupled in
the control problem description would be its first time derivative. We ap-
ply Galerkin POD method to qualify this layout for the Dirichlet boundary
control. This approach results in a new control problem in time scale, which
facilitates computation during the numerical optimization.
The paper is organized as follows. In section 2, we define the Dirichlet
boundary control problem subjects to the Navier-Stokes equations and quote
the well-posedness of the state equations. The optimality system based on a
Lagrangian technique including adjoint equation and variational inequality is
derived in section 3. Section 4 is devoted briefly to theory of Galerkin POD,
which associates the Dirichlet boundary control problem with the modified
one into an optimal control problem in time scale. We illustrate the feature of
the POD subspace, comparison full and reduced order simulations and provide
numerical results of the optimization with Galerkin POD in section 5.
2 The Dirichlet Boundary Flow Control Problem
The incompressible fluid flow described by the Navier-Stokes equations in a
Ω⊆R2throughout [0, T] is characterized by the following quantities:
u:Q:= Ω×[0, T]→R2velocity field,
p:Q:= Ω×[0, T]→Rpressure.
The boundaries of the spatial domain Ωconsist of the inflow boundary Γi,
outflow boundary Γoand solid wall Γs:= Γ\(Γi∪Γo). A controllable design
parameter will be set on the inflow boundary Γi. For the solid wall Γsand
outflow Γo, we apply the nonslip boundary and open boundary condition, the
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 3
latter was detailed in [15].
We intend to find an optimal control zfor the following Dirichlet boundary
control problem
min J(u, z) = 1
2
T
Z0ku(t, ·)−Udk2
Hdt +σ1
2
T
Z0
z2(t)dt
subject to
ut−1
Re∆u+ (u·∇)u+∇p= 0 in Q
∇·u= 0 in Q
u(x, t) = z(t)g(x) on Σi:= Γi×[0, T]
pν −1
Re
∂u
∂ν = (0,0) on Σo:= Γo×[0, T]
u(x, t) = (0,0) on Σs:= Γs×[0, T]
u(x,0) = u0(x) in Ω
in the set of admissible controls
Uad ={z∈L2(0, T) : za(t)≤z(t)≤zb(t) a.e. on (0, T)}.
2.1 Preliminary results
First of all, we present the existence and uniqueness of weak solution for the
inhomogeneous Navier-Stokes equations. Let H,Vand V0be solenoidal spaces
H:= {v∈L2(Ω)2:∇·v= 0}, V : = {v∈H1(Ω)2:∇·v= 0}and
V0: = {v∈H1
0(Ω)2:∇·v= 0}.
The constraint ∇ ·v= 0 is equivalent to hv,∇pi= 0 for all p∈H1(Ω). By
identifying Hand its dual H0, we obtain the well-defined Gelfand triple
V ,→H=H0,→V0,
each embedding being continuous and dense.
The inner product in V0is given by a symmetric bounded, coercive bilinear
form a:V0×V0→R:
a(u,v) = h∇u,∇viL2(Ω)2=Z
Ω
2
X
i=1 ∇ui·∇vidx.(1)
Moreover, we introduce trilinear form b:H1(Ω)2×H1(Ω)2×H1(Ω)2→R
as the weak formulation of the Navier-Stokes nonlinearity (u·∇)uby
b(u,v,w) = h(u·∇)v,wi=Z
Ω
2
X
i,j=1
ui(Divj)wjdx,(2)
and bsatisfies b(u,v,w) = −b(u,w,v) for all u∈V0and v,w∈H1(Ω)2.
To deal with the time derivative in the Navier-Stokes equations, we turn to
the space of functions u, whose time derivative utexists as abstract function
in
W(0, T) := W2(0, T ;V0) = {u∈L2(0, T ;V0) : ut∈L2(0, T ;V0
0)}
4 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
endowed with norm
kukW(0,T )=kukL2(0,T ;V0)+kutkL2(0,T ;V0
0).
W(0, T) is Hilbert space and continuous embedded in C([0, T], V0), for the
detail we refer to [17]. Since the state equation (1) due to boundary control
must not pose homogeneous Dirichlet boundary all the time, the further space
definition is therefore necessary to be initiated by following [8]
W2(0, T ;V) := {u∈L2(0, T ;V) : ut∈L2(0, T;V0
0)},
endowed with norm
kukW2(0,T ;V)=kukL2(0,T ;V)+kutkL2(0,T ;V0
0).
To describe all solutions of the unsteady Navier-Stokes equations with the
admissible inhomogeneous Dirichlet boundary conditions, we specify
WΣi:= {g=τˆg: ˆg∈W2(0, T;V)}(3)
endowed with norm
kgkWΣi= inf{kˆgkW2(0,T ;V):τˆg=g, ˆg∈W2(0, T ;V)}.(4)
Here τ:W2(0, T;V)→L2(H1/2(Γi)2) is the trace operator given by
(τg)(t) = g(t, ·)¯¯Γifor almost every t∈[0, T ]. (5)
Lemma 1. For every ui=z(t)g(x)∈WΣi, there exists ˜
u∈W2(0, T ;V)
which achieves the infimum in (4).
For the proof of Lemma 1 and the detailed properties of the space WΣi, we
refer the reader to [8].
Theorem 1. For every ui∈WΣiwith ˜
u∈W2(0, T;V)(see Lemma 1) and
every divergence free u0∈Hwith u0−˜
u(0) ∈V0, there exists a unique weak
solution u∈W2(0, T;V)for inhomogeneous Navier-Stokes equations, namely
hut,viL2(V0
0),L2(V0)+1
Reh∇u,∇vi+h(u·∇)u,vi= 0 for all v∈L2(V0),
τu=uiin WΣi,
u(0,·) = u0in H,
where L2(V0)denotes the L2(0, T ;V0)and L2(V0
0) = L2(0, T;V0
0).
For the existence and uniqueness of solution for the inhomogeneous Navier-
Stokes equations and the corresponding proof, we refer to [8, Theorem 1.2].
Here we do not repeat the relevant context from [8] for the existence and
uniqueness of optimal solution for the Dirichlet boundary control problem.
3 The Equivalently Modified Dirichlet Boundary
Control Problem
Differing from the Dirichlet boundary control problem given above, we pro-
ceed equivalently a modified one such that the high smoothness of the design
parameter zcan be obtained. We can achieve this goal by restricting the time
dependent function z(t) on Σiof the Navier-Stokes flow (1) as follows
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 5
z(t) =
t
Z0
v(s)ds +z0,(6)
which implies z(t) to satisfy
(z0(t) = v(t) on [0, T],
z(0) = z0.(7)
Modification of the Dirichlet boundary control problem is to permit v(t) as
the design parameter instead of z(t). It seems to obtain a more complicated
control problem, which consists of one cost functional and two state equations,
namely
min J(u, z, v) = 1
2
T
Z0ku(t, ·)−Udk2
Hdt +σ1
2
T
Z0
z2(t)dt +σ
2
T
Z0
v2(t)dt (8)
subject to
ut−1
Re∆u+ (u·∇)u+∇p= 0 in Q,
∇·u= 0 in Q,
u(x, t) = z(t)g(x) on Σi,
pν −1
Re
∂u
∂ν = (0,0) on Σo,
u(x, t) = (0,0) on Σs,
u(x,0) = u0(x) in Ω,
z0(t) = v(t) on [0, T ],
z(0) = z0.
