Optimal Boundary Control of the Stokes Fluids with Point Velocity Observations.
This paper studies constrained linear-quadratic regulator (LQR) problems in distributed boundary control systems governed by the Stokes equation with point velocity observations. Although the objective function is not well defined, we are able to use hydrostatic potential theory and a variational inequality in a Banach space setting to derive a first-order optimality condition and then a characterization formula of the optimal control. Since matrix-valued singularities appear in the optimal control, a singularity decomposition formula is also established, with which the nature of the singularities is clearly exhibited. It is found that in general, the optimal control is not defined at observation points. A necessary and sufficient condition that the optimal control is defined at observation points is then proved.
Introduction
.
In this paper, we are concerned with the problems in boundary control of fluid
flows. We consider the following constrained optimal boundary control problems in
the systems governed by the Stokes equation with point velocity observations.
Let\Omega ae R 3 be a bounded domain with smooth boundary \Gamma, \Gamma 1 an open subset
of
min
Z
subject to
div ~
where
~
w(x) is the velocity vector of the fluid at x 2 \Omega\Gamma
p(x) is the pressure of the fluid at x 2 \Omega\Gamma
w)(x) is the surface stress of the fluid along \Gamma defined by
Received by the editors XX XX, 19XX; accepted by the editors XXXX XX, 19XX.
y Department of Mathematics, Texas A&M University, College Station, 77843. Supported in
part by NSF Grant DMS-9404380 and by an IRI Award of Texas A&M University.
z Department of Aerospace Engineering, Texas A&M University, College Station, Current
address: Department of Mathematical Sciences, University of Nevada-Las Vegas, Las Vegas, NV
89154-4020.
P.You,Z.Ding and J.Zhou
~n(x) is the unit outnormal vector of \Gamma at x;
~g is a given (surface stress) Neumann boundary data (B.D.) on
U is the (surface stress) Neumann boundary control on the surface
U is the admissible control set to be defined later for well-posedness of the
problem and for applications;
are given weighting factors;
are prescribed "observation points";
are prescribed "target values" at
-, a positive quantity, is the kinematic viscosity of the fluid. For simplicity,
throughout this paper we assume that and the density of the fluid is the
constant one.
Let
which is the subspace of the rigid body motions in R 3 . Multiplying the Stokes equation
by ~a integration by parts yield the compatibility condition of the
Stokes system, i.e.,
Z
or
For q - 1, let A be a subspace of (L q (\Gamma)) 3 and denote
The Stokes equation (1.1) describes the steady state of an incompressible viscous
fluid with low velocity in R 3 . It is a frequently used model in fluid mechanics. It
is also an interesting model in linear elastostatics due to its similarities. During the
past years, considerable attention has been given to the problem of active control of
fluid flows (see [1, 2, 7, 18, 19] and references therein). This interest is motivated by
a number of potential applications such as control of separation, combustion, fluid-structure
interaction, and super maneuverable aircraft. In the study of those control
problems and Navier-Stokes equations, the Stokes equations, which describe the slow
steady flow of a viscous fluid, play an important role because of the needs in stability
analysis, iterative computation of numerical solutions, boundary control and etc. The
theoretical and numerical discussion of the Stokes equations on smooth or Lipschitz
domains can be found from [14, 16, 17, 22, 25, 26, 27].
Our objective of this paper is to find the optimal surface stress ~u(x) on \Gamma 1 , which
yields a desired velocity distribution ~
w(x), s.t. at observation points
the observation values ~
are as close as possible to the target values Z k with
a least possible control cost
Z
which arise from the contemporary fluid
control problems in the fluid mechanics.
Notice that point observations are assumed in the problem setting, because they
are much easier to be realized in applications than distributed observations. They
can be used in modeling contemporary "smart sensors".
OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 3
Sensors can be used in boundary control systems (BCS) governed by partial differential
equations (PDE) to provide information on the state as a feedback to the
systems. According to the space-measure of the data that sensors can detect, sensors
can be divided into two types, point sensors and distributed sensors. Point sensors are
much more realistic and easier to design than distributed sensors. In contemporary
"smart materials", piezoelectric or fiber-optic sensors (called smart sensors) can be
embedded to measure deformation, temperature, strain, pressure,.,etc. Each smart
sensor detects only the average of the data in between the sensor and its size can
be less than in any sense, they should be treated as point
sensors. As a matter of fact, so far distributed sensors have not been used in any
real applications, to the best of our knowledge. However, once point observations on
the boundary are used in a BCS, singularities will appear and very often the system
becomes ill-posed. Mathematically and numerically, it becomes very tough to handle.
