SECTORIAL PARSITY IN O CONTROL OF ELLIPTIC EQUATIONS · Annular and Sectorial Sparsity in Optimal Control Herzog, Obermeier, Wachsmuth Jacobians of these mappings as J P (r, j) =

ANNULAR AND SECTORIAL SPARSITY INOPTIMAL CONTROL OF ELLIPTIC

EQUATIONS

Roland Herzog∗ Johannes Obermeier† Gerd Wachsmuth‡

March 18, 2014

Optimal control problems are considered with linear elliptic equations inpolar coordinates. The objective contains L1-type norms, which promotesparse optimal controls. The particular iterated structure of these normsgives rise to either annular or sectorial sparsity patterns. Optimality condi-tions and numerical solution approaches are developed.

1 INTRODUCTION

In this paper we consider optimal control problems in which a certain L1-type normof the control appears in the objective. Problems of this type are of interest for at leasttwo reasons. Firstly, the L1 norm of the control is often a natural measure of the controlcost. Secondly, this term promotes sparsely supported optimal controls, i.e., controlswhich are zero on substantial parts of its domain of definition. Consequently, controlactuators need not be placed everywhere, but only where the control is most effective.

Optimal control problems with partial differential equations (PDEs) and sparsity pro-moting terms were first considered in Stadler [2009], who studied optimality condi-tions, parameter dependence and a semismooth Newton method in the convex case

∗Technische Universität Chemnitz, Faculty of Mathematics, Professorship Numerical Mathematics (Par-tial Differential Equations), 09107 Chemnitz, Germany, [email protected],http://www.tu-chemnitz.de/herzog

†Technische Universität Chemnitz, Faculty of Mathematics, Professorship Numerical Mathematics (Par-tial Differential Equations), 09107 Chemnitz, Germany

‡Technische Universität Chemnitz, Faculty of Mathematics, Professorship Numerical Mathematics (Par-tial Differential Equations), 09107 Chemnitz, Germany, [email protected],http://www.tu-chemnitz.de/mathematik/part_dgl/people/wachsmuth

mailto:[email protected]

http://www.tu-chemnitz.de/herzog

mailto:[email protected]

http://www.tu-chemnitz.de/mathematik/part_dgl/people/wachsmuth

Annular and Sectorial Sparsity in Optimal Control Herzog, Obermeier, Wachsmuth

governed by a linear elliptic PDE. A priori and a posteriori error estimates for thiscase were provided in Wachsmuth and Wachsmuth [2011]. In a sequence of papersCasas et al. [2012b,c], the authors proved second-order necessary and sufficient opti-mality conditions for the non-convex case governed by a semilinear elliptic equation,and provided a priori finite element error estimates for different choices of the controldiscretization. More general problem settings involving measure-valued controls wereinvestigated in Clason and Kunisch [2011, 2012]. The subsequent paper Casas et al.[2012a] provides a priori finite element error estimates for the convex case involving alinear elliptic equation.

All of the aforementioned papers consider cases in which there is no apparent prefer-ence for, nor any control over the shape of the support that the optimal control functionmay have. This issue was first addressed in Herzog et al. [2012], where striped (di-rectional) sparsity patterns were enforced by way of the iterated norm L1(L2) in placeof the plain L1 norm. An alternative setting which involves the norm L2(0, T;M(Ω))was studied in Casas et al. [2013] for problems governed by linear parabolic equations,whereM(Ω) is the space of regular Borel measures on Ω.

The novelty of the present paper is that we consider a norm of the control in the ob-jective which promotes annular or sectorial sparsity patterns. To achieve this, we use aniterated norm as in Herzog et al. [2012], but switch to a polar coordinate system. To beprecise, we consider an optimal control problem with the following objective,

12

R∫0

2π∫0

|y− yd|2 dϕ r dr +α

2

R∫0

2π∫0

|u|2 dϕ r dr + β

R∫0

( 2π∫0

|u|2 dϕ)1/2

r dr. (1.1)

The parameters α and β are positive constants, and yd represents a desired state.

The first and second term in (1.1) are standard. The third term is β times the L1 norm (inradial direction) of the L2 norm (in angular direction) of the control. Since the L1 normpromotes the sparsity of what’s inside, we expect to see optimal controls which arezero on entire annuli centered at the origin. This result will follow from the optimalitysystem proved in Section 2, and we refer to this case as the case with annular sparsitypatterns. As a motivation, we mention that annular actuators of piezoelectric type arein use, for instance, in structural health monitoring, to control precision valves, and forthe purpose of sound generation or noise cancellation on vibrating structures, see vanNiekerk et al. [1995], Coorpender et al. [1999], Raghavan and Cesnik [2004], Yeum et al.[2011], Li et al. [2009] and the references therein. These papers describe the optimizationof the actuator geometry as one topic of interest. We emphasize that an approach basedon (1.1) would not fix a priori the optimal design to a single annulus but would find themost effective topology simultaneously with the optimal dimensions.