With an assumption σ1= 0 throughout the paper, the set of admissible
controls should be altered as
Uad ={v∈L2(0, T ) : va(t)≤v(t)≤vb(t) a.e. on (0, T)}.
We devote to derive the necessary optimality condition with help of the
formal Lagrangian Multiplier. The Lagrangian technique is known to deliver in
general the correct first order optimality conditions. For that a mathematically
rigorous proof is not presented.
Define the Lagrangian function
L(u, p, z, v, {ηi}3
i=1, π, ζ)
:= J(u, v)−ZZ
Q£ut−1
Re∆u+ (u·∇)u+∇p¤·η1dxdt
−ZZ
Q
π£−∇·u¤dxdt −ZZ
Σi£z(t)g(x)−u¤·η2ds(x)dt (9)
−ZZ
Σo£1
Re
∂u
∂ν −pν ¤·η3ds(x)dt −Z
Ω£u(0,x)−u0(x)¤·η1(0,x)dx
−
T
Z0£z0(t)−v(t)¤ζ(t)dt −¡z(0) −z0¢ζ(0)
6 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
with the Lagrangian multipliers {ηi}3
i=1,πand ζrespectively to the three
states u, p and z. According to the Lagrangian principle, we expect the fol-
lowing equations and variation inequality to be valid at the local minimal
denoted by (u∗, p∗, z∗, v∗,{ηi}3
i=1, π, ζ)
DuL(u∗, p∗, z∗, v∗,{ηi}3
i=1, π, ζ)·u= 0,∀u,
DpL(u∗, p∗, z∗, v∗,{ηi}3
i=1, π, ζ)p= 0,∀p,
DzL(u∗, p∗, z∗, v∗,{ηi}3
i=1, π, ζ)z= 0,∀z,
DvL(u∗, p∗, z∗, v∗,{ηi}3
i=1, π, ζ) (v∗−v)≥0,∀v.
(10)
A similar calculation can be found in [21]. Here we give the adjoint equation
which is derived by summarizing η¯¯Σo=η3and otherwise η=η1
−ηt−1
Re∆η + (∇u∗)η−(u∗·∇)η+∇π=u∗−Udin Q(11)
∇·η= 0 in Q(12)
πν −1
Re
∂η
∂ν = 0 on Σo(13)
η= 0 on Σ\Σo(14)
η(·, T) = 0 in Ω(15)
η2=1
Re
∂η
∂ν −πν on Σi(16)
ζ0(t) = g(x)·η2on [0, T] (17)
ζ(T) = 0 (18)
and the variation inequality
T
Z0£σv(t)−ζ(t)¤(v∗(t)−v(t))dt ≥0,∀v∈ Uad .
It is quite expensive to solve such an optimization system iteratively.
For each iteration one must deal with the parabolic nonlinear PDE (1), the
parabolic linearized PDE (11-15) and the ODE (17-18). Insisting on the opti-
mal solution for this modified control problem without so much computational
cost, we could however still employ an order reduction approach, e.g., Galerkin
POD method.
4 The Proper Orthogonal Decomposition
Briefly a general aspect of the POD subspace is introduced, which contains
understanding, finding and using the POD basis functions. The principle be-
hind the proper orthogonal decomposition is to capture the POD basis as an
orthogonal basis system in a certain finite space by minimizing a least-square
error formula. To find them numerically, one may utilize the singular value de-
composition theorem to facilitate the computation. At end of this section we
introduce the way to validate the modified Dirichlet boundary control prob-
lem merely in time scale.
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 7
4.1 The discrete POD
Given n∈Nand 0 = t1≤t2≤, ..., ≤tn≤T, for convenience suppose
equivalent time difference. Then the solution of the Navier-Stokes equation
{u(tj)}n
j=1, which are also called snapshots, can be either obtained numerically
or experimentally with reference to this time discretization {tj}n
j=1. Let us
denote the snapshots ensemble by Vn=span{u(t1), ..., u(tn)} ⊂ V, and 1 ≤
dim Vn≤n, i.e., at least one of the snapshots is nonzero.
Let {φk}n
k=1 denote the orthonormal basis for Vn. Then each member of
the ensemble can be expressed as follows
u(tj) =
n
X
k=1hu(tj), φkiVφkfor j= 1, ..., n .
It is expected that a few of the orthonormal basis {φk}n
k=1 can represent a
typical structure of the ensemble. One way to solve the problem is with M
orthonormal basis of {φk}n
k=1 to yield the maximal projection of the snap-
shots, i.e., {φk}n
k=1 spanned subspace. In mathematical language, the task is
to find M≤n, such that the orthonormal system {φk}M
k=1 of Vnminimizes
the following least-square error:
min
{φk}M
k=1
n
P
j=1
αjku(tj)−
M
P
k=1hu(tj), φkiφkk2
V,
subject to hφi, φjiV=δij,1≤i, j ≤M ,
(19)
where {αk}n
k=1 are positive weights chosen for the purpose of integration, e.g.,
α1=∆t
2, αj=∆t, j = 2, ..., n −1, αn=∆t
2.
Then we define a linear mapping Yn∈L(Rn,Vn) with Yn(ek) = uk=
u(tk), where {ek}n
k=1 denote the canonical basis in Rn. For all u∈Rn
Yn(u) =
n
X
j=1
αjhu, ejiRnuj(20)
where the inner scalar product in Rnis defined as
hv, wiRn=
n
X
k=1
αkvkwk.
Assume that Y∗
n:Vn→Rnis the adjoint of Yn, then it follows for all φ∈ Vn
Y∗
nφ=£hu1, φiV··· hun, φiV¤T.(21)
Define Rn:= YnY∗
n∈L(Vn) and Kn:= Y∗
nYn∈L(Rn). Summarizing (20)
for Ynand (21) for Y∗
nyields
Rn=
n
X
k=1
αkhuk,·iVuk(22)
Kn=£hu1,Yn(·)iV··· hun,Yn(·)iV¤T(23)
where
8 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
huk,Yn(·)iV=h
huk,u1i
.
.
.
huk,uni
,·iRn.
Using a Lagrangian framework, the first order optimality condition for the
least-square problem (19) is
Rnφi=λiφiλ1≥λ2≥ ··· ≥ λn.(24)
For a fuller treatment we refer the reader to [20]. By solving (24) we can
capture the orthonormal basis {φk}n
k=1 that satisfies the first order optimality
condition and is thus the local minimum for the least-square problem (19).