On the other hand, when point observations are used in the problem setting, the
state variable has to be continuous, so the regularity of the state variable stronger
than the one in the case of distributed observations is required. The fact is that in
the literature of related optimal control theory, starting from the classic book [23] by
J.L. Lions until recent papers [3],[4] by E. Casas and others, distributed observations
are always assumed and the optimal controls are characterized by an adjoint system.
The system is then solved numerically by typically a finite-element method, which
cannot efficiently tackle the singularity in the optimal control along the boundary.
On the other hand, since it is important in the optimal control theory to obtain a
state-feedback characterization of the optimal control, with the bound constraints in
the system, the Lagrange-Kuhn-Tucher approach is not desirable because theoretically
it cannot provide us with a state-feedback characterization of the optimal control
which is important in our regularity/singularity analysis of the optimal control and
numerically it leads to a numerical algorithm to solve an optimization problem with a
huge number of inequality constraints. A refinement of the boundary will double the
number of the inequality constraints, so the numerical algorithm will be sensitive to
the partition number of the boundary. Since the BCS is governed by a PDE system in
R 3 , the partition number of the boundary can be very large, any numerical algorithm
sensitive to the partition number of the boundary may fail to carry out numerical
computation or provide reliable numerical solutions.
Recently in the study of a linear quadratic BCS governed by the Laplace equation
with point observations, the potential theory and boundary integral equations (BIE)
have been applied in [20],[10],[11], [12] to derive a characterization of the optimal
control in terms of the optimal state directly and therefore bypass the adjoint system.
This approach shows certain important advantages over others. It provides rather
explicit information on the control and the state, and it is amenable to direct numerical
computation through a boundary element method (BEM), which can efficiently tackle
the singularities in the optimal control along the boundary.
In [10],[11],[9] several regularity results are obtained. The optimal control is characterized
directly in terms of the optimal state. The exact nature of the singularities
in the optimal control is exhibited through a decomposition formula. Based on the
characterization formula, numerical algorithms are also developed to approximate the
optimal control. Their insensitivity to the discretization of the boundary and fast
uniform convergences are mathematically verified in [12],[31].
The case with the Stokes system is much more complicate than the one with the
Laplace equation due to the fact that the fundamental solution of the Stokes system
4 P.You,Z.Ding and J.Zhou
is matrix-valued and has rougher singular behaviors. In this paper, we assume that
the control is active on a part of the surface and the control variable is bounded by
two vector-valued functions. A Banach space setting has been used in our approach,
we first prove a necessary and sufficient condition for a variational inequality problem
which leads to a first order optimality condition of our original optimization
problem. A characterization of the optimal control and its singularity decomposition
formula are then established. Our approach can be easily adopted to handle
other cases and it shows the essence of the characterization of the optimal control,
through which gradient related numerical algorithms can be designed to approximate
the optimal control.
The organization of this paper is as follows: In the rest of Section 1, we introduce
some basic definitions and known regularity results that are required in the later
development; In Section 2, we first prove an existence theorem for an orthogonal
projection, next we derive a characterization result for a variational inequality which
serves as a first order optimality condition to our LQR problem; then a state-feedback
characterization of the optimal control is established. Section 3 will be devoted to
study regularity/singularity of the optimal control. Since the optimal control contains
a singular term, we first derive a singularity decomposition formula for the optimal
control, with which we find that in general the optimal control is not defined at observation
points. A necessary and sufficient condition that the optimal control is defined
at observation points is then established. Some other regularities of the optimal control
will also be studied in this section. Based upon our characterization formulas a
numerical algorithm, in a subsequent paper, we design a Conditioned Gradient Projection
Method (CGPM)) to approximated the optimal control. Numerical analysis
for its (uniform) convergence and (uniform) convergence rate are presented there. We
show that CGPM converges uniformly sub-exponentially, i.e., faster than any integer
power of 1
n . Therefore CGPM is insensitive to discretization of the boundary.
The insensitivity of our numerical algorithm to discretization of boundary is a significant
advantage over other numerical algorithms. Since the fundamental solution of
the Stokes system is matrix valued with a very rough singular behavior, numerical
analysis is also much more complicated than the case with scalar-valued fundamental
solution, e.g., the Laplacian equation.
