on linear-quadratic elliptic control problems of semi-infinite type

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On linear-quadratic elliptic controlproblems of semi-infinite typePedro Merino a , Ira Neitzel b & Fredi Tröltzsch ba Departamento de Matemática Ladrón de Guevara, EscuelaPolitécnica Nacional , E11-253 Quito , Ecuadorb Fakultät II - Mathematik und Naturwissenschaften , TechnischeUniversität Berlin , Str. des 17. Juni 136, D-10623 Berlin ,GermanyPublished online: 09 Sep 2010.

To cite this article: Pedro Merino , Ira Neitzel & Fredi Tröltzsch (2011) On linear-quadratic ellipticcontrol problems of semi-infinite type, Applicable Analysis: An International Journal, 90:6,1047-1074, DOI: 10.1080/00036811.2010.489187

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Applicable AnalysisVol. 90, No. 6, June 2011, 1047–1074

On linear-quadratic elliptic control problems of semi-infinite type

Pedro Merinoa, Ira Neitzelb* and Fredi Troltzschb

aDepartamento de Matematica Ladron de Guevara, Escuela Politecnica Nacional,E11-253 Quito, Ecuador; bFakultat II -Mathematik und Naturwissenschaften,Technische Universitat Berlin, Str. des 17. Juni 136, D-10623 Berlin, Germany

Communicated by B. Mordukhovich

(Received 2 February 2010; final version received 15 April 2010)

We derive a priori error estimates for linear-quadratic elliptic optimalcontrol problems with pointwise state constraints in a compact subdomainof the spatial domain � for a class of problems with finite-dimensionalcontrol space. The problem formulation leads to a class of semi-infiniteprogramming problems, whose constraints are implicitly given by theFE-discretization of the underlying PDEs. We prove an order of h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

pfor the error j �u� �uh

j in the controls, and show that it can be improved to anorder of h2jlog hj under certain assumptions on the structure of the activeset. Numerical experiments underline the proven theoretical results.

Keywords: elliptic optimal control problem; state constraints; errorestimates; finite element discretization; semi-infinite optimization

AMS Subject Classifications: 49J20; 80M10; 49N05; 41A25; 90C34

1. Introduction

In this article, we derive a priori error estimates for optimal control problems withpointwise state constraints in an interior subdomain of the spatial domain � that aregoverned by linear elliptic PDEs and finitely many control parameters. Theconsidered problem class is interesting for practical and theoretical reasons. On theone hand, finitely many controls appear quite frequently in practice, since controlfunctions that can vary arbitrarily in space are more difficult to implement. On theother hand, the question of error estimates for pointwise state constraints is quitechallenging. We aim at extending the results from [1], where optimal controlproblems with finite-dimensional control space as well as finitely many pointwisestate constraints have been analysed and an order of h2jlog hj has been proven.

The state-constraints of optimal control problems with finitely many controlsoften become only active in finitely many points, and hence, similar to [1], anassumption on activity in only finitely many points will play an important role whenderiving our error estimates. However, in semi-infinite problems the location of theactive points is generally not known, and it is necessary to consider the constraints in

*Corresponding author. Email: [email protected]

ISSN 0003–6811 print/ISSN 1563–504X online

� 2011 Taylor & Francis

DOI: 10.1080/00036811.2010.489187

http://www.informaworld.com

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a domain. This fact is in some sense the main difference between this article and the

results in [1], where state constraints were given only in finitely many given points in

the domain. This difference becomes immediately obvious noting that the problems

considered in [1] were equivalent to a finite-dimensional nonlinear programming

problem, whereas the infinite number of constraints in our problem formulation only

allows the formulation as a semi-infinite programming problem. As an effect, a

change of the location of the discrete active points in each refinement level of the

discretized problem is possible, in turn allowing the controls to vary more freely. As

a consequence, we will eventually show that the error estimate of order h2jlog hj can

only be extended to this problem class under certain strong assumptions. In order to

put our work into perspective, let us mention that the theory of semi-infinite

optimization is quite well-established. We point out, for example, [2–4] and the

references therein for an overview, as well as [5–7] where numerical aspects of semi-

infinite programming are discussed.However, in our problem formulation we find that the objective function as well

as the constraints are implicitly defined via the solution of a PDE. Hence, aspects of

finite-element error analysis have to be considered when analysing the convergence

of discrete solutions that are not usually found in semi-infinite programming. On the

other hand, most publications in the optimal control community that are concerned

with discretization error estimates consider control functions, whose discretization

adds an additional component to the error estimate even in the purely control

constrained case (cf. [8–10]).For state constrained problems, only few results are known. We refer to [11,12]

where convergence is shown for distributed, respectively boundary control problems

with finitely many state constraints and piecewise constant approximation of the

control functions. Error estimates for elliptic state constrained distributed control

functions have been derived in [13] and, with additional control constraints, in [14].

In both papers, an order of h1�" in a two-dimensional domain, and h12�" in a three-

dimensional setting have been obtained. Recently, orders hjlog hj and h12 were shown

for two and three dimensions, respectively (cf. [15]).In this article, we discuss error estimates for semi-infinite control problems,

where we treat distributed and boundary control problems in a quite similar manner,

expanding in detail the ideas presented in [16] for a related setting. We benefit from

the fact that the controls are vectors of real numbers and hence no aspects of control

discretization need to be considered. In the next section, we lay out the setting for

distributed control problems. In Section 3, we set up a discrete problem formulation,

then derive an error estimate of order hffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

p, and finally improve it to an order of

h2jlog hj under additional conditions. In Section 4, we then briefly sketch the

differences encountered in boundary control, and explain how the results of Section

3 can be extended under reasonable assumptions. Finally, Section 5 is devoted to

numerical experiments to complement our theory.

2. Problem setting and analysis

In this section, we lay out the setting for the considered problem class. Let us therefore

consider a model problem of tracking type with control u :¼ (u1, . . . , uM)T2RM

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and state y, given by

ðPÞ

minu2Uad

Jð y, uÞ ¼1

2

Z�

ð y� yd Þ2dxþ

�

2juj2

subject to AyðxÞ ¼PMi¼1

uieiðxÞ in �

yðxÞ ¼ 0 on �,yðxÞ � b 8x2K,

8>>>>>><>>>>>>:

where we rely on the following assumptions and notations.

ASSUMPTION 2.1 Let ��R2 be a convex polygonal spatial domain with boundary

�¼ @�, and denote by K(� a compact subset whose interior is a domain. Moreover,consider a regularization parameter �2R

þ and a natural number M� 1, a givendesired state yd2L

2(�), as well as fixed basis functions ei2C0,�(�), i¼ 1, . . . ,M, for

some 05�51.The differential equation is based upon a uniformly elliptic and symmetric

differential operator

AyðxÞ ¼ �X2i, j¼1

@j ðaijðxÞ@iyðxÞÞ þ a0ðxÞ yðxÞ

with coefficients aij2C1þ�(�), 05�51, and a02C

0,�(�), a0(x)� 0 in �. The set ofadmissible controls is defined by

Uad :¼ u2RMj ua � ui � ub for i ¼ 1, . . . ,M

� �,

where ua2R[ {�1} and ub2R[ {1} are given bounds with ua5ub. The constrainton the state is given by a bound b2R. For completeness, we denote the feasible set by

Ufeas :¼ u2Uad j yuðxÞ � b 8x2K� �

,

where yu denotes the state associated with the control u. Hereafter we assume that Ufeas

is not empty.Let us introduce the following short notation: By k�k, we denote the natural norm in

L2(�), and (�, �) will denote the associated inner product. Likewise, the Euclidean normin R

M will be denoted by j�j, and the associated inner product by h�, �i. Last, we denoteby Mð ��Þ the space of regular Borel measures on ��.

We begin by summarizing some results on the underlying PDEs.

Definition 2.2 Let e2L2(�) be given. A function ye 2H10ð�Þ is said to be a weak

solution of the equation

AyeðxÞ ¼ e in �, yeðxÞ ¼ 0 on �, ð2:1Þ

if it fulfilsP2

j,k¼1ðajk@jye, @k�Þ þ ða0ye,�Þ ¼ ðe,�Þ for all �2H10ð�Þ.

LEMMA 2.3 For each function e2L2(�), there exists a unique weak solutionye 2H

10ð�Þ \H

2ð�Þ of (2.1), and the mapping e � ye is continuous fromL2(�)!H2(�). Moreover, if e2C0,�(�), then y2C2,�(�) is satisfied.