However, by following [20] it can be proved that there is no orthogonal system
with Mbasis in the snapshots ensemble which solves the least square problem
better than the POD basis does and {φk}M
k=1 is thus global optimum with
respect to a fixed M. By choosing M, the least-square error formula (19) can
be evaluated as in the following theorem.
Theorem 2. Let λ1≥λ2≥ ··· ≥ λn≥0denote the non-negative eigenval-
ues of Rnwith the associated orthonormal eigenvectors {φk}n
k=1 in Vn. Let
M¿n, then {φk}M
k=1 is orthonormal with rank M, and the least-square error
formula (19) satisfies:
n
X
j=1
αjku(tj)−
M
X
k=1hu(tj), φkiφkk2
V=
n
X
j=M+1
λj.
The proof is straightforward by utilizing the definitions of Rnand its property
(24). In the following we give an example of the snapshots ensemble to examine
the above mentioned properties of the operators Rnand Kn.
Example 1. Let n∈Nand n<N, where Nis the number of spatial grids.
The snapshots ensemble is
Vn=span{u(t1), ..., u(tn)}.
The operators Rnand Knare obtained by the definitions (22) and (23)
Rnφ=
n
X
k=1
αkhuk, φiVuk=
n
X
k=1
αkukuT
k
| {z }
:=R
φ
Knu=
hu1,u1i ··· hu1,uni
.
.
.....
.
.
hun,u1i ··· hun,uni
| {z }
:=K
u
One captures the POD basis system {φk}n
k=1 to minimize the least-square
error formula (19) by solving (24), which is for this example equivalent to
solve eigenvalue problem of the new defined R, namely
Rφi=λiφiλ1≥λ2≥ ··· ≥ λn>0, λn+1 =···=λN= 0 .
Note that this by nsnapshots generated N×Nmatrix Rhas nnonzero
eigenvalues and Northonormal eigenvectors {φi}N
i=1. For M¿n<N, the
MPOD basis system {φk}M
k=1 can minimize the least-square error (19), which
can be also evaluated according to Theorem 2.
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 9
Actually we would rather possibly solve eigenvalue problem for the matrix K
with dimension nthan for Rwith N, since Rpossesses only nnon-zero eigen-
values in total and the number of time distances nis usually much smaller
than the number of spatial grids N. By the theorem of singular value decom-
position, the eigenvector of Rcan be also captured implicitly, i.e., with aid of
the eigenvector of Kand a linear mapping from L(Rn,Vn). For that we find
firstly an orthonormal basis {uk}n
k=1 in Rnsuch that for k= 1, ..., n
Kn(uk) = λkuk.
The solution for the least-square problem (19) should be an orthonormal basis
{φk}M
k=1 in Vn. It is not difficult to utilize the linear mapping Ynsuch that
Yn(uk) = pλkφki.e., φk=1
√λkYn(uk) (25)
for a fixed Mand k= 1, ...., M.
For an exactly mathematical discussion of the continuous POD, which is
essential for error estimates of Galerkin POD approximation, we refer to [10]
and [18].
4.2 Construction of the POD basis functions
To capture the POD basis with the given snapshots, we discuss the practi-
cal algorithm based on the finite-dimensional POD. It is comparable to the
snapshots method, which has been used by Ravindran in [13]. Note that from
now on the POD basis will be written as {Φi}n
i=1 and {Φi}M
i=1 in order to
emphasize POD basis function as vector field.
We begin by making an ansatz for the solution of the Navier-Stokes equa-
tion
u(x, t) = um(x) +
M
X
i=1
βi(t)Φi(x) + z(t)uz(x),(26)
where
um=1
n
n
X
j=1
u(x, tj),(27)
v(x, t) =
M
X
i=1
βi(t)Φi(x).(28)
It is noticed the steady flow uz(x) remains unspecified. In view of qual-
ifying the POD models for Dirichlet boundary fluid flow control, z(t)uz(x)
in the ansatz (26) is designed such that all spatial behaviors in the time in-
terval [0, T ] are extracted into the steady flow uz(x) and the scalar z(t) is
then assigned as time dependent control for accommodating z(t)uz(x) to the
Navier-Stokes solution with certain specified Dirichlet boundary condition.
Therefore, in such a way the time dependent boundary control z(t) is sepa-
rated from the unsteady solution of the Navier-Stokes equations.
According to the definition (3) of WΣiand Lemma 1, for every on Σi
well-defined Dirichlet boundary ui∈WΣithere exist at least one solution
˜
u∈W2(0, T;V) for the Navier-Stokes system (1), which satisfies the inflow
Dirichlet boundary condition uion Σi, i.e.,
˜
u(x, t)¯¯Σi=ui(x, t) = z(t)g(x).
10 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
Let us make the next important assumption
˜
u(x, t) = z(t)uz(x) in Q .
Obviously if one captures a steady flow ug∈Vwhich satisfies the pre-specified
inflow Dirilchlet boundary g(x), then equality holds in (29)
uz=ug,(29)
where g(x) is defined in L2(H1/2(Γi)2) according to the definition (5) of trace
operator τ. In the following we give another construction of uz, and assume
the time dependent function z(t) is given as a linear profile.
Example 2. Let
z(t) = zst+z0, t ∈[0, T] and zs, z0∈R.(30)
The inflow boundary condition of the Navier-Stokes equations is defined for
instance
z(t)g(x) = z(t)µsin( y−y0
y1−y0π)
0¶on Σi,
where y0and y1are the inflow bounds in y-axis. The other settings for the
Navier-Stokes equations remain unchanged as in the Dirichlet boundary con-
trol problem. For this system one can also construct the steady flow solution
uzby
uz(x) = uz1(x)−uz0(x)
z1−z0
,(31)
where z1=zsT+z0and z0is given as above. Note that uz1is the solution
of the steady Navier-Stokes equation regarding the inflow boundary condition
z1g(x) on Γiand uz0is the ones with z0g(x) on Γias the inflow boundary
condition.
It is also known that uz1has the same inflow boundary condition z1g(x)
on Γias the steady flow z1ug, where ugis the steady flow regarding the inflow
boundary condition g(x) on Γi. It is allowed to substitute the steady flow uz1
with the steady flow z1ugand uz0with z0ugin (31), since both compatible
pairs (uz1, z1ug) and (uz0, z0ug) have the same inflow boundary conditions
on Γirespectively. Then one yields
uz(x) = uz1(x)−uz0(x)
z1−z0
=z1ug(x)−z0ug(x)
z1−z0
=ug(x),
which shows no conflict to (29).
Another advantage of using the above construction (31) lies in the fact
that the graph of z(t) for t∈(0, T ) can be skipped without breaking (29)
and thus z(t) must not constrained to be a linear function for t∈(0, T ). We
will fix up a suitable numerical example to confirm the independence between
z(t) for t∈(0, T ) or for t∈[t0, t1]⊂(0, T ) and uzin Ω, although they
are expected by multiplying with each other to match the Dirichlet boundary
condition z(t)g(x) on Σi.