Let us now briefly recall some hydrostatic potential theory, BEM and some known
regularity results. Throughout of this paper, for a sequence of elements in R n , we
use superscript to denote sequential index and subscript to denote components, e.g.,
We may also use ~x k to emphasize that x k is
a vector. We may write ~
w(x; ~u) to indicate that the velocity ~
w depends also on ~u.
Unless stated otherwise, we assume p ?
is the Euclidean
norm in R n and k \Delta k is the norm in (L h (\Gamma)) n (h - 1).
Let fE(x; -); ~e(x;
be the fundamental solution
of the Stokes systems, i.e.
ae
div x E(x;
is the unit Dirac delta function at I 3 is the 3 \Theta 3 identity
matrix. It is known [22] that
OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 5
where ffi i;j is the Kronecker symbol.
Remark 1. The significant difference between the case with point observations
and the case with distributed observations is as follows: for a given vector ~
the function
has a singularity of order O( 1
however it may oscillate between
so it is very tough to deal with. Whereas the function
Z
E(-; x) ~
is well-defined and continuous.
On the other hand, if E(P k ; x) in (1.4) and (1.5) is replaced by the fundamental
solution of the Laplace equation, in this case, E(P k ; x) becomes scalar-valued, then
(1.4) has the same order O( 1
of singularity at but the limit as
exists (including \Gamma1 or +1). So the singularity can be easily handled.
It is then known that the solution ( ~
w; p) of the Stokes equation (1.1) has a simple-
layer representation
~
Z
Z
for some constants ~a; ~ b 2 R 3 and a 2 R. ~j is called the layer density and ~a
represents a rigid body motion. By the jump property of the layer potentials, we
obtain the boundary integral equation
Z
Z
where
With a given Neumann B.D., the layer density ~j can be solved from the above BIE
(1.8). Once the layer density is known, the solution ( ~
w(x); p(x)) can be computed
from (1.6) and (1.7). The velocity solution ~
w(x) is unique only up to a rigid body
motion and the pressure solution p(x) is unique up to a constant.
6 P.You,Z.Ding and J.Zhou
In BEM, the boundary
divided into N elements with nodal points
Assume that the layer density ~j(x) is piecewise smooth, e.g. piecewise constant,
then the BIE (1.8) becomes a linear algebraic system. This
system can be solved for ~j(x i ) and then ( ~
w(x); p(x)) can be computed from a discretized
version of (1.6) and (1.7) for any x 2 \Omega\Gamma
For each ~
we define the simple layer potential of velocity
f) by
Z
E(x; -) ~
For each ~
we define the boundary operators K and K   by
K( ~
Z
Z
Z
Z
where
Next we collect some regularity results on S v ; K and K   into a lemma. Let
which represents the set of all layer densities corresponding to the zero Neumann
B.D., with which the Stokes system has only a rigid body motion. Hence we have
I
Lemma 1.1.
Let\Omega ae R 3 be a bounded simply connected domain with smooth
boundary \Gamma.
(a) (R 3 linear operator for p ? 2 and
(b) For any 1 - is a bounded linear operator
and K (K   ) is the adjoint of K   (K);
(c) For
is a bounded linear
(d) For
?M0 is invertible,
?N is invertible.
OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 7
is a bounded linear operator.
Therefore K
?N is invertible.
Proof. (a)-(d) can be found from [5],[8], [13], [14] and [22].
To prove (e), since \Gamma ae R 3 is a compact set, it suffices to prove (e) for
Then we have 1
q \Gamma1. There exists an " 2 (0; 1), s.t. 1
q \Gamma1.
are the conjugates of
It can be verified that
s
ff
r
s
;ff
s
s
r
Note
and
'Z
where M is a constant independent of x 2 \Gamma. Let
f)(x). Applying H-older's
inequality twice, we get
'Z
'Z
'Z
''Z
ff doe -
'Z
''Z
r
'Z
'Z
Thus
'Z
s
s
'Z
Z
s
s
This proves the first part of (e). The second part follows from (c).
To prove (f), by (1.10), Q ij (x; -) is weakly singular for 1 - 3. Thus K
is an integral operator with weakly singular kernel. By Theorem 2.22 in [21], K is
a compact operator from (C (\Gamma)) 3 to (C (\Gamma)) 3 . The rest follows from the Fredholm
alternative (see [21], p.44).