Proof This follows from the existence and regularity results in [17,18], since � isconvex. g

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Clearly, due to the linearity of the underlying state equation, we can apply the

superposition principle to obtain an equivalent semi-infinite formulation

ðPÞminu2Uad

f ðuÞ :¼1

2

XMi¼1

uiyi � yd

��2

þ�

2juj2

subject toPMi¼1

uiyiðxÞ � b 8x2K,

8>>><>>>:

where the yi, i¼ 1, . . . ,M, are the solutions to (2.1) for e :¼ ei. Obviously, thefollowing proposition holds.

PROPOSITION 2.4 For each u2RM, there exists a unique weak solution y :¼

yu 2H10ð�Þ \H

2ð�Þ \ C2,�ð�Þ of the state equation

AyðxÞ ¼XMi¼1

uieiðxÞ in �, yðxÞ ¼ 0 on �,

and the mapping u � y is continuous from RM to H2(�).

PROPOSITION 2.5 Under Assumption 2.1, there exists a unique optimal control �u2Uad

with associated optimal state �y for Problem (P).

Proof From �40, we obtain that limjuj!1 Jð y, uÞ ¼ 1. Therefore, it is sufficient toconsider a compact subset Ufeas\B�(0) for minimization, and the Weierstrass

theorem yields the assertion. g

Next, we make the standard assumption of a Slater condition.

ASSUMPTION 2.6 There exist a control u� 2Uad and a real number �40 such that

y�ðxÞ ¼ yu� ðxÞ � b� � 8x2K:

To conclude this section, let us now state first-order necessary optimality

conditions. We introduce the Lagrange function L : RM�Mð ��Þ ! R for

Problem (P) by

Lðu,�Þ :¼ f ðuÞ þ

ZK

XMi¼1

uiyiðxÞ � b

!d�ðxÞ:

Then, the following Karush–Kuhn–Tucker conditions are obtained by the theory ofconvex programming problems in Banach spaces (cf. [19]).

PROPOSITION 2.7 Let Assumptions 2.1 and 2.6 be satisfied. If �u is the optimal control

of (P) with associated optimal state �y ¼PM

i¼1 �uiyi, then there exists a regular Borelmeasure ��2MðK Þ such that, with z �u2R

M defined by z �u,i :¼ ð �y� yd, yiÞ, the following

optimality system is satisfied:

� �uþ z �u, v� �uh i þXMi¼1

ZK

ðvi � �uiÞ yi d �� 0 8v2Uad, ð2:2Þ

ZK

�y� bð Þd �� ¼ 0, �� 0:


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3. Finite-element discretization and convergence

In the first part of this section, we will describe the discretization of Problem (P)based on a finite element discretization of the underlying state equation. Next, anintermediate error estimate is derived, followed by a more detailed analysis of thestructure of the discretized problem (Ph). Finally, under certain assumptions, anerror estimate for the controls is derived that corresponds to the finite element errorin the state equation.

3.1. Discretization of the state equation

As a first step, let us discretize the state equation. More precisely, we discretize thestates yi and obtain the approximate states yhi as follows: we consider a family ofmeshes {T h}h40 consisting of triangles T2T h such that

ST2T h T ¼ ��: By

Nh

:¼ fxh j xh is a node of T hg,

we denote the set of nodes defining the triangulation, and for later use we introducethe set of nodes contained in K by

NhK :¼ fxh 2K j xh is a node of T h

g:

For each triangle T2T h, we introduce the diameter �o(T ) of T, and the diameter�i(T ) of the largest circle contained in T. The mesh size h is defined byh ¼ maxT2T h �oðT Þ. We impose the following regularity assumption on the grid:

ASSUMPTION 3.1 There exist positive constants �o and �i such that

�oðT Þ

�iðT Þ� �i and

h

�oðT Þ� �o 8T2T

h

are fulfilled for all h40.

Definition 3.2 Associated with the given triangulation T h, we introduce the discretestate space as the set of piecewise linear and continuous functions

Yh ¼ fvh 2Cð ��Þ j vhjT 2P1ðT Þ 8T2Th, vh ¼ 0 on �g,

where P1(T ) denotes the set of affine real-valued functions defined on T. Moreover,we define the discrete states yhi 2Y

h, i¼ 1, . . . ,M, as the unique function of Yh thatsatisfies

X2j,k¼1

ðajkðxÞ@jyhi , @k�

hÞ þ ða0yhi ,�

hÞ ¼ ðei,�hÞ 8�h 2Yh:

PROPOSITION 3.3 Under Assumptions 2.1 and 3.1, the following accuracy of theapproximation is obtained with a constant c40 not depending on h:

k yhi � yik � ch2, ð3:1Þ

k yhi � yikL1ðK Þ � ch2j log hj: ð3:2Þ


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Proof The first estimate is known, for example, from [20], whereas the secondestimate follows from [21]. g

Note that the error estimate from Proposition 3.3 remains valid for any linearcombinations yu ¼

PMi¼1 uiyi and yhu ¼

PMi¼1 uiy

hi for any fixed u2R

M. We obtain thediscretized problem formulation

ðPhÞ

minu2Uad

f hðuÞ :¼1

2

XMi¼1

uiyhi � yd

��2

þ�

2juj2

subject toPMi¼1

uiyhi ðxÞ � b 8x2K:

8>>><>>>:

The pointwise state constraints are still prescribed in the whole subdomain K ratherthan in finitely many discrete points. We will eventually derive a completelyfinite-dimensional problem formulation in the following. First, however, let usanalyse Problem (Ph) with respect to the existence and uniqueness of solutions aswell as first-order optimality conditions. For that, we define the set of feasiblecontrols for (Ph) by

Uhfeas :¼ u2Uad

��XMi¼1

uiyhi ðxÞ � b 8x2K

( ):

LEMMA 3.4 Under Assumptions 2.1 and 2.6, there exists h040 such that for all h5h0the feasible set Uh

feas is not empty.

Proof The proof follows in a standard way noting that the convex combinationuh :¼ �uþ t(u�� u)2Uad, 05t51 is feasible for (Ph) for all sufficiently small h: InKwe find

XMi¼1

uhi yhi ¼ ð1� tÞ

XMi¼1

�uiyhi þ t

XMi¼1

u�i yhi

¼ ð1� tÞXMi¼1

�uiyi þ tXMi¼1

u�i yi þ ð1� tÞXMi¼1

�uið yhi � yiÞ þ t

XMi¼1

u�i ð yhi � yiÞ

� ð1� tÞbþ tðb� �Þ þ cð1� tÞh2j log hj þ cth2j log hj

� b� t� þ ch2j log hj � b

due to Proposition 3.3. g

Consequently, by standard arguments we obtain that there exists a uniqueoptimal control �uh of Problem (Ph), with associated optimal state

�yh :¼XMi¼1

�uhi yhi :

To formulate first-order necessary optimality conditions, we proceed as in thecontinuous case and define the discrete Lagrange function as

Lhðuh,�hÞ :¼ f hðuÞ þ

ZK

XMi¼1

uhi yhi ðxÞ � b

!d�hðxÞ:


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LEMMA 3.5 Let Assumptions 2.1 and 2.6 be satisfied. For all sufficiently small h40,the Slater point u� with associated state yh� :¼ yhðu�Þ from Assumption 2.6 satisfies

yh�ðxÞ � b��

28x2K,

i.e. u� is also a Slater point for Problem (Ph).

We omit the proof, which follows in a straightforward manner similar to theproof of Lemma 3.4.

PROPOSITION 3.6 Let Assumptions 2.1 and 2.6 be satisfied. If �uh is the optimal controlof (Ph) with associated optimal state �yh ¼

PMi¼1 �uhi y

hi and h is sufficiently small, there

exists a regular Borel measure ��h 2MðK Þ such that, with z �u h2RM defined by

z �uh,i :¼ ð �yh � yd, yhi Þ, the following system is satisfied:

� �uh þ z �uh , v� �uh� �

þXMi¼1

ZK

ðvi � �uhi Þ yhi d ��h � 0 8v2Uad, ð3:3Þ

ZK

�yh � b

d ��h ¼ 0, ��h � 0: ð3:4Þ

3.2. Intermediate convergence analysis

In this part of the section we derive an intermediate error estimate of order hffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

p,

which is derived with the help of a quadratic growth condition. This estimate canlater be improved under an additional assumption, whereas in certain cases it issharp.

3.2.1. Convergence of the controls

We begin this section by constructing auxiliary controls that serve for proving ourfirst convergence result.