We mainly elaborate on the annular case in this paper. The reciprocal situation (the secto-rial case), in which the roles of r and ϕ are reversed, is also studied, but presented morebriefly. The material is organized as follows. In the rest of this section, we introduce thenecessary notation and the precise formulation of the annular problem. Then Section 2

2


is devoted to the analysis of the problem, in particular to its optimality conditions. InSection 3 we address the numerical solution of the annular problem by semismoothNewton algorithms, and appropriate finite element discretization. Problems with andwithout control constraints are treated. Finally, we summarize parallel results and pointout the main differences in the proofs for the sectorial case in Section 4. A conclusionand an outlook are given in Section 5.

We anticipate that the discussion of the optimal control problems involving (1.1) astheir objective requires weighted Lebesgue spaces, and the transformation of the stateequation to polar coordinates calls for appropriately transformed Sobolev spaces. Thisis the main point we focus on in our analytic and algorithmic treatment of the problem.We therefore keep all other ingredients of the problem simple, i.e., we consider onlycircular domains Ω = x ∈ R2 : |x| < R, and we deal with Poisson’s equationonly. We comment on some of the numerous interesting extensions of this setting inSection 5.

1.1 NOTATION AND PRELIMINARIES

DOMAINS AND TRANSFORMATION

Our control problem will be posed on the circle Ω = x ∈ R2 : |x| < R, R > 0. Wewill use a transformation to Cartesian coordinates such that the transformed domain isthe rectangle Ω = (0, R)× (0, 2 π).

The transformation from polar coordinates to Cartesian coordinates is given by

(x, y) = P(r, ϕ) = (r cos(ϕ), r sin(ϕ)), (1.2)

see also Figure 1.1. Note that P maps Ω one-to-one and onto the slit domain Ω \

0 R

2π

Ω R

R

Ω

P

P−1

Figure 1.1: The transformation P from polar coordinates to Cartesian coordinates.

[0, R) × 0. The inverse of P is denoted by P−1. For later reference, we state the

3


Jacobians of these mappings as

JP(r, ϕ) =

(cos(ϕ) −r sin(ϕ)sin(ϕ) r cos(ϕ)

), JP−1(x, y) =

( x√x2+y2

y√x2+y2

− yx2+y2

xx2+y2

). (1.3)

Note that we have J−1P = JP−1 P by the inverse function theorem. We also recall that

det JP(r, ϕ) = r and det JP−1(x, y) = (x2 + y2)−1/2.

MEASURES

Due to the transformation P, we will work with two different measures on the rectangleΩ . Both measures will be defined as product measures. We denote by µ and µϕ theLebesgue measures on (0, R) and (0, 2 π), respectively. By µr we denote the measureon (0, R) defined by

µr(A) =∫

Ar dµ(r) (1.4)

for all Lebesgue measurable subsets A ⊂ (0, R). The two product measures we will useon the rectangle Ω are µ× µϕ and µr × µϕ.

Finally, we denote by η the Lebesgue measure on the circle Ω .

LEBESGUE SPACES

We need several (Bochner-)Lebesgue spaces in order to pose our optimal control prob-lem. First, we define the L2 spaces

L2(Ω )def= L2(η) and L2

r (Ω )def= L2(µr × µϕ).

By using substitution, we now show that these spaces are isometrically isomorphic.

Lemma 1.1. The mapping

L2(Ω ) 3 v 7→ v P ∈ L2r (Ω )

is an isometric isomorphism.

Proof. Let v ∈ L2(Ω ) be given. Since P : Ω : Ω \ [0, R) × 0 is bijective anddifferentiable, v P is Lebesgue measurable by [Fremlin, 2003, Thm. 263D(iii)]. Thesubstitution rule shows∫

Ω|v|2 dη =

∫Ω|v P|2 |det JP|d(µ× µϕ),

see [Fremlin, 2003, Thm. 263D(v)]. Using det JP = r yields the isometry of the mapping.

Using the same arguments for the mapping g 7→ g P−1 yields the surjectivity.