8 P.You,Z.Ding and J.Zhou
For a given Neumann B.D. ~g 2 (L p (\Gamma 0 our control bound constraints
to the entire boundary \Gamma by
ae
and
ae Bu(x) x
with
where ~
vector depending on ~g and will be specified later. Define
the feasible control set
stands for the compatibility condition of the Neumann B.D. in the
Stokes system (1.1). It is clear that U is a closed bounded convex set in (L p (\Gamma)) 3 .
According to Lemma 1.1 (a), for each given Neumann B.D. ~u 2 U , the Stokes
system (1.1) has a solution ~
w in
(C(\Omega to a vector ~a
~
where
~
That is, for each given ~u, the velocity state variable ~
w is multiple-valued, so the
objective function J(~u) is not well-defined. However among all these velocity solutions,
there is a unique solution ~
~ h2M0
A direct calculation yields that ~
must satisfy
Since such a ~
w is unique and continuous, the point observations ~
our LQR
problem setting make sense and the LQR problem is well-posed.
From (1.14) and Lemma 1.1, we know
where C is a constant depending only on \Gamma. Let us observe (1.16). If we notice that
~
is linear in ~u, then we have
Lemma 1.2. Let ~a 3 be the unique solution to
OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 9
Then for
~
and
where C is a constant depending only on \Gamma.
2. Characterization of the Optimal Control.
We establish an optimality condition of the LQR problem through a variational
inequality problem (VIP). The characterization of the optimal control is then derived
from the optimality condition.
In optimal control theory it is important to obtain a state-feedback characterization
of the optimal control, i.e., the optimal control is stated as an explicit function
of the optimal state. So the optimal control can be determined by a physical measurement
of the optimal state. Our efforts are devoted to derive such a result.
For each ~
we define the vector-valued truncation function
~
f
Bl
Let h\Delta; \Deltai be the pairing on ((L q our feasible control set U
defined in (1.11) is a convex closed bounded set in (L p (\Gamma)) 3 , it is known that ~u   is an
optimal control of the LQR problem if
For any ff ? 0, (2.1) is equivalent to
To derive an optimality condition, we need to find a characterization of a solution to
the above variational inequality.
Theorem 2.1. For each f 2 (L q (\Gamma)) 3 , u f is a solution to the variational inequal-
ity
if and only if
Bl
where z f 2 M 0 such that [f
Theorem 2.2 for the existence of
such a z f ).
Moreover, (2.3) is well-defined in the sense that if z 1 and z 2 are two vectors in
then
Bl a.e. x 2 \Gamma:
(2.
P.You,Z.Ding and J.Zhou
Proof. By Theorem 2.2, there exists z
Bl . We have for each u 2 U ,
Z
on
doe x
where the last inequality holds since each integrand, the product of two terms, is
nonnegative.
Next we assume that u f is a solution to the VIP, i.e.,
Take
Bl , which is in U , we obtain
By the first part, we have
Taking
Combining (2.5) with (2.7) gives us
Thus
The proof of the second part of the theorem follows directly from taking z
Bl in (2.8).
In a Hilbert space setting, the above theorem is called a characterization of pro-
jection. When U is a convex closed subset of a Hilbert space H , for each f 2 H , u f
is a solution to the VIP if and only if
i.e., u f is the projection of f on U . This characterization is used to derive a first
order optimality condition for convex inequality constrained optimal control prob-
lems. However, this result is not valid in general Banach spaces. Instead we prove a
characterization of truncation, which is a special case of a projection. Note that in a
Hilbert space setting, a projection maps a point in the space into a subset of the same
space. However our truncation is a projection that maps a point in (L q (\Gamma)) 3 into a
subset of (L p
1). It crosses spaces. This characterization gives a
connection between the truncation and the solution to VIP, in our case, an optimality
OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 11
condition in terms of the gradient. That is, by our characterization of truncation,
~u   2 U is a solution to the VIP (2.2) if and only if
where ~z   2 M 0 is defined in Theorem 2.2 s.t.
To prove the existence of a rigid body motion z f in (2.3), we establish the following
existence theorem for an orthogonal projection, which is given in a very general case
and plays a key role in establishing the optimality condition. It can be used to
solve LQR problems governed by PDE's, e.g., the Laplacian, the Stokes, the linear
elastostatics, .,etc. where the PDE has multiple solutions for a given a Neumann
type boundary data satisfying certain orthogonality condition.