LEMMA 3.7 Let �u and �uh be the optimal controls of (P) and (Ph), respectively, and letu� 2Uad be the Slater point from Assumption 2.6. There exist sequences {ut}t(h) andfuhgðhÞ of controls that are feasible for (P

h) and (P), respectively, and that converge to �u

and �uh, respectively, with order h2jlog hj.

Proof Define ut :¼ �uþ t(h)(u�� u) with t(h) tending to zero as h tends to zero, to bedefined below. Obviously, {ut}t(h) converges to �u as h tends to zero, and the order ofconvergence is defined by t(h). For brevity, we write t instead of t(h) in the following.Let us prove the feasibility of ut for (P

h). By the feasibility of �y for (P) and the errorestimate of Proposition 3.3, we obtain

XMi¼1

ut,iyhi ¼ ð1� tÞ

XMi¼1

�uiyi þ ð1� tÞXMi¼1

�uið yhi � yiÞ þ t

XMi¼1

u�i yhi

� ð1� tÞbþ ð1� tÞch2j log hj þ tXMi¼1

u�i yhi :


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Now, by the Slater point properties of Lemma 3.5, we find that

XMi¼1

ut,iyhi � ð1� tÞbþ ð1� tÞch2j log hj þ t

�b�

�

2

�� b� t

�

2þ ð1� tÞch2j log hj:

Choosing t ¼ tðhÞ ¼ ch2j log hjch2j log hjþ�=2

¼ Oðh2j log hjÞ; we obtainPM

i¼1 ut,iyhi � b. Similarly,

we proceed to show the existence of uh tending to �uh. We take ðhÞ ¼ch2j log hj

ch2j log hjþ�¼ Oðh2j log hjÞ and uh :¼ �uh þ ðhÞðu� � �uhÞ, and obtain the existence of

the second sequence. g

THEOREM 3.8 Let �u be the optimal solution of Problem (P) and let �uh be the optimalcontrol for (Ph). Then, with a constant c40 independent of h, there holds

j �u� �uhj � chffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

pfor all sufficiently small h40.

Proof By optimality of �uh for (Ph) and feasibility of ut for (Ph), we obtain

f hð �uhÞ � f hðutÞ � j fhðutÞ � f hð �uÞj þ j f hð �uÞ � f ð �uÞj þ f ð �uÞ:

Then, by the uniform Lipschitz property of f h and the fact that

jut � �uj þ j f hð �uhÞ � f ð �uhÞj � ch2j log hj,

we find

f hð �uhÞ � f ð �uÞ þ c1h2j log hj: ð3:5Þ

Moreover, from the linear quadratic structure of Problem (P), it is clear that aquadratic growth condition of the form

f ð �uÞ � f ðvÞ � !jv� �uj2 8v2Ufeas

holds with an !40. For v :¼ uh from Lemma 3.7 this condition reads f ð �uÞ � f ðuhÞ�!juh � �uj2: From j �uh � uh j � ch2j log hj and f ðuhÞ � f ð �uhÞ � cjuh � �uhj we deduce

f ð �uÞ � f ðuhÞ � !juh � �uj2 � f ð �uhÞ þ c2h

2j log hj �!

2j �uh � �uj2: ð3:6Þ

Combining the inequalities (3.5) and (3.6) yields !j �u� �uhj2� ch2jlog hj, and hence the

assertion. g

We point out that this error estimate is true without any other assumption thanour general Assumption 2.1 and the Slater condition from Assumption 2.6. In orderto improve it, additional assumptions are necessary.

COROLLARY 3.9 As a consequence of the last theorem, we obtain that �yh convergesuniformly to �y in K as h tends to zero.

This follows from (3.2) and the convergence of �uh to �u using the last theorem,noting that k y �u � yh

�uhkL1ðK Þ � k y �u � y �uhkL1ðK Þ þ k y �uh � yh

�uhkL1ðK Þ.

3.2.2. Properties of Problem (Ph)

In order to improve the error estimate from Theorem 3.8, we will have to rely onadditional assumptions on the structure of the active set. To motivate this, weconsider the following lemma.


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LEMMA 3.10 Let the functions ei, i¼ 1, . . . ,M, and a0 be linearly independent on

every open subset of �. Moreover, assume that �y 6¼ 0. Then the active set of �y cannot

contain any open subset of K.

Proof Let us assume the contrary, i.e. there exists an open subset �b�K such that�yðxÞ ¼ b for all x2�b. Then, we obtain

A �y ¼ a0ðxÞb ¼XMi¼1

�uiei 8x2�b:

This yields b¼ 0 as well as �ui¼ 0 for all i¼ 1, . . . ,M by the linear independence

assumption. This in return implies �y ¼ 0, which contradicts our assumption. g

Still, the set of active points might be fairly irregular. We will, however, rely on

the following standard situation.

ASSUMPTION 3.11 The optimal state �y is active in exactly N points �x1, . . . , �xN 2 intK,

i.e. �yð �xj Þ ¼ b. Moreover, there exists 40 such that

�h�,r2 �yð �xj Þ�i � j�j2 8�2R

n8j ¼ 1, . . . ,N: ð3:7Þ

We proceed by exploring some consequences of Assumption 3.11.

LEMMA 3.12 There exists a real number R140 such that for each j2 {1, . . . ,N},

�yðxÞ � b�

4jx� �xj j

2 8x2K with jx� �xj j � R1 ð3:8Þ

is satisfied. Moreover, there exists a �40 such that

�yðxÞ � b� � 8x2K n[Nj¼1

BR1ð �xj Þ: ð3:9Þ

Proof By Taylor expansion, we obtain for a fixed active point �xj, j2 {1, . . . ,N}, and

an x� ¼ �xj þ �ðx� �xj Þ with 05�51,

�yðxÞ ¼ �yð �xj Þ þ hr �yð �xj Þ, x� �xji þ1

2hx� �xj,r

2 �yðx�Þðx� �xj Þi

¼ bþ1

2hx� �xj,r

2 �yð �xj Þðx� �xj Þi þ1

2hx� �xj, ðr

2 �yðx�Þ � r2ð �yð �xj ÞÞÞðx� �xj Þi

� b�

2jx� �xj j

2 þc

2jx� �xj j

�jx� �xj j2

by Assumption 3.11 and the Holder continuity of r2 �y. Notice that r �yð �xj Þ vanishes,

since �xj is a local maximum of �y. Hence, there exists a real number Rj40 such that

estimate (3.8) is satisfied. Outside of all the balls BRjð �xj Þ, �y is inactive by assumption.

Since the problem admits only finitely many active points, we define R1 as the

minimum over all Rj, and by the continuity of �y as well as Assumption 3.11 we can

conclude that there exists a �40 such that (3.9) is satisfied. g

LEMMA 3.13 There exists h040 such that, for all h� h0, we have

�yhðxÞ � b� �=2 8x2K n[Nj¼1

BR1ð �xj Þ: ð3:10Þ


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Moreover, if �xhj 2BR1ð �xj Þ is an active point of the optimal state �yh of (Ph), there exists a

constant c40 such that

j �xhj � �xj j � ch12j log hj

14: ð3:11Þ

Proof The first inequality follows directly from the uniform convergence stated inCorollary 3.9 and from Assumption 3.11 on the structure of the active set. Thisimplies that the discrete state can only be active in a neighbourhood of thecontinuous active points �xj with radius R1, j¼ 1, . . . ,N. To prove the secondestimate, we assume that �xhj 2BR1

ð �xj Þ is an active point of (Ph) and observe

b ¼XMi¼1

�uhi yhi ð �x

hj Þ ¼

XMi¼1

�uhi yið �xhj Þ þ

XMi¼1

�uhi yhi ð �xhj Þ � yið �x

hj Þ

�

�XMi¼1

�uiyið �xhj Þ þ

XMi¼1

ð �uhi � �uiÞ yið �xhj Þ þ ch2j log hj

� �yð �xhj Þ þ chffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

pþ ch2j log hj � b�

4j �xhj � �xj j

2 þ chffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

pby Proposition 3.3, Theorem 3.8 and estimate (3.8). It follows that

j �xj � �xhj j � ch12j log hj

14,

which implies the assertion. g

We now improve estimate (3.11) for the distance of the discrete and continuousactive points. This is the main ingredient in the final error estimate for the control.We split this proof into several parts.

