Technische Universit¨at Berlin
Institut f¨ur Mathematik
L∞-ESTIMATES FOR
APPROXIMATED OPTIMAL
CONTROL PROBLEMS
Christian Meyer, Arnd R¨osch
Preprint 2004/22
Preprint-Reihe des Instituts f¨ur Mathematik
Technische Universit¨at Berlin
Report 2004/22 September 2004
An optimal control problem for a 2-d elliptic equation is investigated with pointwise control con-
straints. This paper is concerned with discretization of the control by piecewise linear functions.
The state and the adjoint state are discretized by linear finite elements. Approximation of order
hin the L∞-norm is proved in the main result.
AMS subject classifications. 49K20, 49M25, 65N30
L∞-ESTIMATES FOR APPROXIMATED OPTIMAL CONTROL PROBLEMS ∗
C. MEYER †AND A. R ¨
OSCH ‡
Abstract. An optimal control problem for a 2-d elliptic equation is investigated with pointwise control con-
straints. This paper is concerned with discretization of the control by piecewise linear functions. The state and the
adjoint state are discretized by linear finite elements. Approximation of order hin the L∞-norm is proved in the
main result.
Keywords: Linear-quadratic optimal control problems, error estimates, elliptic equations, numer-
ical approximation, control constraints.
AMS subject classification: 49K20, 49M25, 65N30
1. Introduction. The paper is concerned with the discretization of the 2-d elliptic optimal
control problem
J(u) = F(y, u) = 1
2ky−ydk2
L2(Ω) +ν
2kuk2
L2(Ω) (1.1)
subject to the state equations
Ay +a0y=uin Ω
y= 0 on Γ (1.2)
and subject to the control constraints
a≤u(t, x)≤bfor a.a. x∈Ω,(1.3)
where Ω is a bounded domain with boundary Γ; Adenotes a second order elliptic operator of the
form
Ay(x) = −
2
X
i,j=1
Di(aij (x)Djy(x))
where Didenotes the partial derivative with respect to xi, and aand bare real numbers. Moreover,
ν > 0 is a fixed positive number. We denote the set of admissible controls by Uad:
Uad ={u∈L2(Ω) : a≤u≤ba.e. in Ω}.
We discuss here the full discretization of the control and the state equations by a finite element
method. The asymptotic behaviour of the discretized problem is studied.
The approximation of the discretization for semilinear elliptic optimal control problems is discussed
in Arada, Casas, and Tr¨oltzsch [1]. The optimal control problem (1.1)–(1.3) is a linear-quadratic
counterpart of such a semilinear problem.
∗supported by the DFG Research Center ”mathematics for key technologies” (FZT 86) in Berlin
†Technische Universit¨at Berlin, Fakult¨at II Mathematik und Naturwissenschaften, Straße des 17. Juni 136,
‡Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences,
1
The discretization of optimal control problems by piecewise constant functions is well investigated,
we refer to Falk [7], Geveci [8]. Piecewise constant and piecewise linear discretization in space are
discussed for a parabolic problem in Malanowski [10]. Theory and numerical results for elliptic
boundary control problems are contained in Casas and Tr¨oltzsch [6] and Casas, Mateos, and
Tr¨oltsch [5]. All papers are mainly focussed on L2-estimates. However, in Arada, Casas, and
Tr¨oltzsch [1] we find also an L∞-estimate of order hfor piecewise constant functions.
Piecewise linear control discretizations for elliptic optimal control problems are studied by Casas
and Tr¨oltzsch, see [6]. In an abstract optimization problem, piecewise linear approximations are
investigated in R¨osch [13]. In all papers, the convergence is mainly discussed in the L2-norm.
In this paper, we will show that also for piecewise linear controls the approximation order hcan
be obtained in the L∞-norm. Such type of result can not be obtained with one of the above
mentioned methods. The L∞-estimate is obtained in two main steps. We prove in the first step
that the discretized solutions violate a pointwise projection formula only in an order h. The
L∞-estimates for grid points and later for arbitrary points are derived in the second step.
The paper is organized as follows: In section 2 the discretizations are introduced and the main
results are stated. Section 3 contains auxiliary results. The proofs of the approximation result is
placed in section 4. The paper ends with numerical experiments shown in section 5.
2. Discretization and main result. Throughout this paper, Ω denotes a convex bounded
open subset in IR2of class C1,1. The coefficients aij of the operator Abelong to C0,1(¯
Ω) and
satisfy the ellipticity condition
m0|ξ|2≤
2
X
i,j=1
aij (x)ξiξj∀(ξ, x)∈IR2ׯ
Ω, m0>0.