4


We observe that the third term in (1.1) is the norm of the Bochner-Lebesgue space

L1,2r (Ω )

def= L1(µr; L2(µϕ)). (1.5)

The dual space of this space is L∞(µr; L2(µϕ)), see [Diestel and Uhl, 1977, Thm. IV.1.1].Since µr and µ possess the same null sets, the space L∞(µr; L2(µϕ)) coincides with

L∞,2(Ω )def= L∞(µ; L2(µϕ))

with the unweighted Lebesgue measure µ instead of µr.

SOBOLEV SPACES

We denote by H10(Ω ) the usual Sobolev space on the circle Ω with homogeneous

Dirichlet boundary conditions incorporated. We define

H10(Ω )

def= v : Ω → R : v P−1 ∈ H1

0(Ω ). (1.6)

By Lemma 1.1, we find H10(Ω ) ⊂ L2

r (Ω ) = L2(µr × µϕ). The following lemma showsthat v ∈ H1

0(Ω ) possesses weak derivatives. Moreover, these weak derivatives can becomputed from the weak derivatives of v P−1 ∈ H1

0(Ω ).

Lemma 1.2. Let v ∈ H10(Ω ) be given. Then, v possesses weak derivatives of first order

and they are∇v(r, ϕ) = J>P (r, ϕ)

[∇(v P−1 )

](P(r, ϕ)).

Proof. Let ψ ∈ C∞0 (Ω ) be given. We find by substitution∫

Ωv∇ψ d(µ× µϕ) =

∫Ω

v P−1 (∇ψ) P−1 |det JP−1 |dη.

Using the chain rule (for classical derivatives), we find

∇(ψ P−1) = J>P−1

[(∇ψ) P−1].

This showsJ−>P−1 ∇(ψ P−1) = (∇ψ) P−1.

5


Since ψ P−1 ∈ C∞0 (Ω ) and v P−1 ∈ H1

0(Ω ), we can use integration by parts. Thisyields∫

Ωv P−1 (∇ψ) P−1 |det JP−1 |dη =

∫Ω

v P−1 [J−>P−1 ∇(ψ P−1)]|det JP−1 |dη

= −∫

Ωdiv

[v P−1 J−>P−1 |det JP−1 |

]ψ P−1 dη

= −∫

Ωv P−1 div

[J−>P−1 |det JP−1 |

]ψ P−1 dη

−∫

ΩJ−>P−1∇(v P−1 ) |det JP−1 |ψ P−1 dη,

where div denotes the row-wise divergence of a matrix. We have

J−>P−1 |det JP−1 | =( x

x2+y2y

x2+y2

− y√x2+y2

x√x2+y2

),

Hence, we finddiv

[J−>P−1 |det JP−1 |

]= 0.

This yields∫Ω

v P−1 (∇ψ) P−1 |det JP−1 |dη = −∫

ΩJ−>P−1∇(v P−1 ) |det JP−1 |ψ P−1 dη.

By using substitution, we get∫Ω

v∇ψ d(µ× µϕ) = −∫

ΩJ>P[∇(v P−1 ) P

]ψ dη

for all ψ ∈ C∞0 (Ω ), which concludes the proof.

1.2 ANNULAR FORMULATION OF THE OPTIMAL CONTROL PROBLEM

STATE EQUATION

For clarity of the presentation, we consider only the case where the state is given as thesolution of Poisson’s equation on the circular domain Ω with distributed control. Thatis, given a control u0 ∈ L2(Ω ), the state y0 ∈ H1

0(Ω ) is the unique solution of∫Ω∇y0 · ∇v0 dη =

∫Ω

u0 v0 dη for all v0 ∈ H10(Ω ). (1.7)

6


Now, we are going to transform this variational problem into a variational problemposed in H1

0(Ω ). To this end, we define the bilinear form a on H10(Ω )

a(y, v) =∫

Ω∇y>

(1 00 r−2

)∇v d(µr × µϕ). (1.8)

Lemma 1.3. Let u ∈ L2r (Ω ) and y ∈ H1

0(Ω ) be given. Then y solves

a(y, v) =∫

Ωu v d(µr × µϕ) for all v ∈ H1

0(Ω ) (1.9)

if and only if y0 := y P−1 solves (1.7) with right-hand side u0 = u P−1.

Proof. The proof is an application of Lemma 1.2, using the definition (1.6) of H10(Ω )

and

J−1P (r, ϕ) =

(cos(ϕ) sin(ϕ)

− sin(ϕ) r−1 cos(ϕ) r−1

), J−1

P (r, ϕ) J−>P (r, ϕ) =

(1 00 r−2

).

The unique solvability of (1.9) in H10(Ω ) follows from the unique solvability of (1.7) in

H10(Ω ). Alternatively, we may introduce the norm

‖v‖H10 (Ω ) =

√a(v, v) = ‖v P−1‖H1

0 (Ω ). (1.10)

It is easy to check that H10(Ω ) endowed with this norm is a Hilbert space. Now, an

application of the Lemma of Lax-Milgram yields the unique solvability of (1.9) directly.