LEMMA 3.14 Let �uh denote the discrete optimal control and let ~yh be the auxiliaryfunction defined by

~yh :¼XMi¼1

�uhi yi:

Then, for each active point �xj, j¼ 1, . . . ,N, of the original problem (P), there exists aunique local maximum ~xhj 2BR1

ð �xj Þ of ~yh for h and R1 sufficiently small. Moreover, theestimate

j �xj � ~xhj j � chffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

pis satisfied for a constant c40 independent of h.

Proof We define Fðx, uÞ :¼PM

i¼1 uiryiðxÞ and note that Fð �xj, �uÞ ¼ 0 for allj¼ 1, . . . ,N, since �y admits for all j a local maximum in �xj due to Assumption3.11. Moreover, by the same assumption, we know that the matrix @F

@x ð �xj, �uÞ ¼PMi¼1 �uir

2yið �xj Þ is not singular. Hence, by applying the implicit function theorem, weobtain the existence of �, , c40 such that for all u2R

M with ju� �uj � , there existsa unique ~xj ðuÞ 2B�ð �xj Þ with Fð ~xj ðuÞ, uÞ ¼ 0 and j ~xj ðuÞ � �xj j � cju� �uj: Applying thisto u :¼ �uh yields the existence of ~xhj :¼ ~xj ð �u

hÞ with

j ~xhj � �xj j � chffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

p,


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by the convergence result of Theorem 3.8. For h small enough, we hence have~xhj 2BR1

ð �xj Þ. Moreover, we obtain

�r2 ~yhð ~xhj Þ ¼ �XMi¼1

�uhi r2yið ~x

hj Þ

¼ �XMi¼1

�uir2yið ~x

hj Þ �

XMi¼1

ð �uhi � �uiÞr2yið ~x

hj Þ: ð3:12Þ

Multiplying this equation from the left- and right-hand side by �2R2, the first item

in (3.12) can be estimated from below by j�j2 by inequality (3.7). Moreover, from

Lemma 3.14 we know that ~xhj tends to �xj as h tends to zero, from which we conclude

that r2yið ~xhj Þ is bounded. Hence, with the convergence result of Theorem 3.8, we

obtain for the second term in (3.12) that it can be estimated by

�ð �uhi � �uiÞh�,r2yið ~x

hj Þ�i � �ch

ffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

pj�j2. Combining both estimates yields that

�h�,r2 ~yhð ~xhj Þ�i � ð � chffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

pÞj�j2 �

2j�j2 ð3:13Þ

is satisfied for h sufficiently small. This implies the coercivity of the Hessian matrix

�r2 ~yhð ~xhj Þ so that ~yh admits a strict local maximum in ~xhj . Using the coercivity

derived above, there is a small ball around ~xhj such that this local maximum is unique

in this ball. Without the limitation of generality we can assume that R1 was taken

small enough such that this also holds in BR1ð �xj Þ. g

We point out that ~yh may violate the constraints.

LEMMA 3.15 There exist positive real numbers R2 and c such that for all sufficiently

small h the auxiliary function ~yh defined in Lemma 3.14 satisfies the following

properties:

~yhðxÞ � bþ ch2j log hj for all x2K ð3:14Þ

~yhðxÞ � b��

2for all x2K n

[Nj¼1

BR1ð �xj Þ ð3:15Þ

~yhðxÞ � ~yhð ~xhj Þ �

8jx� ~xhj j

2 for all x2BR2ð �xj Þ, j ¼ 1, . . . ,N: ð3:16Þ

Proof The proof is straightforward. We observe that

~yhðxÞ ¼XMi¼1

�uhi yiðxÞ ¼XMi¼1

�uhi yhi ðxÞ þ

XMi¼1

�uhi ð yiðxÞ � yhi ðxÞÞ

¼ �yhðxÞ þXMi¼1

�uhi ð yiðxÞ � yhi ðxÞÞ � bþ ch2j log hj

by Theorem 3.8, which proves (3.14). From �uh! �u we also know that ~yh converges

uniformly towards �y as h tends to zero. Hence, we obtain (3.15) as an analogue to

(3.9) and (3.10). Now, notice that r ~yhð ~xhj Þ ¼ 0, since ~yh admits a maximum in all of


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the ~xhj . Then, by Taylor expansion in ~xhj we obtain

~yhðxÞ ¼ ~yhð ~xhj Þ þ1

2hx� ~xhj ,r

2 ~yhðx j Þðx� ~xhj Þi

� ~yhð ~xhj Þ þ1

2hx� ~xhj ,r

2 ~yhðxhj Þðx� ~xhj Þi þc

2jx� ~xhj j

�jx� ~xhj j2,

with some x j ¼ xþ ð ~xhj � xÞ, 2 (0, 1), and �2 (0, 1) by Holder continuity of r2 ~yh.Hence, invoking the coercivity of �r2 ~yð ~xhj Þ from inequality (3.13), we obtain theexistence of a sufficiently small real number R240 not depending on h such that forarbitrary x2K with jx� ~xhj j � R2

~yhðxÞ � ~yhð ~xhj Þ �

8jx� ~xhj j

2

if h is small enough. g

LEMMA 3.16 Let �uh be optimal for (Ph). Then, for any discrete active point�xhj 2BR1

ð �xj Þ and the associated ~xhj maximizing ~yh, we obtain for some c40

j �xhj � ~xhj j � chffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

p:

Proof Note that by Lemmas 3.13 and 3.14 we find with the help of �xj that

j �xhj � ~xhj j � j �xhj � �xj j þ j �xj � ~xhj j � ch

12j log hj

14

is satisfied. Therefore, we have �xhj 2BR2ð ~xhj Þ for h small enough. We observe that

�yhðxÞ ¼ ~yhðxÞ þXMi¼1

�ui,hð yhi ðxÞ � yiðxÞÞ � ~yhðxÞ þ ch2j log hj 8x2K,

from which we deduce that �yhðxÞ ¼ b can only hold for x2K, if

~yhðxÞ � b� ch2j log hj

holds. By the uniform estimate ~yhðxÞ � b� �=2 stated in Lemma 3.15, this can onlybe true inside the balls BR1

ð �xj Þ. If now �xhj 2K is an arbitrary active point of �yh inBR1ð �xj Þ, we obtain from the last inequality of Lemma 3.15 that a necessary condition

for �yhð �xhj Þ ¼ b is given by

b� ch2j log hj � ~yhð �xhj Þ � ~yhð ~xhj Þ �

8j �xhj � ~xhj j

2,

which yields that

j �xhj � ~xhj j2 � ch2j log hj þ

8

ð ~yhð ~xhj Þ � bÞ

� ch2j log hj þ8

XMi¼1

�uhi yið ~xhj Þ � b

!

� ch2j log hj þ8

XMi¼1

�uhi yhi ð ~x

hj Þ � bþ

XMi¼1

�uhi ð yið ~xhj Þ � yhi ð ~x

hj ÞÞ

!

� ch2j log hj:

The last inequality follows from Proposition 3.3 and the fact that �yhð ~xhj Þ � b. g


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Note that we have not yet discussed conditions that guarantee the existence of

points where the discrete state �yh is active. We therefore proceed by introducing the

concept of strong activity of constraints, which is defined via positivity of the

associated Lagrange multipliers. With respect to the state constraints, let us first

point out that, under Assumption 3.11, any Lagrange multiplier �� associated with

Problem (P) has the form �� ¼PN

j¼1 ��j� �xj , where � �xj denotes the Dirac measure at the

active point �xj. However, �� need not be unique, and hence the associated coefficients��j 2R need not be unique. In what follows, we identify the multiplier ��2MðK Þ withthe associated N-vector of coefficients, which we denote by �� as well.

Remark 1 The discussion of whether or not Lagrange multipliers are uniquely

determined can be quite involved. We do not comment on this further, since we will

not make use of unique dual variables in the following. When imposing assumptions

on the multipliers in the following, we will simply do so for all of them.