Moreover, we require aij (x) = aji(x) and yd∈Lp(Ω) for some p > 2. For the function a0∈L∞(Ω),
we assume a0≥0. Next, we recall some results from Bonnans and Casas [2].
Lemma 2.1. [2] For every p > 2and every function g∈Lp(Ω), the solution yof
Ay +a0y=gin Ω, y|Γ= 0,
belongs to H1
0(Ω) ∩W2,p(Ω). Moreover, there exists a positive constant c, independent of a0such
that
kykW2,p(Ω) ≤ckgkLp(Ω).
We introduce the adjoint equation
Ap +a0p=y−ydin Ω
p= 0 on Γ (2.1)
Due to Lemma 2.1, the state equation and the adjoint equation admit unique solutions in H1
0(Ω)∩
W2,p(Ω), if yd∈Lp(Ω) for p > 2. This space is embedded in C0,1(¯
Ω).
We call the solution yof (1.2) for a control uassociated state to uand write y(u). In the same
way, we call the solution pof (2.1) corresponding to y(u) associated adjoint state to uand write
p(u).
Introducing the projection
Π[a,b](f(x)) = max(a, min(b, f(x))),
2
we can formulate the necessary and sufficient first-order optimality condition for (1.1)–(1.3).
Lemma 2.2. A necessary and sufficient condition for the optimality of a control ¯uwith correspond-
ing state ¯y=y(¯u)and adjoint state ¯p=p(¯u), respectively, is that the equation
¯u(x) = Π[a,b](−1
ν¯p) (2.2)
holds.
Since the optimal control problem is strictly convex, we obtain the existence of a unique optimal
solution. The optimality condition can be formulated as variational inequality
(ν¯u+ ¯p, u −¯u)U≥0 for all u∈Uad.
A standard pointwise a.e. discussion of this variational inequality leads to the above formulated
projection formula, see [10].
We are now able to introduce the discretized problem. We define a finite-element based approxima-
tion of the optimal control problem (1.1)–(1.3). To this aim, we consider a family of triangulations
(Th)h>0of ¯
Ω. With each element T∈Th, we associate two parameters ρ(T) and σ(T), where
ρ(T) denotes the diameter of the set Tand σ(T) is the diameter of the largest ball contained in T.
The mesh size of the grid is defined by h= max
T∈Th
ρ(T). We suppose that the following regularity
assumptions are satisfied.
(A1) There exist two positive constants ρand σsuch that
ρ(T)
σ(T)≤σ, h
ρ(T)≤ρ
hold for all T∈Thand all h > 0.
(A2) Let us define ¯
Ωh=S
T∈Th
T, and let Ωhand Γhdenote its interior and its boundary, respectively.
We assume that ¯
Ωhis convex and that the vertices of Thplaced on the boundary of Γhare points
of Γ. From [12], estimate (5.2.19), it is known that
|Ω\Ωh| ≤ Ch2,
where |.|denotes the measure of the set.
(A3) For simplicity, we require 0 ∈[a, b].
Assumption (A3) allows a simple discussion of the set Ω \Ωh. The main part of the presented
results is independent from this assumption. However, the discussion of the general case leads to
very technical investigations on extensions of controls to Ω \Ωh. For a clear presentation of the
ideas and the results, we decide to discuss here only the case 0 ∈[a, b].
Next, to every boundary triangle Tof Thwe associate another triangle ˆ
Twith curved boundary
as follows: The edge between the two boundary nodes of Tis substituted by the corresponding
curved part of Γ. We denote by ˆ
Ththe union of these curved boundary triangles with the interior
triangles to Ω of Th, such that ¯
Ω = S
ˆ
T∈ˆ
Th
ˆ
T. Moreover, we set
Uh=Vh={yh∈C(¯
Ω) : yh∈ P1for all T∈Th,and yh= 0 on ¯
Ω\Ωh}, Uad
h=Uh∩Uad,
where P1is the space of polynomials of degree less or equal than 1. The definition of the space Uad
with homogeneous boundary values is motivated by the projection formula (2.2) and the homo-
geneous boundary condition (2.1) of the adjoint equation. Here, we benefit from the assumption
(A3).