Corollary 1.4. For every u ∈ L2r (Ω ), the state equation (1.9) possesses a unique so-

lution y ∈ H10(Ω ). Moreover, the mapping u 7→ y is continuous from L2

r (Ω ) intoitself.

Remark 1.5. The transformation of the state equation (1.7) to polar coordinates (1.9)is not essential for the analysis of the continuous problem in Section 2. However, itfacilitates significantly the numerical implementation, see Section 3 and in particularrelation (3.6).

STATEMENT OF THE OPTIMAL CONTROL PROBLEM

Using the derived state equation (1.9), we may transform the optimal control problemto Ω . With the objective (1.1) and control constraints, the resulting optimal control

7


problem becomes

Minimize12‖y− yd‖2

L2r (Ω ) +

α

2‖u‖2

L2r (Ω ) + β ‖u‖L1,2

r (Ω )

such that (y, u) satisfy (1.9)and ua ≤ u ≤ ub a.e. on Ω .

(P)

Here, α > 0 and β ≥ 0 are given constants and the bounds ua, ub ∈ L2r (Ω ) are assumed

to satisfy ua < 0 < ub almost everywhere. We define the set of admissible controls as

Uad = u ∈ L2r (Ω ) : ua ≤ u ≤ ub. (1.11)

We will also consider the unconstrained case −ua = ub = ∞. In this case, Uad =L2

r (Ω ).

Note that the assumption ua < 0 < ub is made in order to render u = 0 an admissiblecontrol, i.e., in order to allow the desired sparsity. While for this purpose it would beenough to require ua ≤ 0 ≤ ub, some of the results would have to be modified underthis relaxed assumption. In particular, the uniqueness of λ of Theorem 2.3 would belost, and Corollary 2.4 no longer holds. Due to the non-uniqueness, we also expectmodifications to be necessary in the numerical solution of the optimality conditions.

2 ANALYSIS OF THE ANNULAR FORMULATION

Using standard arguments, one infers the unique solvability of (P), owing to the uni-form convexity of the reduced objective w.r.t. the norm ‖u‖L2

r (Ω ). The optimality con-

ditions will involve the subdifferential of the L1,2r (Ω )-norm on L2

r (Ω ), which will bedetermined using the following lemma.

Lemma 2.1. Let (X, ‖·‖) be a normed linear space and let |·| : X → [0, ∞) be anothernorm on X. Then the subdifferential of |·| at x ∈ X is given by

∂|·|(x) = x? ∈ X? : |x?|? ≤ 1 and 〈x, x?〉 = |x|, (2.1)

where X? is the dual space of X and |·|? : X? → [0, ∞] is defined by

|x?|? = sup|x|≤1〈x, x?〉. (2.2)

Proof. “⊂”: Let x? ∈ ∂|·|(x) be given. That is,

|x|+ 〈y− x, x?〉 ≤ |y| (2.3)

8


holds for all y ∈ X. By taking y = 0 and y = 2 x, we infer

|x| ≤ 〈x, x?〉 and |x| ≥ 〈x, x?〉.

Hence, |x| = 〈x, x?〉. Using this identity in (2.3), we get 〈y, x?〉 ≤ |y| for all y ∈ X,which shows |x?|? ≤ 1.

“⊃”: Let x? ∈ X? be given such that |x?|? ≤ 1 and 〈x, x?〉 = |x|. This shows 〈y, x?〉 ≤|y| for all y ∈ X and hence (2.3) is satisfied.

Note that the assertion of Lemma 2.1 is standard in case ‖·‖ = |·|, see, e.g., [Ioffe andTichomirov, 1979, p. 20].

We are going to apply Lemma 2.1 with the setting

X = L2r (Ω ), ‖·‖ = ‖·‖L2

r (Ω ), |·| = ‖·‖L1,2r (Ω ).

Using the density of L2r (Ω ) in L1,2

r (Ω ), it follows that the dual norm (2.2) is just thenorm of the dual of L1,2

r (Ω ), i.e., the L∞,2(Ω )-norm. Lemma 2.1 now yields

∂‖·‖L1,2r (Ω )(v) =

w ∈ L2

r (Ω ) : ‖w‖L∞,2(Ω ) ≤ 1 and∫

Ωv w d(µr × µϕ) = ‖v‖L1,2

r (Ω )

.