With Remark 1 in mind, we say that a state constraint is strongly active in an

active point �xj, j2 {1, . . . ,N}, if the associated component of any corresponding

Lagrange multiplier is strictly positive. Likewise, it is possible to consider strongly

active control constraints. For that purpose, we introduce nonnegative Lagrange

multipliers ��a, ��b 2Rn for the control constraints on the continuous level, rather than

expressing the constraints by the set of admissible controls, Uad. With these

multipliers the KKT conditions admit the form

� �ui þ ð �y� yd, yiÞ þXNj¼1

��jyið �xj Þ þ ��b,i � ��a,i ¼ 0,

ðua � �uiÞ ��a,i ¼ ð �ui � ubÞ ��b,i ¼ 0,

for all i¼ 1, . . . ,M instead of the variational inequality (2.2) for �u. We point out that

just like ��, the multipliers ��a, ��b need not be unique, either. This becomes clear noting

that ��a, ��b depend on ��, and �� is not necessarily unique. Then, similar to the state

constraints, we call a control constraint strongly active in a component of �u if the

associated component of all corresponding Lagrange multipliers is strictly positive.We define the index set of the strongly active control constraints associated

with �u by

A �u ¼ i2 f1, . . .M g j ��a,i 4 0 or ��b,i 4 0 for all Lagrange multipliers ��a and ��b� �

,

and denote by MA¼ ]A�M the number of strongly active control constraints. By

the index set

I �u ¼ f1, . . .Mg n A �u,

we cover the remaining (inactive or weakly active) constraints. Accordingly, we say

that the state constraint b is strongly active in one of the finitely many active �xj 2K,

j¼ 1, . . . ,N, if �yð �xj Þ ¼ b and all possible associated Lagrange multipliers ��j are

strictly positive, i.e. ��j 4�0 for some �040. We then call �xj 2K a strongly active

point of Problem (P), and define NA�N as the number of strongly active points of

Problem (P). Moreover, let

A �y ¼ f j2 f1, . . . ,Ng j ��j 4�0 for all Lagrange multipliers ��g


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denote the index set for the strongly active state constraints, and define

I �y ¼ f1, . . .Ng n A �y:

Next, we make use of the strong activity property to show the existence of activepoints of �yh in the neighbourhoods of all strongly active points �xj, j2A �y, for allsufficiently small h. To prepare this statement, we proceed with an intermediateresult.

LEMMA 3.17 Let {hn}n2N be a sequence of positive real numbers converging to zeroas n tends to infinity. Any sequence f ��hngn2N of Lagrange multipliers for (Ph) isbounded in M(K).

Proof This is a standard conclusion from the Slater condition. The proof is givenfor convenience. We have already pointed out in Lemma 3.5 that the Slater point u�

is also a Slater point for Problem (Ph). For simplicity, we omit the subscript n in hnand the associated optimal controls, states and Lagrange multipliers. Insertingu� 2Uad into the variational inequality (3.3) for �uh and making use of thecomplementary slackness condition (3.4) and the positivity of �h, we find

0 � hrf hð �uhÞ, u� � �uhi þ

ZK

ð yhu� � �yhÞd ��h

¼ hrf hð �uhÞ, u� � �uhi þ

ZK

ð yhu� � bÞd ��h þ

ZK

ðb� �yhÞd ��h

� hrf hð �uhÞ, u� � �uhi ��

2

ZK

1 � d ��h,

for all h sufficiently small, which yields �2RK 1 � d ��h � hrf hð �uhÞ, u� � �uhi:Moreover, the

sequence { �uh} is bounded as h tends to zero. This follows either directly from theboundedness of Uad, or by �40. Therefore, we obtain k ��hkMðK Þ ¼

RK d ��h � 2 c

� : g

LEMMA 3.18 Under Assumptions 2.1–3.11, for each strongly active point �xj, j2A �y,the state �yh has at least one active point �xhj 2BR1

ð �xj Þ i.e. �yhð �xhj Þ ¼ b.

Proof Let {hn}40 be a sequence of mesh sizes converging to zero, and denote by �un,�yn, and ��n the control, state, and an associated Lagrange multiplier associated withhn, respectively. Now let us assume the contrary: then, there exists an index j2A �y

and for all n a positive hn 5 1n such that �ynðxÞ :¼ �yhn ðxÞ5 b holds for all

x2K \ BR1ð �xjÞ. Consequently, we can assume that

�ynðxÞ5 b 8x2BR1ð �xj Þ:

By the complementary slackness condition (3.4), we have

��njBR1 ð �xj Þ¼ 0 ð3:17Þ

for all n. By Lemma 3.17, the sequence f ��ng is bounded in M(K). Therefore, we canselect a sub-sequence converging weakly� to some �2MðK Þ. Let, w.l.o.g., f ��ng bethis sub-sequence. Moreover, we already know that �un! �u in R

M and �yn! �y inC(K). We now verify that � is a Lagrange multiplier associated with �y: We have

hrf hnð �unÞ, v� �uni þ

ZK

ð yv � y �un Þd ��n � 0


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for all v2Uad and all n2N. Passing to the limit yields

hrf ð �uÞ, v� �ui þ

ZK

ð yv � �yÞd� � 0,

i.e. � satisfies the variational inequality (2.2). Moreover, we obviously have � � 0.

Finally, passing to the limit in ZK

ð y �un � bÞd ��n ¼ 0,

the complementary slackness condition is fulfilled by �. Therefore, � fulfils all

conditions to be satisfied by a Lagrange multiplier. Selecting a y2Cð ��Þ with y(x)¼ 1

in BR12

ð �xj Þ and y(x) 0 in K n BR1ð �xj Þ, we findZBR1

2

ð �xj Þ

1 d ��n ¼ 0

for all n by (3.17). Passing to the limit, we find that the restriction of � to BR12

ð �xj Þ

vanishes, contradicting our assumption of strict positivity of all Lagrange multipliers

associated with �yð �xj Þ. Therefore, the assertion of the lemma is obtained. g

Note that the actual number of discrete active points in the neighbourhood of an

associated active point for Problem (P) might be quite large.As a consequence of the last results, we directly deduce the following lemma.

LEMMA 3.19 For any j2A �y, there exists some c40 and at least one point�xhj 2BR1

ð �xj Þ of Problem (Ph) where �yh is active, i.e. �yhð �xhj Þ ¼ b, with

j �xj � �xhj j � chffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

p: ð3:18Þ

Proof The existence of at least one active point �xhj 2BR1ð �xj Þ follows from Lemma

3.18. The estimate (3.18) is a consequence of Lemmas 3.14 and 3.16. g

Let us briefly comment on the further effects of strong activity.

Remark 2 We point out that a strongly active control constraint in a component of�u ensures the existence of an h040 such that for all h5h0 the associated constraint is

active in the discrete optimal control �u. This can be proven similarly to the last

lemma. Hence, the corresponding component of the discrete optimal solution is

known and it is possible to remove such components from the formulation of (P) and

(Ph). Then, a reduced formulation with onlyM�MA control parameters is obtained.

We implicitly assume thatMA5M, i.e. the control constraints are not strongly active

in all components of �u.

LEMMA 3.20 Let Assumptions 2.1, 2.6 and 3.11 be satisfied. Then, the following

inequality holds for M, MA, NA:

NA þMA �M:

Proof From Remark 2 we know that we can disregard the strongly active control

components and consider a problem formulation with only M�MA control

parameters. Then, from [4, Prop. 4.92], we deduce that there exists a Lagrange


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multiplier �2MðK Þ, which is nonnegative in at most M�MA finitely many

components. This implies the assertion noting that strong activity requires positivity

of each Lagrange multiplier associated with �u. g

Note that there might be components of the control �u that are active, but not

strongly active, as well as points �xj, j¼ 1, . . . ,N, where the optimal state �y is only

weakly active. We do not make use of these controls and points, since for each h� h0,

the associated discrete control component may switch between active and inactive, as

well as there may or may not exist active points of �yh in a small neighbourhood of

such �xj.Still, even the structural Assumption 3.11 will not necessarily guarantee a better

error estimate than proven in Theorem 3.8. It is, however, possible to improve the

estimate under yet an additional assumption, as we will see in the next section.

Beforehand, we derive a completely discrete formulation of Problem (P), i.e. a

formulation with the constraints given in only finitely many points.