Definition 1. Let us define a modified ensemble set
V0=span{vi}n
i=1 ,
where viis vector field
v(x, t) = u(x, t)−um(x)−z(t)uz(x)defined at t=ti.
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 11
For the numerical algorithm one assumes that every POD basis Φhas the
form
Φi=
n
X
j=1
w(i)
jvj(32)
in terms of {vj}n
j=1, where w(i)is to be determined such that Φminimizes
the least square formula (19). By (20) and (25) in the finite-dimensional case,
the eigenfunctions {Φi}n
i=1 of Rnhave the form
Φi=1
√λiYn(u(i)) = 1
√λi
n
X
j=1
u(i)
jvj,(33)
where u(i)
jis the j-th component of the i-th eigenvector of Kn. To coincide
(32) with (33), we let
w(i)
k=1
√λi
u(i)
k,for k= 1, ..., n.
To determine the orthonormal basis {uk}n
k=1 for Kn, one can solve the
following eigenvalue problem,
1
nKn
|{z}
C
u(i)=λi
n
|{z}
Λi
u(i)
Cu(i)=Λiu(i)i= 1, ..., n, (34)
where Cis the correlation matrix according to the Kn’s definition (23) and
Example 1 with components
hCiij =1
nhvi,vjiV,for i, j = 1, ..., n.
It follows from the fact that Cis a positive semi-definite symmetric matrix,
which has a complete set of orthogonal eigenvectors {u(i)}n
i=1 along with a set
of non-negative eigenvalues {λi}n
i=1.
By Theorem 2 it is known that the least-square error can be evaluated as
the sum of eigenvalues, i.e., Pn
i=M+1 λi. It should be thus as small as possible.
From that one can specify an energy level 0 < e < 1 to be captured, and then
seek M¿nsuch that
M
P
i=1
λi
n
P
i=1
λi
> e and 0 ≤
n
P
i=M+1
λi
n
P
i=1
λi
<1−e
Obviously e= 1 implies the full order models. We choose eto be close to 1
in the numerical computation such that the expected energy level ecan be
reached with M. The Algorithm 1 for construction of POD basis is given on
page 12.
Utilizing the Galerkin projection of the Navier-Stokes equations onto the
modified ensemble V0(see Definition 1) yields a nonlinear ODE, which will
be summarized in the following.
Lemma 2. Let Vndenote the snapshots ensemble, all snapshots of which are
determined by solving the Navier-Stokes equations (1) in a time interval [0, T ]
12 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
with the pre-specified inflow boundary condition. V0is the modified ensem-
ble defined in Definition 1. The POD basis {Φi}M
i=1 spans the POD subspace
Vpod ⊂ V0. Then the POD reduced order models for the Navier-Stokes equa-
tions (1) is obtained for all t∈(0, T ]
˙
X(t) =
10
β0(t)
z0(t)
=
0
−1
Re AX(t)−K(X(t))
0
+v(t)
0
˜
B
1
(35)
X(0) = (1 β(0) z0)T
where for i= 1, ..., M and j= 0, ..., M + 1
F(X) =
0
−1
Re AX(t)−K(X(t))
0
,
hAii,j =h∇Φi,∇ΦjiL2(Ω)2,
Ki(X(t)) = X(t)TPiX(t),hPiik,l =h(Φk·∇)Φl,ΦiiL2(Ω)2,
B=
0
˜
B
1
,h˜
Bii=−hΦM+1,ΦiiL2(Ω)2,
hβ0ii=hu0−um−z0uz,ΦiiL2(Ω)2
and z0(t) = v(t)with z(0) = z0, where z(t)is the time dependent term of the
inflow boundary condition for the Navier-Stokes equations (1).
The explicit proof we rely on [21]. In fact, the POD reduced order models
consist of the POD weak solution of the Navier-Stokes equations, namely
Algorithm 1: Construction of POD basis
begin
•Define for i= 1, ..., n
vi(x) = v(x, ti) = u(x, ti)−um(x)−z(ti)uz(x),
where um(x) is defined as (27) and uz(x) constructed as (31).
•Compute the symmetric correlative matrix C. Its entries are given
hCii,j =1
nZ
Ω
vi(x)vj(x)dx, i, j = 1, ..., n.
•Solve the eigenvalue problem C U =Λ U.
•Prescribe an energy level ein percentage and find minimal
M¿n, such that
M
P
i=1
λi
n
P
i=1
λi
> e ,
where λi=nΛiand λ1≥... ≥λM≥λM+1 ≥... ≥λn≥0.
•Obtain the POD basis with the expression
Φi=1
√λi
n
X
j=1
u(i)
jvj, i = 1, ..., M.
end
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 13
β0(t) = −1
ReAX(t)−K(X(t)) + z0(t)˜
B ,
which is derivable by restricting the Navier-Stokes equation’s POD ansatz (26)
to the POD subspace Vpod with L2(Ω2) scalar product. It is not complete as
a solvable differential equation, since there are M+ 1 unknowns but only M
equations. Therefore the POD weak solution is extended as in (35) by utilizing
the second state equation of the modified Dirichlet boundary control problem
z0(t) = v(t) with z(0) = z0.
For generating the snapshots we must pre-specify a Dirichlet boundary
condition on Σifor the Navier-Stokes equations, which is included by the
time dependent function z(t) on [0, T]. Once z(t) is specified, then z0(t), i.e.,
v(t) is also fixed. While solving the POD reduced order models (35), we have
thus known which function as v(t) on [0, T ] should be taken for the POD basis
set.
The numerical recast of fluid flow is completed by solving the nonlinear
ODE system (35) for X(t) and subsequently applying the POD ansatz (26)
with X(t) and POD basis.
4.3 The POD Boundary Flow Control
It is convenient now to search a suboptimal triple {v(t),u(x, t), z(t)}in the
admissible set, the snapshots ensemble and L2([0, T ]) respectively, since both
state equations of the modified Dirichlet boundary control problem can be
substituted with the POD reduced order models (35). To find equivalently a
optimal pair (v(t),X(t)), we also need to revise the cost functional J(u, z, v)
by projecting it onto the snapshots ensemble.
The new cost functional Jshould be with respect to a control parameter v
and its associated state Xby assuming σ1= 0 in (8). Except for substituting
u(·, t) in the cost functional (8) by the POD ansatz (26), Xdmust be captured
for a given expected flow pattern in the POD subspace. Let Udthe target flow,
then induces
Ud(x) = Φ0(x) +
M
X
i=1 hUd(x)−Φ0(x)−CdΦM+1(x),Φi(x)i
| {z }
:=βd,i
Φi(x)
+CdΦM+1(x),(36)
where {βd,i}M
i=1 and Cdare summarized in a vector as
Xd:= [1, βd,1, ... , βd,M , Cd]T.
It is not difficult to calculate {βd,i}M
i=1 numerically, e.g., with QR decompo-
sition. For a unsteady target flow, Xd(t) can be generated analogously with
(36) for every t∈[0, T ].