3
For each uh∈Uh, we denote by yh(uh) the unique element of Vhthat satisfies
a(yh(uh), vh) = ZΩ
uhvhdx ∀vh∈Vh,(2.3)
where a:Vh×Vh→IR is the bilinear form defined by
a(yh, vh) = ZΩ
a0(x)yh(x)vh(x) +
2
X
i,j=1
aij (x)Diyh(x)Djvh(x)
dx.
In other words, yh(uh) is the approximated state associated with uh. Because of yh=vh= 0
on ¯
Ω\Ωhthe integrals over Ω can be replaced by integrals over Ωh. The finite dimensional
approximation of the optimal control problem is defined by
inf J(uh) = 1
2kyh(uh)−ydk2
L2(Ω) +ν
2kuhk2
L2(Ω) uh∈Uad
h.(2.4)
The adjoint equation is discretized in the same way
a(ph(uh), vh) = ZΩ
(yh(uh)−yd)vhdx ∀vh∈Vh.(2.5)
Now, we are able to state the main result.
Theorem 2.3. Let ¯uand uhbe the optimal solution of (1.1) and (2.4), respectively. Then, there
exists a positive constant Cindependent of hwith
k¯u−uhkL∞(Ω) ≤Ch. (2.6)
The proof of Theorem 2.3 is contained in section 4. Moreover, the constant Cis specified in that
section.
3. Auxiliary results. We start with a L2-estimate corresponding to Theorem 2.3.
Lemma 3.1. Let ¯uand uhbe the optimal solution of (1.1) and (2.4), respectively. Then an
estimate
k¯u−uhkL2(Ω) ≤C2h(3.1)
holds true with a positive constant C2.This statement can easily proved by the arguments of
Casas and Tr¨oltzsch [6]. It is a special case of a new general result of Casas [4].
This implies easily the following L∞-estimate
k¯p−p(uh)kL∞(Ω) ≤ck¯p−p(uh)kH2(Ω) ≤ch. (3.2)
Lemma 3.2. The inequality
k¯p−ph(uh)kL∞(Ω) ≤κh (3.3)
is valid with a positive constant κ.
Proof. First, we recall a L∞-estimate for the finite element solution
kp(uh)−ph(uh)kL∞(Ω) ≤ch, (3.4)
4
see Braess [3]. Using (3.2), we find
k¯p−phkL∞(Ω) ≤ k¯p−p(uh)kL∞(Ω) +kp(uh)−phkL∞(Ω) ≤κh.
Next, we introduce a new notation for the piecewise linear functions. Let Eibe an arbitrary vertex
of the triangulation Th. Then, we define a basis function ei∈Uhby
ei(Ej) = δij ,
where δij is the Kronecker symbol. Therefore, we can represent the functions uhand ph(uh) by
uh(x) = X
Ei
uiei(x)
(ph(uh))(x) = X
Ei
piei
with ui=uh(Ei) and pi= (ph(uh))(Ei).
We denote the set of neighbouring vertices of Ei, i.e. (ei, ej)6= 0 and i6=j, by N(Ei).
Lemma 3.3. For every jwith Ej∈N(Ei)we have
|pi−pj| ≤ (L+ 2κ)h, (3.5)
where Ldenotes the Lipschitz constant of ¯p.
Proof. Because of Lemma 2.1, ¯pbelongs to W2
p(Ω) for a certain p > 2. Therefore ¯pis lipschitz and
we have
|¯p(Ei)−¯p(Ej)| ≤ Lh.
Combining this inequality with (3.3), we obtain
|pi−pj| ≤ |pi−¯p(Ei)|+|¯p(Ei)−¯p(Ej)|+|¯p(Ej)−pj|
≤κh +Lh +κh.
Later, we need a similar inequality
1
ν|pi−pj| ≤ Dh. (3.6)
with
D=L+ 2κ
ν.
Next, we recall a property concerning the mass matrix.
Lemma 3.4. For every basis function ei
(ei, ei)U≥X
Ej∈N(Ei)
(ei, ej)U(3.7)
is valid.
5
Proof. The element mass matrix of the reference element is given by
Mr=1
24
2 1 1
1 2 1
1 1 2
which has the desired property with equality. Clearly, every linear transformation preserves this
property. This holds also for the summation over all triangles. The inequality sign is obtained if
the support of eicontains at least one boundary point.
Next, we want to investigate the following quantity
M:= max
i
ui−Π[a,b](−1
νpi)
.(3.8)
In all what follows, the index idenotes a fixed vertex where this maximum is attained. Moreover,
we will assume that M > 0. Otherwise, the main results of the paper can be easily derived.