Theorem 2.2. Let \Gamma be a bounded closed set in R n and \Gamma be a subset s.t.
be given s.t.
~
where ~
is given by (2.17) and
~
Assume that M 0 is an m-dimensional subspace in (L q (\Gamma)) n (q - 2; 1
and
g, then a necessary and sufficient condition that for each
~
Bl
is that
Moreover the set of all solutions ~z f in (2.10) is locally uniformly bounded in the sense
that for each given ~
exist
with k ~
Bl
we have
Proof. Case 1: dim
be an
orthonormal basis in M 1 (in M 0 as well). To prove the first part of the theorem, we
have to show that for each ~
~
Bl
12 P.You,Z.Ding and J.Zhou
For each ~
by
~
Bl
Then to prove the first part, it suffices to show that for each ~
exists
It is easy to check that for any ~
exist two
constants depending only on \Gamma and the basis y s.t.
is a bounded (depends on Bl and Bu) Lipschitz continuous map.
To show that T f has a zero, we prove that there exists a constant R ? 0 s.t. when
Once (2.15) is verified, we have
By Altman's fixed point theorem [15], the map C has a fixed point
(BR is the ball of radius R at the origin), i.e.,
it remains to verify (2.15). Define
It suffices to show that there exists R ? 0 s.t. for t ? R,
In the following, we prove that for each given ~
exist
have
So the second part of the theorem also follows. For each C 2 D, we denote
~y C
OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 13
It is obvious that
Z
j~y C (x)jdoe x
is continuous in C and positive on the compact set D, hence
C2D
f
Z
and we set
R
For any given " ? 0, we assume
For each C 2 D; t ? 0,
Bl
Z
Bl
Z
~
Bl
Z
I C
Z
where for
I C
Z
Let
We have
lim
I C
Z
Z
Z
jy C
Thus
lim
Z
jy C
Z
Z
jy C (x)jdoe x
Z
14 P.You,Z.Ding and J.Zhou
where m y given by (2.16) is independent of C. From (2.14), we see that T f (C) \Delta C is
continuous in both ~
f and C, therefore there exist R C
Since D is compact, there exist C
Let
So we only need to take
~
and
~
Bu; a.e. on
Case 2: be an orthonormal
basis in M 0 , where (~y
By the proof in Case 1, for each ~
s.t.
Bl
Then for any c f
by (2.18), we have
~
Bl
~
Bl
On the other hand, when by (2.18), we have
Bl
Therefore
~
Bl
OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 15
if and only if
i.e., (2.11) is satisfied. The proof is complete.
Remark 2. In the above theorem,
(1) when rigid body motion is considered,
we have dim(M all the conditions in the theorem are satis-
fied. So for each ~
there is ~a f such that
Bl
(2) if
the conclusion still holds for each ~
an m-dimensional
subspace of (L q (\Gamma)) n where q - 1, 1
(3) the vector C in (2.13) represents the rigid body motion in our case. From the
above theorem, we can see that the solution C f such that T f (C f
unique.
The following error estimate contains an uniqueness result, which will also be
used in proving the uniform convergence rate in a subsequent paper.
Theorem 2.3. Let us maintain all the assumptions in Theorem 2.2. Let ~
given in (L 1 respectively two zeros of T f and T h defined by (2.13). If
where
meas
c h
then
where the constant fl is independent of C f and C h .
P.You,Z.Ding and J.Zhou
Proof. We may assume that
For T f (C), we denote
where
y C
Write
Since T f (C) is Lipschitz continuous in C, a direct calculation leads to the Frechet
derivative
hy k
m\Thetam
a.e. C
a Gram-matrix, which is symmetric positive semi-definite, i.e., for any nonzero vector
(b
where "?" holds strictly if
because f~y is linearly independent.
On the other hand, we have
hy k
m\Thetam
hy k
m\Thetam
hy k
m\Thetam
where the Gram-matrix
hy k
m\Thetam
is also symmetric positive semi-definite. Therefore
where "!" holds strictly in the first inequality if meas (\Gamma C holds strictly
in the second inequality if meas and two
respectively, we let
OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 17
Since T f (C) is Lipschitz continuous in C, once meas (\Gamma C f
It follows that T 0
positive definite matrix with
Therefore
defines a symmetric positive definite matrix with
For any 0 ! - ! 1, we have
for some
into account, we arrive at
Consequently, we have
and the proof is complete.