The new cost functional is now derived as
J(X(t), v(t)) =
T
Z0
1
2(X(t)−Xd)T·Ψ·(X(t)−Xd) + σ
2v2(t)dt , (37)
where
hΨii,j =Z
Ω
hΦi,Φjidxfor i, j = 0, ..., M + 1 .(38)
14 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
The modified Dirichlet boundary control problem given in section 3 with
the two state equations can be converted into the POD reduced order flow
control problem, namely:
min J(X(t), v(t))
subject to: (PM)
X0(t) = F(X) + v(t)B ,
X(0) = X0.
The set of admissible controls Uad remains
Uad ={v∈L2(0, T ) : va(t)≤v(t)≤vb(t) a.e. on (0, T)}.
We devote to investigate the existence of the optimal solution for (PM) as
well as the assumptions, for a detailed representative standard sample refer
to [17].
Definition 2. The Sobolev space is defined as
H1(0, T )M+2 := {X(t)∈L2(0, T )M+2 :∃X0(t)∈L2(0, T)M+2}.
Remark 1. H1(0, T) is compactly embedded in C(0, T ). This proof can be
found in [6, Satz 3.2.4].
Theorem 3. Let [0, T ]be a fixed interval and σ > 0, the admissible control
set Uad be non-empty. Let f0be convex and continuous in v. Assume that all
successful states on [0, T ]satisfy an apriori bound
|X(t;X0, v)| ≤ Cfor all v∈ Uad,almost everywhere on [0, T ].(39)
Then the POD optimal control problem (PM)has optimal solution v∗∈ Uad
with associated state X∗∈H1(0, T )M+2.
Proof. The non-empty set of admissible controls is bounded in L2(0, T ). For
every v∈ Uad and suitable X0, there exists a unique solution of the state
equation. Since Jis bounded from below, J(X(t), v(t)) ≥0 and Jhas an
infimum
0≤j:= inf J(X, v)≤ ∞.
There is a minimizing sequence (Xn, vn) of admissible pairs such that
J(Xn, vn)→jas n→ ∞.
Since the set of admissible controls Uad is bounded, and all states with respect
to the admissible controls are bounded, i.e., the apriori bound (39) (see apriori
estimates in [10, Theorem 4.2, Theorem 4.7, Theorem 5.1, Theorem 5.2]), there
exists a subsequence
(Xnk, vnk)*(X∗, v∗)∈H1(0, T )M+2 ×L2(0, T).(40)
Uad is convex and closed in L2(0, T). It follows that Uad is weakly sequence
closed and consequently v∗∈ Uad. It remains to prove that the state limit X∗
is the solution of the state equation with respect to v∗. Since
Xnk*X∗∈H1(0, T)M+2
and H1(0, T) is compactly embedded in C(0, T ), there exists a subsequence
{Xnkl}∞
l=1 of the subsequence {Xnk}∞
k=1 such that as l→ ∞
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 15
Xnkl→X∗in C(0, T )M+2 and X0
nkl*(X∗)0in L2(0, T)M+2. (41)
Since Fis continuous in Xand Xnkl→X∗in C(0, T)M+2, it follows as l→ ∞
F(Xnkl)→F(X∗) in C(0, T)M+2.
It is known that C(0, T )M+2 ,→L2(0, T)M+2, then
F(Xnkl)*F(X∗) in L2(0, T)M+2. (42)
Without loss of generality, we substitute the circumstantial index nklwith n
such that
(Xn, vn)*(X∗, v∗)∈H1(0, T )M+2 ×L2(0, T),as n→ ∞.(43)
Summarizing (40), (41) and (42) in the state equation of (PM) and combining
the notation of (43) yields
X0
n
|{z}
*(X∗)0in L2(0, T )M+2
=F(Xn)
| {z }
*F(X∗) in L2(0, T )M+2
+Bvn
|{z}
*Bv∗in L2(0, T )M+2
.
Note that B∈RM+2 is a constant vector with respect to the POD basis
system {Φi}M
i=1. Then as n→ ∞, we obtain that
(X∗)0=F(X∗) + Bv∗in L2(0, T)M+2.
That is to say, the state limit X∗in L2(0, T )M+2 is the solution of the state
equation with respect to the optimal control v∗with an initial condition X0.
It is also known that f0is convex and continuous in v, then Jis weakly
lower-semicontinuous (see [5, Theorem 1.1]) and
j= lim
k→∞ inf J(Xnk, vnk)≥J(X∗, v∗).
Since jis the infimum, i.e., j6> J(X∗, v∗), it follows that j=J(X∗, v∗).
Next we need the first order necessary condition that enables us to distinguish
the optimal control from the other controls in the set of admissible controls
Uad. We remark that the formal Lagrangian technique can be employed again
for the first order optimality condition, however the Pontryagin maximum
principle is well-known as a standard result, which was proved by L.S. Pon-
tryagin in 1956 and can be applied for the task.
We define the Hamiltonian by
H:= p0f0+
M+2
X
i=1
pifi.
The integrand of the (PM) cost functional (37) is accordingly notated due to
convenience as follows
f0,1:= 1
2(X(t)−Xd)T·Ψ·(X(t)−Xd), f0,2:= σ
2v2(t), f0:= f0,1+f0,2.
The right hand side of the (PM) state equation, i.e., F(X) + v(t)Bcan be
rewritten as the scalar fields {fi}M+2
i=1 respectively. In general p0=−1, and
we derive the adjoint equation,
˙p0=−∂H
∂X0
= 0,
16 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
˙pi=−∂H
∂Xi
=∂f0,1
∂Xi−
M+2
X
j=1
pj
∂fj
∂Xi
,for i= 1, ..., M + 2.
Note that the transversality condition becomes
(p1(T), ..., pM+2(T))T=0,
if the state X(t) at t=Tis completely unspecified. The adjoint equation can
be composed in vectorial notation style and due to numerical execution of the
backward system set −w(t) = p(t)
(AD)(−˙w=£f0,1¤X+£F¤T
Xw ,
w(T) = 0.
Let v∗be optimal, it is necessary that Hattains its maximum at v∗for
almost every t∈[0, T ].
sup
v∈Uad
H(X, v, p)⇔sup
v∈Uad ³−σ
2v2(t) + BTp·v(t)´
= sup
v∈Uad ³−σ
2v2(t)−BTw·v(t)´
= min
v∈Uad ³σ
2v2(t) + BTw·v(t)´,if σ > 0.
It is known that v∗solves the above minimizing problems, i.e., v∗∈L2(0, T)
solves the following optimization problem for σ > 0
min
v∈Uad
T
Z0³σ
2v2(t) + v(t)BTw´dt .
Then the variation inequality holds for σ > 0:
T
Z0
(σv∗+BTw)(v−v∗)dt ≥0∀v∈ Uad .