LEMMA 3.21 The control �uh is optimal for (Ph) if and only if it is optimal for

ðPhÞ

minu2Uad

f hðuÞ :¼1

2

��XMi¼1

uiyhi � yd

��2

þ�

2juj2

subject toPMi¼1

uiyhi ðxj Þ � b 8xj 2N

hK:

8>>><>>>:

Proof The idea of the proof is to show that it is not relevant for the optimality of �uh

whether the constraints are considered in xj 2NhK or in all x2K. In any T�K, we

have �yhðxÞ � b for all x2T if and only if �yhðxj Þ � b for all corners xj of Th, since �yh is

linear in T. The triangles in � nK do not have to be considered. All remaining

triangles T intersect @ K and we can assume that T \ K � K nSN

j¼1 BR1ð �xj Þ for small

h. Therefore, we have �yhðxÞ � b� �=2 for all x2T\K (cf. (3.10)). By the continuity

of �y and the uniform convergence of �yh towards �y, we find that �yhðxÞ � b� �=4 for

x2T nK. Hence, even if constraints are imposed in these triangles lying outside K

they will remain inactive if h is sufficiently small. g

All previously shown convergence and structural results remain valid for Problem

ðPhÞ. For simplicity we will therefore denote ðPhÞ by (Ph) in the following. Note, in

particular, that the discrete active points �xhj from Lemma 3.18 can be assumed to be

nodes. This becomes clear when considering any triangle T�K and a point x2T

with �yhðxÞ ¼ b. Then, by piecewise linearity of �yh and the fact that �yhðxÞ � b for all

x2K, we obtain either �yhðxÞ ¼ b for all x2T, or on an edge of T, or x is a corner

of T. In either case, we obtain �yhðxÞ ¼ b in at least one corner of T.

3.3. Improved error estimate

In this section, we improve the intermediate error estimate from Theorem 3.8 under

an additional assumption.

ASSUMPTION 3.22 If MA and NA denote the number of strongly active control

constraints and strongly active state constraints, respectively, and M is the number of


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controls, the following inequality is fulfilled:

M ¼MA þNA:

Assumption 3.22 implies that there are exactly as many strongly activeconstraints as there are controls (cf. also Lemma 3.20).

We point out that we are interested in a setting where NA40, and implicitlyassume this in the following. Then, we define the NA�NA-matrix Y with entriesYik, jk ¼ yikð �xjkÞ, ik2I �u, jk 2A �y.

THEOREM 3.23 Let �u be the optimal solution of Problem (P), let �uh be optimal for(Ph), and let the Assumptions 2.1–3.22 be satisfied. Moreover, let the matrix Y beregular. Then, there exists h040 such that the following estimate is true for a constantc40 independent of h:

j �u� �uhj � ch2j log hj 8h � h0:

Proof By Remark 2, we know that j �ui � �uhi j ¼ 0 for i2A �u. Now, consider the NA

strongly active points �xjk , jk 2A �y, k¼ 1, . . . ,NA, and for each such point choose oneassociated discrete active point �xhjk 2BR1

ð �xjkÞ, which exists according to Lemma 3.18,fulfilling the error estimate

j �xjk � �xhjk j � chffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

pfrom Lemma 3.19. We obtain

XMi¼1

�uiyið �xjkÞ ¼ b ¼XMi¼1

�uhi yhi ð �x

hjkÞ ¼

XMi¼1

�uhi ð yhi ð �x

hjkÞ � yið �x

hjkÞÞ þ �uhi yið �x

hjkÞ

�,

and hence��XMi¼1

�uiyið �xjkÞ � �uhi yið �xhjkÞ

�� XMi¼1

j �uhi jj yhi ð �x

hjkÞ � yið �x

hjkÞj � ch2j log hj,

for each �xjk , since �uh is bounded and j yhi ð �xhjkÞ � yið �x

hjkÞj � ch2j log hj by Proposition 3.3.

This inequality can be rewritten as

ch2j log hj �XMi¼1

ð �ui � �uhi Þ yið �xjk Þ þ �uhi ð yið �xjkÞ � yið �xhjkÞÞ

��:

We proceed by Taylor expansion of yið �xhjkÞ at �xjk , which yields

ch2j log hj �XMi¼1

ð �ui � �uhi Þ yið �xjk Þ � �uhi ryið �xjkÞð �xhjk� �xjkÞ

��þOðj �xhjk � �xjk j

2Þ

¼XMi¼1

ð �ui � �uhi Þ yið �xjk Þ þ ð �ui � �uhi Þryið �xjkÞð �xhjk� �xjkÞ

��þOðj �xhjk � �xjk j

2Þ

using r �yð �xjk Þ ¼PM

i¼1 �uiyið �xjkÞ ¼ 0 implied by Assumption 3.11. With Theorem 3.8and Lemma 3.19 we obtain

XMi¼1

ð �ui � �uhi Þ yið �xjkÞ

�� ch2j log hj 8jk 2A �y, k ¼ 1, . . . ,NA,


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which implies that

XMi¼1


�� ¼

Xi2A �u

ð �ui � �uhi Þ yið �xjkÞ þXi2I �u


��

� ch2j log hj 8jk 2A �y, k ¼ 1, . . . ,NA: ð3:19Þ

Noting that �ui ¼ �uhi for all i2A �u, we hence obtain

Xi2I �u


�� ch2j log hj:

Defining �uI �u¼ ( �ui1, . . . �uiNA

)T, ik2I �u and �uhI �u¼ ð �uhi1 , . . . , �uhiNA

ÞT, this inequality can be

written as jYð �uI �u� �uhI �u

Þj � ch2j log hj. By the regularity of Y, we obtain

�uI �u� �uhI �u

�� ch2j log hj:

Noting again that �ui ¼ �uhi for all i2A �u, the assertion is obtained from estimate(3.19). g

Assumption 3.22 seems quite restrictive at first glance, and the question isinteresting, whether it is indeed necessary for the optimal error estimate of Theorem3.23. It is, for example, satisfied in cases where no control constraints are active andthe number of strongly active state constraints is equal to the number of controls. In

[16], where a simplified version of the problem setting has been discussed, we haveanalysed simple counter examples with an order of the error lower than h2jlog hj,with and without relation to PDEs. In Section 5, we will show a numerical examplewith more controls than active points, where only the convergence rate of Theorem3.8 can be observed numerically. In addition, we will show experiments that show theimproved error estimate under Assumption 3.22.

4. Boundary control problems

It is fairly obvious that the ideas of the former sections can be extended to the caseof boundary control. Let us briefly sketch the necessary changes. We consider aproblem with controls in a Dirichlet boundary conditions. The Neumann case can be

handled analogously. We discuss the problem

ðPDÞ

minu2Uad

Jð y, uÞ :¼1

2

Z�

ð y� yd Þ2 dxþ

�

2juj2

subject to AyðxÞ ¼ 0 in �

yðxÞ ¼PMi¼1

uieiðxÞ on �,

yðxÞ � b 8x 2K,

8>>>>>><>>>>>>:

where we rely essentially on the assumptions stated in Assumption 2.1, yet with thefollowing adaption.

ASSUMPTION 4.1 The fixed basis functions ei, i¼ 1, . . . ,M, are restrictions to � ofC2-functions gi :O!R defined on an open set O ��.


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This fairly strong requirement is made to simplify the presentation and yet toobtain a sufficiently high order of the error. An L2-error estimate for a semilinearelliptic equation with rough inhomogeneous Dirichlet boundary data was proven in[22] for the particular case of the Laplace operator. The error k y� yhk � chsþ

12 was

proven for given boundary data in L1(�)\Hs(�). This reference also contains aselection of other results on approximation and interpolation.

There are two main differences to the discussion of the distributed problem (P).The first lies in the assumptions ensuring the desired regularity of yu for each vectoru2R

M. The second concerns the error estimates for the finite-element discretizationof the state equation. The following existence and regularity result follow directlyfrom [18].

LEMMA 4.2 Let functions f2C0,�(�), 05�51, and e2C(�) are given. Then thereexists a unique classical solution y2Cð ��Þ \ C2,�ð�Þ to

ðAyÞðxÞ ¼ f ðxÞ in �, yðxÞ ¼ eðxÞ on �: ð4:1Þ

Proof Our domain � is convex and hence satisfies an exterior sphere condition.Therefore, we can apply Theorem 6.13 of [18] to get a unique solution with thesmoothness stated in the theorem. g

From the superposition principle and this result, we now obtain for each u2RM

the existence of a unique state yu ¼ yðuÞ ¼PM

i¼1 uiyi 2Cð��Þ \ C2,�ðK Þ of the state

equation of (PD). Since the states yi, i¼ 1, . . . ,M, are precomputable as for thedistributed control problems, we again arrive at a semi-infinite programmingproblem, given by

ðPDÞ

minu2Uad

f ðuÞ :¼1

2

��XMi¼1

uiyi � yd

��2

þ�

2juj2

subject toPMi¼1

uiyiðxÞ � b 8x2K:

8>>><>>>:

This problem has exactly the same form as the semi-infinite formulation of Problem(P). Hence, all results shown in the previous sections remain valid for this class ofboundary control problems, since none involved a further discussion of theunderlying PDE.