As a direct consequence of Theorem 2.3, we obtain the following uniqueness result.
Corollary 2.4. Let us maintain all the assumptions in Theorem 2.2. For given
~
is a zero of T f with
then C f is the unique zero of T f .
Now we present a state-feedback characterization of the optimal control.
Theorem 2.5.
Let\Omega ae R 3 be a bounded domain with smooth boundary \Gamma. The
LQR problem has a unique optimal control ~u   2 U and a unique optimal velocity
state ~
P.You,Z.Ding and J.Zhou
and
Bl
\Theta ~x is defined in Theorem 2.2 s.t. ~u   ? M 0 and M 0 is given in (1.2).
Proof. Let
?M0 . Since our objective function J(~u) is strictly convex
and differentiable, and the feasible control set U is a closed bounded convex subset
in the reflexive Banach space X , the existence and uniqueness of the optimal control
are well-established. Equation (2.20) is just a copy of (1.16). By our characterization
of truncation, Theorem 2.1 with
Bl
is defined in Theorem 2.2 s.t.
Bl
To prove (2.21), we only need to show
Applying (1.9), i.e., M
and then
Since rJ(~u) defines a bounded linear functional on X , for any ~ h 2 X , take (1.12)
into account, we have
Z
I +K   )
Z
So (2.22) is verified and the proof is complete.
OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 19
3. Regularities of the Optimal Control.
It is clear that (2.21) is a feedback characterization of the optimal control. To
obtain such a characterization,
in (2.9) is crucial. Later on we will see that
is also crucial in proving the uniform convergence of our numerical algorithms
in a subsequent paper. Observe that when corresponds
to the LQR problem without constraints on the control variable, the optimal solution,
if it exists, becomes
\Theta ~x is defined in Theorem 2.2 s.t. ~u   ? M 0 (see Remark 2). But according
to Lemma 1.1(d) such a solution ~u   is only in (L q (\Gamma)) 3 (q ! 2), since E(P k ; \Delta) is only
in (L q (\Gamma)) 3 . So it is reasonable to apply bound constraints Bl and Bu on the control
variable ~u. However we notice that the optimal control still contains a singular term
which is not computable at . In order to carry out the truncation by Bl and
Bu, we have to know the sign of this singular term. Hence we derive a singularity
decomposition formula of (2.21), in which the singular term is expressed as continuous
bounded terms plus a simple dominant singular term and a lower order singular
term. With the simple dominant singular term, the nature of the singularity is clearly
exposed.
Theorem 3.1. For the optimal control ~u   given in (2.21), let
~
f
Then
f
where in the singular part, the second term 4K ~
f   (x) is dominated by the first term
f   (x) whose nature of singularity can be determined at each P k and the regular term
f   (x) is continuous on \Gamma.
Proof. For given ~g 2 (L q (\Gamma)) 3
?N with q
I +K)
I \Theta ~x:
Let
~
f
By (2.23), ~
?N for every q ! 2, thus (3.1) follows. The first part of
Lemma 1.1 (e) states that the singularity in 2 ~
f   dominates the one in 4K ~
f   . While
20 P.You,Z.Ding and J.Zhou
the second part of Lemma 1.1 (e) and (f) imply that ( 1I +K) \Gamma1 ffiK ffiK ~
f   is continuous.
The above singularity decomposition formula plays an important role in our singularity
analysis and also in our numerical computation. It is used to prove the
uniform convergence and to estimate the uniform convergence rate of our numerical
algorithms in a subsequent paper.
Note that the fundamental velocity solution
E(-;
is not defined when in the sense that when x some of
the entries may oscillate between \Gamma1 and +1. So if we look at the simple dominant
singular term in the singularity decomposition formula of the optimal control, we
can see that in general, the optimal control ~u   (x) is not defined at P k even with
the truncation by Bl and Bu. This is a significant difference between systems with
scalar valued fundamental solution and with matrix-valued fundamental solution. For
the formal case, e.g., the Laplacian, the optimal control is continuous at every point
where Bl and Bu are continuous. Of course, if Bl(P k
which means the control is not active at P k , then trivially ~u
prescribed value. This is the case when a sensor is placed at P k , then a control device
can not be put at the same point P k . However, in general point observation case, the
control may still be active at P k . The above analysis then states that the optimal
control is not defined at P k unless some other conditions are posed. This is the nature
of point observations. Notice that a distributed parameter control is assumed in our
problem setting, theoretically the values of the control variable at finite points will not
affect the system. But, in numerical computation we can only evaluate the optimal
control ~u   at finite number of points. The observation points P k 's usually are of the
most interest. On the other hand, the optimal velocity state ~
w   is well-defined and
continuous at P k , no matter ~u   (P k ) is defined or not. So if one does want the optimal
control ~u   to be defined at P k , when Bl(P k m, it is clear that
is defined at each P k . When Bl(P m, then we
have the following necessary and sufficient condition.