Equation (3.8) means, that one of the following cases (A) and (B) occurs:
(A)M=ui−Π[a,b](−1
νpi)
(B)M=−(ui−Π[a,b](−1
νpi)).
Without loss of generality, we discuss here the case (A). The case (B) can be investigated in the
same way. Since Mis positive and Π[a,b](−1
νpi)∈[a, b] by definition, this implies
M=ui−Π[a,b](−1
νpi)≤ui+1
νpi.(3.9)
and
ui> a.
Consequently, there exists an ε > 0 such that
ui−ε > a.
This means, that the control vh=uh−εeiis admissible.
Corollary 3.5. In the case (B) we obtain that vh=uh+εeiis admissible for an ε > 0.
Lemma 3.6. Let M > 0and ibe the index, where the maximum in (3.8) is attained. Then (A)
implies
ui+1
νpi≤max
Ej∈N(Ei)−(uj+1
νpj).(3.10)
Moreover, if equality holds in (3.10), then we have
ui+1
νpi=−(uj+1
νpj)for all jwith Ej∈N(Ei).
Proof. We start with the optimality condition for uh
(νuh+ph(uh), vh−uh)U≥0 for all vh∈Uad
h.
6
We test this inequality with vh=uh−εei
(νuh+ph(uh),−εei)U≥0.
¿From this, we obtain
(νui+pi)(ei, ei)≤X
Ej∈N(Ei)−(νuj+pj)(ei, ej)U.
Using (3.7), we find
(νui+pi)(ei, ei)≤max
Ej∈N(Ei)−(uj+1
νpj)X
Ej∈N(Ei)
(ei, ej)U≤( max
Ej∈N(Ei)−(uj+1
νpj))(ei, ei).
Division by (ei, ei) yields (3.10). Since the scalar products (ei, ej)Uare positive for all jwith
Ej∈N(Ei), equality can only occur, if
ui+1
νpi=−(uj+1
νpj) for all jwith Ej∈N(Ei).
Corollary 3.7. In the case (B) we find
−(νui+pi)(ei, ei)≤X
Ej∈N(Ei)
(νuj+pj)(ei, ej)U.
Next, we denote the index where the maximum is attained by k
−(uk+1
νpk) = max
Ek∈N(Ei)−(uj+1
νpj) (3.11)
Combining (3.9)–(3.11), we find
M≤ui+1
νpi≤ −(uk+1
νpk).(3.12)
Moreover, we have by definition of M
M≥ |uk−Π[a,b](−1
νpk)|.
Since (3.12) and M > 0, uk+1
νpkis negative and consequently uk−Π[a,b](−1
νpk), too. Therefore,
we have
M≥ −(uk−Π[a,b](−1
νpk)).(3.13)
¿From (3.12) and (3.13), we obtain
−(uk−Π[a,b](−1
νpk)) ≤M≤ui+1
νpi≤ −(uk+1
νpk).(3.14)
This inequality is one of the key point for our results.
Corollary 3.8. In the case (B), we have
uk−Π[a,b](−1
νpk)≤M≤ −(ui+1
νpi)≤uk+1
νpk.
7
Lemma 3.9. There exists an index iwith
M=|ui−Π[a,b](−1
νpi)|
and a corresponding index k,Ek∈N(Ei)with
Π[a,b](−1
νpk)6=−1
νpk.(3.15)
Proof. First, we discuss the case where in inequality (3.14) at least one strong inequality occurs.
Then we have
−(uk−Π[a,b](−1
νpk)) <−(uk+1
νpk).
This implies directly
Π[a,b](−1
νpk)) <−1
νpk(3.16)
and the assertion is proved for this case.
In the other case, we discuss as follows. Here, we know
M=−(uk−Π[a,b](−1
νpk)).
This means that the maximum Mis also attained in the vertex Ek. Consequently, we have the
case (B) for the vertex Ek. ¿From Corollary 3.5, we know that vh=uh+εekis admissible for
sufficiently small ε. Moreover, we obtain
−(νuk+pk)(ek, ek)≤X
Ej∈N(Ek)
(νuj+pj)(ek, ej)U
by Corollary 3.7.
Next we show, that the equality case cannot occur for the index k, too: Here we have
ui+1
νpi=−(uj+1
νpj) for all jwith Ej∈N(Ei)
because of Lemma 3.6. The sign of ui+1
νpiis inverse to the sign of all jwith Ej∈N(Ei). This
holds especially for j=k. But, there exists at least one common neighbouring vertex (El∈N(Ei)
and El∈N(Ek)). Due to our discussion, uk+1
νpkand ul+1
νplhave the same (negative) sign.