The only difference to the discussion of the distributed control problem (P) arisesfrom the finite element error analysis of the inhomogeneous Dirichlet problem that ismore delicate since it is of non-variational type. Let us first set up the discretizedstate equation. To this aim, we introduce a bilinear form a : H1(�)�H1(�)!R by

a½ y,�� :¼

Z�

X2j,k¼1

ajkðxÞ@jyi@k�dx ¼

Z�

ðAðxÞryðxÞÞ>r�ðxÞdx:

Now consider the boundary value problem (4.1) for f¼ 0 and a given function e thatis the restriction to � of a C2-function g : O!R. Then e is continuous and piecewiseC2. The associated classical solution y of (4.1) is also a very weak solution that isdefined in transposition sense. We can split y as y¼wþ g, where w2H1

0ð�Þ is a weaksolution with homogeneous boundary data defined by

a½wþ g,�� ¼ 0 8�2H10ð�Þ:


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Let gh :¼ ihg be the piecewise linear interpolate of g on ��, with ih : Cð ��Þ ! Yh beingthe interpolation operator. Then we define the finite element approximation of yh

by yh :¼whþ gh, where wh is the unique solution to

a½wh þ gh,�h� ¼ 0 8�h 2Yh:

The same can be done for e :¼PM

i¼1 uiei to approximate the solution of the stateequation of (PD). To be independent of the specific details of the FEM and theassumptions on the smoothness of the ei, let us assume that the associated error canbe estimated by

k yu � yhukCðK Þ þ k yu � yhuk � j�ðhÞj juj 8h5 h0, ð4:2Þ

where � : [0, h0]!Rþ is the expression for the order of the error. Under our strong

assumptions on the ei, we are able to show an order �(h)¼ ch2 jlog(h)j for the error.We shall prove this result in a forthcoming paper.

Assuming the general estimate above, we clearly obtain from the previoustheorems.

THEOREM 4.3 Let �u be the optimal solution of the boundary control problem (PD) andlet �uh be the optimal control for ðPh

DÞ. If the error estimate (4.2) is valid for (PD), thenthere holds with a constant c40 independent of h

j �u� �uhj � cffiffiffiffiffiffiffiffiffi�ðhÞ

p8h2 ð0, h0�:

Let, in addition, Assumptions 2.1–3.22 as well as 4.1 be satisfied and let the(NA,NA)-matrix Y with entries yik, jk ¼ ð yikð �xjkÞÞ, ik 2A �u, jk 2A �y, k¼ 1, . . . ,NA, beregular. If the error estimate (4.2) is valid for (PD), then

j �u� �uhj � c�ðhÞ 8h2 ð0, h0�:

Completely analogous, Neumann boundary control problems can be treated. Forconvenience, we mention only two results on the finite-element approximation. For a2D polygonal domain, a semilinear equation with A¼�D, and boundary data ofH1/2(�), an L2-error estimate of order h2 was obtained in [23]. An L1 estimate of theerror y� yh in C(K) was proven recently in a domain with smooth boundary for alinear elliptic equation in [24]: if the Neumann data belong to L1(�), thenky� yhkC(K)� h2 is obtained for an appropriate definition of boundary triangles.Theorem 4.3 remains true for Neumann boundary control with �(h) chosenaccordingly.

5. Numerical experiments

We provide numerical results that show our proven orders of convergence. Weconsider four examples E1�E4 with different properties, and point out that we focuson distributed control problems, only. Each of the examples is transformed into afinite-dimensional quadratic programming problem of the form ðPhÞ, and solved bythe MATLAB routine quadprog provided by the optimization toolbox. We considereach example on a finite sequence of mesh sizes h, starting with one rough meshwhich is iteratively refined by the refinemesh command in MATLAB.


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5.1. Example E1

Our first example is chosen to show that the error estimate of order hffiffiffiffiffiffiffiffiffiffiffiffiffiffij log hj

pis

sharp in certain situations. We also refer to [16], where we have discussed this by

means of a simple, non-PDE-related example. We present an example with a control

vector from R4:

E1

minu2R4

1

2y� yd�� 2 þ 1

2ju� ud j

2R

4

subject to:

�DyðxÞ ¼X4i¼1

uieiðxÞ in � ¼ ð0, 1Þ � ð0, 1Þ

yðxÞ ¼ 0 on � ¼ @�yðxÞ � 14 8x2K :¼ ½0:1, 0:9� � ½0:1, 0:9�,

8>>>>>>>>><>>>>>>>>>:

with the given data

yd ¼ 8y1 þ 2y2 þ 0:5y3 � 2y4, ud ¼ ½128=15 12=5 113=186 �9=4�>,

and basis functions ei, i¼ 1, . . . , 4, that satisfy �Dyi¼ ei with

y1 ¼ 192x1ðx1 � 1Þx2ðx2 � 1Þðx1 þ x2 � 1Þ2,

y2 ¼ sinð2�x1Þ sinð2�x2Þ,

y3 ¼ ðx22 � x2Þðx

21 � x1Þ,

y4 ¼ sinð2�x1Þ2 sinð2�x2Þ

2:

The exact solution of this problem is not known. Therefore, we compute an

approximate solution u�h for h�� 0.002 using the exact state functions yi given above

and obtain

u�h ¼ ½8:077641, 1:870485, 0:6422584, �2:450455�>:

The associated state y�h is active at x�1 � ð0:796875, 0:796875Þ and almost active at

x�2 � ð0:203125, 0:203125Þ, suggesting that the exact optimal state �y admits two active

points. Hence, in this example the number of controls is greater than the number of

active points, with no control constraints given. For this specific setting, our theory

only provides an order of hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij logðhÞj

p. To measure the rate of convergence we use the

quantity

EOC 0 ¼logðjuh � u�hjÞ � logðju�h � uhref jÞ

logðhÞ � logðhrefÞ,

with href� 0.004 and associated reference solution uhref. In Table 1 and Figure 1 we

present our numerical results, i.e. the error in the control variable for different values

of h and the experimental order of convergence computed for two different initial

meshes. On the first mesh, we observe quadratic convergence, while on the second

mesh the rate of convergence is only linear. On the one hand, this underlines that the

estimate from Theorem 3.8 is sharp in this specific situation, but at the same time it

implies that the actual observed order of convergence also depends on other

properties of the mesh than just the mesh size. This fact is already known from [25],

where the discretization of semi-infinite programming is discussed and a criterion on

the mesh has been shown in a different setting. In Figure 2, we show the


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Table 1. Example E1: convergence for two different sets of meshes.

h juh � u�hj EOC0 h juh � u�hj EOC0

0.1250 0.1288 2.05 0.3653 0.8140 1.260.0625 0.0581 2.27 0.1826 0.3644 1.280.0313 0.0070 2.01 0.0913 0.0811 1.060.0156 0.0017 2.02 0.0456 0.0444 1.130.0078 0.0004 2.03 0.0228 0.0210 1.15

Figure 2. Example E1: �yh (a) and �ph (b) computed at h� 0.016.

10−5

10−4

10−3

10−3

10−2

10−2

10−1

10−1

100(a) (b)

10−2

10−1

100

100 10−2 10−1 100

h

Err

or in

u

Quadratic error ofconvergenceComputed error Computed error

h

Err

or in

u

Linear order ofconvergence

Figure 1. Example E1: convergence for two different sets of meshes.


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approximated optimal state �yh as well as an associated adjoint state �ph that can beintroduced in the optimality conditions in the usual way, cf. also [1], for h� 0.016.The adjoint state clearly indicates that there is only one point where the discrete stateis active, which suggests that the exact optimal state �y probably admits only onestrongly active point.