Theorem 3.2. Let Bl(P m, then the optimal
control ~u   is well-defined at the observation points P k if and only if
where for each fixed k and i, the equality holds for at most one j 6= i unless
~
When ~u   is well-defined at P k , we have
ae Bl i
Proof. If we observe the fundamental velocity solution, we can see that the proof
follows from the following argument. For
lim
OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 21
exists (including \Sigma1) if and only if
where at most one equality can hold unless c that when (3.5)
holds, and two equalities hold in (3.5), then
We can make the limit either equal to zero by taking x
to sign (c i )1 by taking x i 6= \Upsilonx j or x i 6= \Upsilonx k and x ! 0. So the limit will not exist.
When lim x!0 - e i (x) exists and
lim
With the above result and the singularity decomposition formula for the optimal
control, the following continuous result can be easily verified.
Theorem 3.3. Let Bu and Bl be continuous on \Gamma 1 . If for each
either or the condition (3.3) holds strictly with ( ~
0, then the optimal control ~u   is continuous on \Gamma 1 . So the equality in (2.4) holds for
every point on \Gamma.
From the state-feedback characterization (2.20), the control can be determined
by a physical measurement of the state at finite number of observation points
m. The question is then asked, will a small error in the measurement of the state
cause a large deviation in the control? Due to the appearance of the singular term
in (2.20), in general the answer is yes, i.e., the state-feedback system is not stable.
However under certain conditions, we can prove that the state-feedback system is
uniformly stable.
Theorem 3.4. Let ~
be the exact velocity state at observation points and
~u p be the control determined from (2.20) in terms of ~
w(P k ). If for each
either Bl and Bu are continuous and equal at P k or Bu and Bl are locally bounded
at P k , the condition (3.3) holds strictly with ( ~
then the state-feedback
system (2.20) is uniformly stable in the sense that for any " ? 0, there is
such that for any measurement ~
where ~u 0 is the control determined from (2.20) in terms of ~
Proof. For each " ? 0. For each fixed and Bu are continuous
and equal at P k , there is d 0
Since the control variable is bounded by Bl and Bu,
If instead the condition (3.3) holds strictly with ( ~
chosen so that when j ~
holds strictly with
Due to the singular term in (2.20) and since Bu and Bl are
22 P.You,Z.Ding and J.Zhou
locally bounded at P k , there is d k ? 0 such that when x
some m, we have
\Gammafl
either ? Bu(x) i
or
After the truncation by Bu and Bl, it follows that
So if we define
then in either case we have
Denote
~
I +K) \Gamma1
~
and
meas
implies that
there is nothing to prove. So we assume that meas (\Gamma CF
Theorem 2.3 can be applied. For x compact set, by using (2.20)
and triangle inequality, we obtain
Bl
Bl
OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 23
Since the operator linear and bounded, and the function E(P k ; \Delta) is
continuous and bounded on the compact set
As for I 2 (x), Theorem 2.3 yields
where the constant fl depends only on \Gamma. Since there is constant C 0 independent of
~
there is
Finally for
The proof is complete.
As a final comment, it is worth while indicating that though in the problem
setting, the governing differential equation, the Stokes, is linear, the bound constraint
on the control variable introduces a nontrivial nonlinearity into the system. This can
be clearly seen in Theorem 2.2. Also our approach can be adopted to deal with certain
nonlinear boundary control problems.



--R

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--TR

--CTR
Zhonghai Ding, Optimal Boundary Controls of Linear Elastostatics on Lipschitz Domains with Point Observations, Journal of Dynamical and Control Systems, v.7 n.2, p.181-207, April 2001
hydrostatic potential;boundary integral equation;stokes fluid;singularity decomposition;distributed boundary control;point observation;LQR