Summarizing the optimality system for the POD boundary control problem
(PM) yields
state equation ⇒(˙
X(t) = F(X) + v(t)B
X(t0) = X0
adjoint equation ⇒
−˙w=Ψ·¡X(t)−Xd¢+hFiT
X·w
w(T) = 0
variation inequality ⇒½T
R0
(σv∗+BTw)(v−v∗)dt ≥0∀v∈ Uad ,
Briefly we give here the algorithm, which provides the optimal Dirichlet
boundary for the Navier-Stokes equations equivalently by solving the above
optimal control system.
Algorithm 2
1. Solve the Navier-Stokes equation (1) at ndifferent time steps for the
snapshots {uj}n
j=1 with z(t) and the associated steady flow u0as initial
condition.
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 17
2. Use Algorithm 1 to calculate the MPOD Basis.
3. Define Φ0=um(x) and ΦM+1(x) = uz(x), which are given in (27) and
(31).
4. Project the Navier-Stokes equation onto POD subspace and construct
a) the correlative POD basis matrix Ψas (38),
b) ˜
B=−hΨii,M+1, for i= 1, ...., M,
c) the stiffness matrix A of POD projection system,
d) the nonlinearity matrix Pi, for i= 1, ..., M and
e) the initial condition X0.
5. Capture Xdor Xd(t) by (36) with QR decomposition.
6. Execute an optimization algorithm
a) initialize k:= 0 and v(0) =z0(t) for t∈[0, T]
b) solve the POD state equation forward for X(t) with respect to v(k)
c) solve the adjoint equation backward for w(t) with respect to X(t)
d) set the search direction, e.g., anti-gradient direction
dk =−(σv(k)+BT·w)
e) determine sn with Armijo step size control
f) v(k+1) =v(k)+sn ×dk
g) r=kv(k+1) −v(k)k
h) repeat (a)-(h) and set k=k+ 1, if ris not sufficient small. Otherwise
stop the optimization, v∗=v(k+1)
7. Solve z0(t) = v(k+1) and z(0) = z0for z∗(t) as the optimal controlled
boundary condition in (1),
8. Calculate the Navier-Stokes equations (1) with respect to the optimal
Dirichlet boundary condition z∗(t) to yield the optimal fluid flow.
5 Computational Result
To illustrate the result of the POD approach in solving the Dirichlet boundary
flow control problem, several numerical examples are carried out in the same
geometry. First of all, the geometry in which the numerical tests are to be
calculated is a rectangle Ω⊆R2containing as in Figure 1 the inflow boundary
Γi, outflow boundary Γoand wall Γ\(Γi∪Γo).
The Navier-Stokes system (1) is solved in the time interval [1,10] for
0
y
x
Γi
Γo
Fig. 1: Domain Ωwith inflow boundary Γiand outflow boundary Γout
generating snapshots ensemble. The sinusoidal function as one choice of the
spatially dependent velocity field on inflow boundary is given
18 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
g(x) = µsin( y−y0
y1−y0π)
0¶and x=µx
y¶.
The linear profile z(t) = z0tfor t∈[1,10] is taken to complete the boundary
setting. Two hundreds snapshots are recorded at Re = 10000 with equivalent
time distance ∆t =T−1
200−1.
The entries of correlative matrix Ccan be computed by trapezoidal inte-
gration schema. Its eigenvalue range in logarithmic scale is shown in Figure 2,
which is solved by using matlab routine eig. As shown the eigenvalues drop
quickly and thus very few models are able to carry the essential dynamic en-
ergy of the snapshots.
0 20 40 60 80 100 120 140 160 180 200
−20
−18
−16
−14
−12
−10
−8
−6
−4
−2
Eigenvalue in Logscale
Eigenvalue Index
Fig. 2: Eigenvalues of the correlative matrix C
Table 1 displays percentage of the full order models energy captured by
the POD reduced order models, e.g., 99.9999 percent of the full order models
energy captured by only six basis functions. These six POD basis are chosen
to represent the modified ensemble for carrying out the further numerical flow
simulation and control experiments.
M 123456
Energy in % 97.9758 99.7095 99.9452 99.9956 99.9998 99.9999
l1Error 0.005370 0.005305 0.005305 0.005305 0.005300 0.005300
Table 1: Percentage of full order model energy and l1-norm difference between full
order and POD reduced order models captured with M= 1, ..., 6
To complete the flow simulation with the POD basis, one should recast
solution of the Navier-Stokes equations with the POD ansatz (26), where
the coefficient function {βi(t)}M
i=1 are gained by decoupling (35) with Crank-
Nicolson and 4th order Runge-Kutta schema. We compare the profiles of the
flow computed with full order models and the POD reduced order models in
Figure 3, which is recorded at t= 9 for various cross-sections of the spatial
domain. Both simulations show quantitatively satisfactory agreement.
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 19
0 0.5 1 1.5
−0.01
0
0.01
0.02
0.03
0.04
0.05
y − axis
u−v velocity
x = 0
0 0.5 1 1.5
−0.01
0
0.01
0.02
0.03
0.04
y − axis
x = 0.6
0 0.5 1 1.5
−0.01
0
0.01
0.02
0.03
y − axis
x = 1.5
0 0.5 1 1.5
−0.01
−0.005
0
0.005
0.01
0.015
0.02
y − axis
u−v velocity
x = 2.1
0 0.5 1 1.5
−5
0
5
10
15
20 x 10−3
y − axis
x = 2.4
0 0.5 1 1.5
−5
0
5
10
15 x 10−3
y − axis
x = 2.7
Fig. 3: Profile comparison of full order model(solid line) and POD reduced order
models solution(dash-dot line); velocities u(red) and v(blue) at t= 9.
Before we start Galerkin POD control stage, the streamlines of the eight
POD basis are shown in Figure 4 and Figure 5. Φ0displays the averaged
flow, and the other POD basis are barely identified with any ordinary steady
flow except for the last two POD basis Φ6and Φ7. However, we remark
they are quantitatively different, though they show very similar in streamline.
Therefore, we compare the Φ6to Φ1−5and to Φ7respectively in horizonal
velocity through various cross section in Figure 6 and Figure 7.
0 1 2 3
0
0.5
1
1.5
φ0
0123
0
0.5
1
1.5
φ1
0 1 2 3
0
0.5
1
1.5
φ2
0123
0
0.5
1
1.5
φ3
Fig. 4: The POD basis Φ0(x, y),Φ1(x, y),Φ2(x, y),Φ3(x, y)
20 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
0 1 2 3
0
0.5
1
1.5
φ4
0123
0
0.5
1
1.5
φ5
0 1 2 3
0
0.5
1
1.5
φ6
0123
0
0.5
1
1.5
φ7
Fig. 5: The POD basis Φ4(x, y),Φ5(x, y),Φ6(x, y),Φ7(x, y)
−0.06 −0.04 −0.02 0 0.02 0.04
0
0.5
1
1.5
x = 0
y
−3 −2 −1 0 1 2 3
x 10−3
0
0.5
1
1.5
x = 1.05
y
−1 0 1 2 3
x 10−3
0
0.5
1
1.5
x = 2.175
y
−0.01 −0.005 0 0.005 0.01
0
0.5
1
1.5
x = 3
y
φ1
φ2
φ3
φ4
φ5
φ6
Fig. 6: The profile comparison between Φ6and Φ1−Φ5
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 21
−0.06 −0.04 −0.02 0 0.02
0
0.5
1
1.5
x = 0
y
0 0.05 0.1 0.15 0.2
0
0.5
1
1.5
x = 1.05
y
0 0.05 0.1 0.15 0.2
0
0.5
1
1.5
x = 2.175
y
0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
1.5
x = 3
y
φ6
φ7
Fig. 7: The profile comparison between Φ6and Φ7
5.1 Control Test I
It remains to allocate the expected flow samples to appraise the control qual-
ity with the Galerkin POD method. As the first test we set a steady flow
Udshown in Figure 8, which can be capture by solving steady Navier-Stokes
equations with the inflow boundary u(x)¯¯¯Γi
=zdg(x)=0.025 g(x) and the
other boundary conditions prescribed by (1).