Hence, we can continue with
−(νuk+pk)(ek, ek)≤X
Ej∈N(Ek)
(νuj+pj)(ek, ej)U<X
Ej∈N(Ek),j6=l
(νuj+pj)(ek, ej)U.
Using again (3.7), an index m,Em∈N(Ek) exists with
−(uk+1
νpk)< um+1
νpm.
This inequality and Corollary 3.8 imply
um−Π[a,b](−1
νpm)≤M≤ −(uk+1
νpk)< um+1
νpm.
8
Consequently, the assumptions for the first case are fulfilled for the indices kand mand we have
−Π[a,b](−1
νpm)) <1
νpm.
Therefore, the assertion is true.
Without loss of generality, we will assume that in inequality (3.14) at least one strong inequality
occurs. In this case, (3.16) is valid.
Lemma 3.10. Assume that
Dh < b −a
is valid. Then, the estimate
M= max
i|ui−Π[a,b](−1
νpi)|< Dh (3.17)
holds true.
Proof. Inequality (3.16) implies directly
b= Π[a,b](−1
νpk)) <−1
νpk.(3.18)
¿From this and (3.6), we obtain
−1
νpi> b −Dh.
By assumption, the value b−Dh is greater than a. ¿From (A)
ui−Π[a,b](−1
νpi) = M > 0
and u≤bwe obtain
−1
νpi≤b.
Consequently, we find
−1
νpi= Π[a,b](−1
νpi)
that implies
ui+1
νpi=ui−Π[a,b](−1
νpi) = M.
Using ui≤band 1
νpi<−(b−Dh), we find
ui+1
νpi< b −(b−Dh) = Dh.
Combining the last two inequalities, the assertion is proved.
Let us shortly comment the exceptional cases. First, for M= 0 the statement of the lemma is true
for arbitrary positive h. Second, for b−a≤Dh Theorem 2.3 holds with C=D. Therefore, we
have not to take care for these two cases.
9
4. Proof of the main result. The proof of Theorem 2.3 is divided in two parts. In the next
lemma we derive a corresponding estimate for the grid points. The estimate for arbitrary points
is obtained in a second step.
Lemma 4.1. The estimate
max
i|uh(Ei)−¯u(Ei)| ≤ (D+κ
ν)h.
is valid.
Proof. ¿From Lemma 3.7, we know
max
i|ui−Π[a,b](−1
νpi)| ≤ Dh
or in other notation
max
i|uh(Ei)−Π[a,b](−1
νph(Ei))| ≤ Dh.
¿From (3.3)
k¯p−phkL∞(Ω) ≤κh,
and the Lipschitz continuity of the projection operator we deduce
kΠ[a,b](−1
ν¯p(ei)) −Π[a,b](−1
νph(Ei))kL∞(Ω) ≤κ
νh.
Using
¯u(Ei) = Π[a,b](−1
ν¯p(Ei))
and the triangle inequality we end up with
max
i|uh(Ei)−¯u(Ei)| ≤ (D+κ
ν)h.
Now, we are able to proof Theorem 2.3.
Proof. A non grid point x∈Tican be expressed by a convex linear combination of the vertices Ej
of the corresponding triangle
x=X
Ej∈Ti
λjEj,X
Ej∈Ti
λj= 1.
Since uhis linear on Ti, we get
|uh(x)−¯u(x)|=|X
Ej∈Ti
λjuh(Ej)−¯u(x)|
≤X
Ej∈Ti
λj|uh(Ej)−¯u(Ej)|+X
Ej∈Ti
λj|¯u(x)−¯u(Ej)|
≤(D+κ
ν)h+X
Ej∈Ti
λj|¯u(x)−¯u(Ej)|
≤(D+κ
ν)h+L
νh.
10
In the final inequality we used the lipschitz continuity of ¯u. Summarizing all results, we obtain
k¯u−uhkL∞(Ωh)≤(D+κ+L
ν)h.
Therefore, the assertion is true for every point x∈Tiwith
C=D+κ+L
ν.