5.2. Example E2

Now, we consider an example with known exact solution, for which we expect to seethe order h2jlog hj in the control error. The problem involves only one controlparameter, but the associated optimal state admits two active points. The problemreads:

ðE2Þ

minu2R

1

2y� yd�� 2 þ 1

2juj2

subject to:

�1

4�2DyðxÞ ¼ ueðxÞ in � ¼ ð0, 1Þ � ð0, 1Þ

yðxÞ ¼ 0 on � ¼ @�yðxÞ � 1 8x2K :¼ ½0:1, 0:9� � ½0:1, 0:9�:

8>>>>>><>>>>>>:

The example is constructed such that the optimal state and control are given by

�y ¼ sinð2�x1Þ sinð2�x2Þ, �u ¼ 2,

respectively. For that matter, we define the basis function e¼ sin(2�x1) sin(2�x2) aswell as the desired state

yd ðxÞ ¼�yðxÞ þ 4�2 if �yðxÞ � 0

�yðxÞ if �yðxÞ5 0:

�

Clearly, the optimal state �y is active at x1¼ (0.25, 0.25) and x2¼ (0.75, 0.75). Let usmention that in the following computations, the active points do not belong to thenodes of the mesh on any level of discretization. Note that the associated Lagrangemultiplier �� ¼ ��1�x1 þ ��2�x2 fulfilling

R�

�y d �� ¼ 2 is not uniquely determined. Also,from Lemma 3.20 we can expect at most one strongly active point. This is verified inour computations, where we observe that on each refinement level the stateconstraint is active in only one discrete point. For this example, with known solution,we determine the experimental order of convergence by

EOC ¼logðj �u� uh1 jÞ � logðj �u� uh2 jÞ

logðh1Þ � logðh2Þ,

where h1 and h2 are consecutive mesh-sizes. We again neglect the logarithmic term.Numerical results are shown in Table 2 as well as Figure 3. We show the convergencebehaviour for the controls, as well as for the active points. For the controls, weobserve a quadratic order of convergence in accordance with our convergence resultfrom Theorem 3.23, while the distance of the active points converges linearly to zero.In Table 2, we observe that on the first two meshes we find a discrete active point inthe neighbourhood of �x1, while on all finer meshes the discrete state becomes activeclose to �x2. For the convergence plot in Figure 3, we, therefore, only consider the


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values on the finer meshes. In Figure 4 we visualize the optimal computed state �yh

and its associated adjoint state �ph for h¼ 0.04. We clearly see the influence of theactive state constraint at the point xh2 � ð0:740, 0:739Þ, where the associated Lagrangemultiplier component is positive.

5.3. Example E3

Now, we consider an example with additional control constraints:

E3

minu2Uad

1

2y� yd�� 2 þ 1

2ju� ud j

2

subject to:�DyðxÞ ¼ u1e1ðxÞ þ u2e2ðxÞ in � ¼ ð0, 1Þ � ð0, 1ÞyðxÞ ¼ 0 on � ¼ @�yðxÞ � 1 8x2K :¼ ½0:1, 0:9� � ½0:1, 0:9�, Uad ¼ fu2R

2 : u2 � �1g:

8>>>>><>>>>>:

For this example we choose

e1ðx1,x2Þ ¼ 4�2 cosð2�ðx1 � x2ÞÞ and e2ðx1, x2Þ ¼ 4�2 cosð2�ðx1 þ x2ÞÞ:

The desired state yd is defined by

yd ðxÞ ¼ 2 sinð2�x1Þ sinð2�x2Þ,

10−3 10−2 10−1 10010−5

10−4

10−3

10−2

10−1

100(a) (b)

h

Err

or in

u

Quadratic order ofconvergenceComputed error

10−3 10−2 10−110−3

10−2

10−1

h

Err

or in

the

activ

e po

ints

Linear order ofconvergenceComputed error

Figure 3. Example E2: error for the control and the active point.

Table 2. Example E2: convergence of controls and active points.

h j �u� �uhj EOC �xhi j �xhi � �xij

0.1589 0.2977 2.03 (0.290, 0.291) 0.05850.0795 0.0729 2.04 (0.228, 0.229) 0.02940.0397 0.0177 1.98 (0.740, 0.739) 0.01460.0199 0.0045 2.10 (0.742, 0.752) 0.00740.0099 0.0010 2.32 (0.749, 0.753) 0.00340.0050 0.0002 2.10 (0.748, 0.750) 0.0013


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and a control shift ud¼ [1 �1]> is considered. We do not construct exact solutions,but proceed as in Example E1. That is, we compare the solutions computed fordifferent mesh sizes with the computed approximate solution

u�h ¼ ½1:000077, �1:000000�>

for h�� 0.0018. We observe a situation similar to Example E1: The associated statey�h is active at x�1 � ð0:250, 0:250Þ

> and admits another maximum atx�2 � ð0:749, 0:749Þ

>, where the state constraints are almost active. This leads tothe assumption that the exact state �y admits two active points, of whom one isstrongly active. In Figure 5 we observe one active constraint at the approximatesolution �yh for h� 0.029. Moreover, the second component of the optimal control isstrongly active, i.e. u�h,2 ¼ �1 with positive associated Lagrange multiplier, so that weexpect Assumption 3.22 to be fulfilled with NA¼MA¼ 1. Hence, we expect aconvergence rate of h2jlog hj.

Figure 4. Example E2: �yh (a) and �ph (b) computed for h� 0.04.



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As before, we measure the experimental rate of convergence using formula (5.1),with href� 0.0037. The results show a quadratic rate of convergence as expected,listed and visualized in Figure 6.

5.4. Example E4

Finally, we consider the case where the number of controls is equal to the number ofpoints, where the state constraint is (strongly) active in the optimal state �y. Theproblem to be computed is similar to the previous one.

E4

minu2R

2

1

2y� yd�� 2 þ 1

2ju� ud j

2

subject to:

�DyðxÞ ¼ u1e1ðxÞ þ u2e2ðxÞ in � ¼ ð0, 1Þ � ð0, 1Þ

yðxÞ ¼ 0 on � ¼ @�

yðxÞ � 0:01, 8x2K :¼ ½0:1, 0:9� � ½0:1, 0:9�:

8>>>>>>><>>>>>>>:

We choose e1 and e2 as follows:

e1ðxÞ ¼ e�100jx� ~x1j2

with ~x1 ¼ ð1=4, 1=4Þ>

e2ðxÞ ¼ e�100jx� ~x2j2

with ~x2 ¼ ð3=4, 3=4Þ>,

and define yd¼ 200, and ud¼ [2 2]>. Once again, the exact solution is unknown,and we take the computed approximate solution at h�� 0.002 instead, which isgiven by

u�h ¼ ð1:215674, 1:215663Þ>:

We calculate the order of convergence from formula (5.1). The reference solution isthe one computed at href� 0.004. The experimental rate of convergence is presentedin Figure 7. The data reflects an order of convergence close to quadratic order, whichis what we expect since the derived order is of h2jlog(h)j. Figure 8 shows the optimalcomputed state and its corresponding adjoint state for h� 0.029, where the activity

h uh− u∗h EOC´

0.4787 0.0336 1.31

0.2394 0.0774 1.73

0.1197 0.0808 2.09

0.0598 0.0266 2.22

0.0299 0.0065 2.28

0.0150 0.0014 2.3210−2 10−1 100

10−4

10−3

10−2

10−1

h

Err

or in

u


Figure 6. Example E3: convergence of the optimal control.


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of two points in K is clearly visible close to the points x�1 � ð0:263, 0:264Þ andx�2 � ð0:735, 0:736Þ, where the state y�h is strongly active.

References

[1] P. Merino, F. Troltzsch, and B. Vexler, Error estimates for the finite element approximation

of a semilinear elliptic control problem with state constraints and finite dimensional control

space, ESAIM: Math. Model. Numer. Anal. 44(1) (2010), pp. 167–188.[2] R. Reemtsen and J.J. Ruckmann (Eds.), Semi-infinite Programming, Kluwer Academic

Publishers, Boston, 1998.[3] F.G. Vazquez, J.J. Ruckmann, O. Stein, and G. Still, Generalized semi-infinite

programming: A tutorial, J. Comput. Appl. Math. 217 (2008), pp. 394–419.[4] F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer,

New York, 2000.

[5] G. Gramlich, R. Hettich, and E. Sachs, Local convergence of SQP methods in semi-infinite

programming, SIAM J. Optim. 5 (1995), pp. 641–658.


h uh−u∗h EOC´

0.4787 0.2570 1.49

0.2394 0.6785 1.97

0.1197 0.1416 1.91

0.0598 0.0422 1.95

0.0299 0.0088 1.84

0.0150 0.0025 1.8610−3 10−2 10−1 100

10−5

10−4

10−3

10−2

10−1

100

h

Err

or in

u


Figure 7. Example E4: error for the control.


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[6] M. Huth and R. Tichatschke, A hybrid method for semi-infinite programming problems,Methods Oper. Res. 62 (1990), pp. 79–90.

[7] G. Still, Generalized semi-infinite programming: Numerical aspects, Optimization 49(2001), pp. 223–242.

[8] N. Arada, E. Casas, and F. Troltzsch, Error estimates for the numerical approximation of asemilinear elliptic control problem, Comput. Optim. Appl. 23 (2002), pp. 201–229.

[9] E. Casas, Using piecewise linear functions in the numerical approximation of semilinear

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