A start guess control v(0) is next required for the optimization. We start
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
Streamline Ud(x,y)
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
Velocity field Ud(x,y)
Fig. 8: The steady target flow with zd= 0.025.
to control with v(0)(t) = z0(t)≡z0for t∈[1,10], which is well-related to the
current POD basis with respect to the snapshots ensemble generated by the
inflow boundary condition z(t) = z0tin [1,10].
In Figure 9, the optimal control v∗(t) in POD system and the correspond-
ing z∗(t) are displayed at different σvalues. The parameter σis found to
regularize the effort in the controller design. Figure 9 presents the relatively
smaller value of σcould achieve the expected flow more quickly due to the
light weighted control-effort term v2(t) in the cost functional J.
22 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
1 2 3 4 5 6 7 8 9 10
0
0.005
0.01
0.015
0.02
0.025
0.03
t
control z(t)
Optimal Boundary Control
1 2 3 4 5 6 7 8 9 10
−0.02
0
0.02
0.04
0.06
0.08
0.1
t
pod control v(t)
Optimal Control in the POD−based System
Fig. 9: Optimal controls of the Navier-Stokes equations and POD-based system for
different values of σ: 10(dotted line), 1/10(solid line), 1/20(dashed line),
1/100(dash-dot line) and expected boundary profile (grey dashed line) re-
spectively.
At different σvalues the integrands f0(X∗(t), v∗(t)) in the cost functional
are evaluated with respect to the by CG method solved optimal controls v∗
and the associated states X∗. Figure 10 presents straightforwardly the optimal
steady flow is reached rapidly as σis set to be 0.05 or smaller.
2 4 6 8 10
5
6
7
8
9
10 x 10−5
t
f0(X*(t),v*(t))
σ = 10
2 4 6 8 10
4
6
8
10
12 x 10−5
t
f0(X*(t),v*(t))
σ = 0.1
2 4 6 8 10
4
6
8
10
12 x 10−5
t
f0(X*(t),v*(t))
σ = 0.05
2 4 6 8 10
4
6
8
10
12 x 10−5
t
f0(X*(t),v*(t))
σ = 0.01
Fig. 10: The evaluated integrand f0(X∗(t), v∗(t)) for t∈[1,10] at different σ
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 23
For a clear sight of the POD captured boundary control effect, Figures
11-12 demonstrate streamlines of the snapshots as uncontrolled flow and the
POD controlled flow, from which we confirm that the POD controlled flow had
no vortex in the left upper corner during all time steps. We refer reader to note
that streamline of the expected steady flow in Figure 8 exhibits also no vortex
there, which is set in the objective function as the pattern flow and should
be imitated by controlling with the optimal Dirichlet boundary condition z∗.
Finally, Figure 12 at t= 10 shows a good effect of the POD boundary control,
meanwhile an extra vortex has grown up there in the uncontrolled flow.
0 1 2 3
0
0.5
1
1.5
at t = 1
Uncontrolled Flow
0 1 2 3
0
0.5
1
1.5
at t = 4
0123
0
0.5
1
1.5
at t = 1
POD Controlled Flow
0123
0
0.5
1
1.5
at t = 4
Fig. 11: POD Controlled and uncontrolled flow streamlines comparison at t= 1
and t= 4.
0 1 2 3
0
0.5
1
1.5
at t = 7
Uncontrolled Flow
0 1 2 3
0
0.5
1
1.5
at t = 10
0123
0
0.5
1
1.5
at t = 7
POD Controlled Flow
0123
0
0.5
1
1.5
at t = 10
Fig. 12: POD Controlled and uncontrolled flow streamlines comparison at t= 7
and t= 10.
24 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
5.2 Control Test II
In the unsteady flow matching case we could devise an analytical design pa-
rameter zd(t) in [0,10] and capture the unsteady target flow pattern Ud(x, t)
by solving (1) with zd(t) on the inflow Σi. By utilizing the eight POD basis
system, the second control test can be performed, in which the unsteady tar-
get flow Ud(x, t) described above is set in the cost functional.
Firstly we give the inflow boundary condition on Σi, for t∈[0,10]
Ud(x, t)¯¯¯Σi
:= zd(t)g(x) = h1
10³1−t´2³1−cos ¡5
6+5t¢´+86
5000ig(x)
where g(x) remain unchanged. Analogously as (36) for every t∈[0,10], Xd(t)
can be figured out with zd(t).
The optimization with the Galerkin POD induces the optimal solutions for
the second control example as in Figure 13, which shows excellent agreement
between the POD optimal solution at σ= 0.001 and analytical target. Note
that these optimal solutions are solved with the known initial condition of
target flow, i.e., steady flow with inflow zd(0)g(x). However it is not always
self-evident to possess the same initial condition as the target flow Ud(0,x).
Next we explore control flexibility of Galerkin POD approach without being
revealed the initial condition of target flow. From Figure 14 we can confirm
the optimal control with a different initial condition succeeds. All POD con-
trolled z∗(t) regarding σstart with z∗(0) = z0at t= 0, which is originally
for generating snapshots. Obviously the optimal trajectories z∗(t) for almost
everywhere in [0,10] accommodates the target zd(t).
In both numerical experiments, eight POD basis functions calculated
0 1 2 3 4 5 6 7 8 9 10
0.01
0.02
0.03
0.04
0.05
0.06
t
control z(t)
Optimal Boundary Control
0 1 2 3 4 5 6 7 8 9 10
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
t
pod control v(t)
Optimal Control in the POD−based System
Fig. 13: POD optimal solutions for different values of σ: 1(dotted line), 1/10(solid
line), 1/100(dashed line), 1/1000(dash-dot line) compared with analytical
function zd(t)(grey thick solid line) and its derivative vd(t) = z0
d(t) (grey
thick solid line).