Until now, we have not used assumption (A3). It remains the part Ω \Ωh. By definition, we have
uh= 0 on this part. ¿From (2.2), we obtain easily ¯u= 0 on Γ. Let x∈Ω\Ωhbe an arbitrary
point. From [12], we know that
min
xΓ∈Γ|x−xΓ| ≤ cΓh2
holds with a certain constant cΓ>0 independent of h. Therefore, we find for x∈Ω\Ωh
|uh(x)−¯u(x)|=|0−¯u(x)|=|¯u(xΓ)−¯u(x)| ≤ cΓL
νh2.
5. Numerical example. We have tested the convergence theory by the following example:
−∆y=uin Ω
y= 0 on Γ (5.1)
with Ω = (0,1) ×(0,1). One can easily verify that this problem fulfills the assumptions mentioned
at the beginning of section 2 except the boundary regularity. Although Γ is not of class C1,1, the
W2,p-regularity of ¯p(see Lemma 2.1) is obtained by an result of Grisvard [9] for convex polygonal
domains.
In [11], we derived an exact solution to (5.1), which is also used here. For convenience of the
reader, we recall this example.
The optimal state is defined by
¯y=ya−yg
with an analytical part ya= sin(π x1) sin(π x2) and a less smooth function yg. The function yg
represents the solution of
−∆yg=gin Ω
yg= 0 on Γ.
Here, gis given by
g(x1, x2) =
ˆu(x1, x2)−a, if ˆu(x1, x2)< a
0 , if ˆu(x1, x2)∈[a, b]
ˆu(x1, x2)−b, if ˆu(x1, x2)> b
with ˆu(x1, x2) = 2 π2sin(π x1) sin(π x2). Due to the state equation (5.1), we obtain for the exact
optimal control ¯u
¯u(x1, x2) =
a, if ˆu(x1, x2)< a
ˆu(x1, x2) , if ˆu(x1, x2)∈[a, b]
b, if ˆu(x1, x2)> b.
11
For the optimal adjoint state ¯p, we find
¯p(x1, x2) = −2π2νsin(π x1) sin(π x2).
To fulfill the necessary and sufficient first oder optimality conditions, the desired state ydis defined
by
yd(x1, x2) = ¯y+ ∆¯p=ya−yg+ 4 π4νsin(π x1) sin(π x2).
The optimization problem was solved numerically by a primal-dual active set strategy. As men-
tioned in section 2, the state equation and the adjoint equation were discretized with linear finite
elements. Here, uniform meshs were used. The resulting linear system of equations was solved
with the CG-method.
To approximate the L∞-norm k¯u−uhkL∞(Ω), we evaluated |¯u(x)−uh(x)|in the grid points, in
the barycenters of the elements and in the midpoints of the edges of the triangulation.
In a first test we chose a=−15 and b= 15. Consequently, assumption (A3) is valid.
Figure 5.1 shows the numerically calculated optimal control uh, for the mesh size h/√2 = 0.02.
00.2 0.4 0.6 0.8 1
0
0.5
1
0
5
10
15
x1
x2
uh
Fig. 5.1.Optimal control uh
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
0
0.2
0.4
0.6
0.8
1
1.2
1.4
2−0.5 h
|| uopt − uh ||L∞(Ω)
Fig. 5.2.k¯u−uhkL∞(Ω)
Table 5.1
h/√2 0.04 0.02 0.01 0.005
k¯u−uhkL∞(Ω) 1.17450 0.26396 0.11536 0.06328
Figure 5.2 and Table 5.1 illustrate the convergence behavior for the first test. As one can see, the
theoretical predictions are fulfilled and one obtains linear approximation order for k¯u−uhkL∞(Ω)
(except on the coarsest grid).
In the second test we chose a= 3 and b= 15. Consequently, 0 6∈ [a, b], i.e. assumption (A3) is not
fulfilled. However, this fact causes no difficulties with extensions because of Ω = Ωh.
12
0
0.5
1
0
0.5
1
0
5
10
15
x1
x2
uh
Fig. 5.3.Optimal control uh
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2−0.5 h
|| uopt − uh ||L∞(Ω)
Fig. 5.4.k¯u−uhkL∞(Ω)
Figure 5.3 show again the numerical calculated optimal control uh, for the mesh size h/√2 = 0.02.
Figure 5.4 and Table 5.2 illustrate the convergence behavior for the second test. The convergience
behavior is similar to the first test and one again obtains linear convergence for k¯u−uhkL∞(Ω).
Table 5.2
h/√2 0.04 0.02 0.01 0.005
k¯u−uhkL∞(Ω) 0.58292 0.30681 0.14813 0.07390
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