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 25
0 1 2 3 4 5 6 7 8 9 10
0
0.01
0.02
0.03
0.04
0.05
t
control z(t)
Optimal Boundary Control
0 1 2 3 4 5 6 7 8 9 10
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
t
pod control v(t)
Optimal Control in the POD−based System
Fig. 14: POD optimal solutions for different values of σ: 1(dotted line), 1/10(solid
line), 1/100(dashed line), 1/1000(dash-dot line) compared with analytical
function zd(t)(grey thick solid line) and its derivative vd(t) = z0
d(t) (grey
thick solid line).
with respect to the snapshot ensemble have been used for the optimiza-
tion (see. Algorithm II). All control candidates v(t)∈L2(0, T) are not cer-
tainly well related to these eight POD basis functions during the optimiza-
tion. However, these POD basis functions enable us to capture the optimal
control v∗∈L2(0, T ). If the POD basis functions and control candidates
v(t)∈L2(0, T) are not well related to each other, i.e., the POD basis func-
tions can not simulate the behavior of the current v(t) related flow, then the
so-called adaptive POD control should be applied, on which the standard
work [14] is for boundary control problems and [1] for the distributed control
problems.
5.3 Control Test III
For the third control example, we attempt to replace the original snapshots
ensemble, which is generated by linear function z(t) on the inflow boundary.
Setting another inflow profile, for instance zd(t) used as target inflow condition
in the second control test, namely
z(t) = 1
10(1 −t)2³1−cos ¡5
6+5t¢´+86
5000 ,
yields the new snapshots ensemble.
As usual we make firstly the POD ansatz, in which uzcan be constructed
by following (31). However, in this case one may capture
uz(x) = uz1−uz0
z1−z0
=uz1−uz0
z(10) −z(0)
| {z }
≈0
.(44)
26 Y. Wang, F. Tr¨oltzsch and G. B¨arwolff
This singularity can be removed by dividing z(t) with respect to tinto different
parts according to its monotone. For t∈[0,1], z(t) is monotonously decreasing
and for t∈[1,10] monotonously increasing. Each monotonous part of z(t) is
suitable for generating uz, and we choose the latter
uz(x) = uz(10) −uz(1)
z(10) −z(1) .
A new target flow Ud(x, t) should subject to the following analytical inflow
boundary condition
Ud(x, t)¯¯¯Σi
=h³1
10¡1−cos( 5
5t+ 6)¢(1 −t)2+1
100´cos(t+ 5
10 ) + 3
250ig(x).
The technique given above does not ruin the result, see Figure 15. As men-
tioned the last POD basis can be built successfully for the purpose of optimal
control, if one gains the snapshots ensemble established by such a nonlinear
inflow time profile.
0 1 2 3 4 5 6 7 8 9 10
0.01
0.02
0.03
0.04
0.05
t
control z(t)
Optimal Boundary Control
0 1 2 3 4 5 6 7 8 9 10
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
t
pod control v(t)
Optimal Control in the POD−based System
Fig. 15: POD optimal solutions for different values of σ: 1(dotted line), 1/10(solid
line), 1/100(dashed line), 1/1000(dash-dot line) compared with analytical
configuration (grey thick solid line).
References
1. K. Afanasiev, M. Hinze. Adaptive control of a wake flow using proper orthog-
onal decomposition. Lecture Notes in Pure and Applied Mathematics: Shape
Optimization and Optimal Design, Marcel Dekker. 216, 317-332, 2001.
2. M. Berggren. Numerical solution of a flow-control problem: Vorticity reduction
by dynamic boundary action. SIAM J. Sci. Comput., Vol 19, No.3, pp. 829-860,
1998.
The POD Dirichlet Boundary Control of the Navier-Stokes Equations 27
3. E. Casas, M. Mateos, F. Tr¨oltzsch. Error Estimates for the Numerical Approx-
imation of Boundary Semilinear Elliptic Control Problems. Computational Op-
timization and Applications, 31: 193-219, 2005.
4. E. Casas and J. P. Raymond. Error Estimates for the Numerical Approxima-
tion of Dirichlet Boundary Control for Semilinear Elliptic Equations. SIAM J.
Control Optim. Vol. 45, No. 5, pp. 1586-1611, 2006.
5. B. Dacorogna. Weak Continuity and Weak Lower Semicontinuity of Non-Linear
Functionals. Lecture Notes in Mathematics 922. Springer-Verlag Berlin Heidel-
berg NewYork, 1982.
6. E. Emmrich. Gew¨ohnliche und Operator-Differentialgleichungen. Vieweg, 2004.
7. M. D. Gunzburger and S. Manservisi. The velocity tracking problem for Navier-
Stokes flows with boundary control. SIAM. J. Control Optim., 39: No. 2, pp.
594-634, 2000.
8. M. Hinze and K. Kunisch. Second order methods for boundary control of the
instationary Navier-Stokes system. ZAMM, 84: 171-187, 2004.
9. L. S. Hou and S. Ravindran. A penalized Neumann control approach for solving
an optimal Dirichlet control problem for the Navier-Stokes equations. SIAM J.
Control Optimization, 36: 1795-1814, 1998.
10. K. Kunisch and S. Volkwein. Galerkin proper orthogonal decomposition methods
for a general equation in fluid dynamics. SIAM J. Numer. Anal., 40: 492-515,
2002.
11. Lumley JL. The structure of inhomogeneous turbulence. Atm. Turb. and Radio
Wave Prop. (Yaglom and Tatarsky eds.) Nauka, Moscow, pp 166-178, 1967
12. E. R. Pinch. Optimal control and the Calculus of Variations. Oxford University
Press, 1993.
13. S. Ravindran. A reduced order approach to optimal control of fluids using proper
orthogonal decomposition. Int. J. Numer. Methods Fluids, 34(5): 425-448, 2000.
14. S. S. Ravindran. Reduced-Order Adaptive Controllers for Fluid Flows Using
POD. J. Sci. Comput., 15(4): 457-478, 2000.
15. R.L. Sani and P.M. Gresho. Resume and remarks on the open boundary condi-
tion mini-symposium. International Joundal of Numerical Methods in Fluids.
18: 983-1008, 1994.
16. F. Tr¨oltzsch and D. Wachsmuth. Second-order sufficient optimality conditions
for the optimal control of instationary Navier-Stokes equations. PAMM ·Proc.
Appl. Math. Mech. 4, 628-629, 2004
17. F. Tr¨oltzsch. Optimale Steuerung partieller Differentialgleichungen. Vieweg,
Berlin, 2005.
18. F. Tr¨oltzsch and S. Volkwein. POD a-posterior error estimates for linear-
quadratic optimal control problems. Accepted by Computational Optimization
and Applications, DOI 10.1007/s110589-008-9224-3.
19. S. Volkwein. Proper orthogonal decomposition and singular value decomposi-
tion. SFB-Preprint No. 153, 1999.
20. S. Volkwein. Optimal Control of a Phase-Field Model Using Proper Orthogonal
Decomposition. ZAMM. Z. Angew. Math. Mech., 81(2): 83-97, 2001.
21. Y. Wang. Reduced order optimal control for Fluid Flow by Using Galerkin POD.
Master Thesis, 2008.