(2.4)Note that these conditions imply that equality holds in the chain of inequalities∫ R

0

∫ 2 π

0v w dϕ dµr ≤

∫ R

0‖v(r, ·)‖L2(µϕ) ‖w(r, ·)‖L2(µϕ) dµr ≤ ‖v‖L1,2

r (Ω )‖w‖L∞,2(Ω ).

By this relation, one obtains the following explicit characterization of the subdifferen-tial.

Lemma 2.2. Let u ∈ L2r (Ω ) be given. Then, λ ∈ ∂‖·‖L1,2

r (Ω )(u) holds if and only if‖λ(r, ·)‖L2(µϕ) ≤ 1 where u(r, ·) ≡ 0,

λ(r, ·) = u(r, ·)‖u(r, ·)‖L2(µϕ)

elsewhere(2.5)

for almost all r ∈ (0, R).

Proof. The case u ≡ 0 follows directly from (2.4).

Let u 6≡ 0 and λ ∈ ∂‖·‖L1,2r (Ω )(u) be given. By (2.4) we obtain ‖λ‖L∞,2(Ω ) = 1 by

Hölder’s inequality. The first assertion in (2.5) follows directly from (2.4). By the calcu-lation following (2.4), we infer∫ 2 π

0u(r, ϕ) λ(r, ϕ)dϕ = ‖u(r, ·)‖L2(µϕ) ‖λ(r, ·)‖L2(µϕ)

9


for almost all r ∈ (0, R). If u(r, ·) 6≡ 0, this equality implies the existence of c(r) ≥ 0with

λ(r, ·) = c(r) u(r, ·)

for almost all r ∈ (0, R). By referring to∫ R

0‖u(r, ·)‖L2(µϕ) ‖λ(r, ·)‖L2(µϕ) dµr = ‖u‖L1,2

r (Ω )‖λ‖L∞,2(Ω )

we find that

u(r, ·) 6≡ 0 implies∣∣‖λ(r, ·)‖L2(µϕ)

∣∣ = ‖λ‖L∞,2(Ω ) = 1 for almost all r ∈ (0, R).

This shows c(r) = ‖u(r, ·)‖−1L2(µϕ)

and, hence, the second relation in (2.5).

Conversely, let λ satisfy (2.5). A direct calculation shows λ ∈ ∂‖·‖L1,2r (Ω )(u), see (2.4).

Theorem 2.3. Let (y, u) be the solution of (P). Then, there exists a unique adjoint statep and a unique subgradient λ ∈ ∂‖·‖L1,2

r (Ω )(u) such that the system

a(v, p)−∫

Ω(yd − y) v d(µr × µϕ) = 0 for all v ∈ H1

0(Ω ) (2.6a)∫Ω

(α u− p + β λ) (u− u)d(µr × µϕ) ≥ 0 for all u ∈ Uad (2.6b)

a(y, v)−∫

Ωu v d(µr × µϕ) = 0 for all v ∈ H1

0(Ω ) (2.6c)

is satisfied.

Proof. The existence of p and λ follows from standard arguments and the Theorem ofMoreau and Rockafellar, see for instance [Ekeland and Temam, 1999, Proposition I.5.6].Moreover, the uniqueness of p follows from (2.6a). On the set where u(r, ·) 6≡ 0,Lemma 2.2 shows that λ is unique. On the complement, (2.6b) shows β λ = p sinceua < 0 < ub. And hence λ is also unique.

Since (P) is convex, the above optimality system is also sufficient.

Corollary 2.4. Let (y, u) be the solution of (P). Denote by p the associated adjoint state.Then

u(r, ·) ≡ 0 ⇔ ‖ p(r, ·)‖L2(µϕ) ≤ β (2.7)

holds for almost all r ∈ (0, R).

10


Proof. Let us denote by λ ∈ ∂‖·‖L1,2r (Ω )(u) the associated subgradient such that the

optimality system (2.6) is satisfied.

Let u(r, ·) ≡ 0 be satisfied for some r ∈ (0, R). By Lemma 2.2 we have ‖λ(r, ·)‖L2(µϕ) ≤ 1and by (2.6b) we find α u(r, ·)− p(r, ·) + β λ(r, ·) = 0 since ua < 0 < ub. Putting thistogether, we get ‖ p(r, ·)‖L2(µϕ) = β ‖λ(r, ·)‖L2(µϕ) ≤ β.

It remains to prove the converse. Let us define N = r ∈ (0, R) : ‖ p(r, ·)‖L2(µϕ) ≤ β.Using the test function (recall ua < 0 < ub)

u(r, ϕ) =

0 r ∈ N,u(r, ϕ) else

in the variational inequality (2.6b) we obtain∫N×(0,2 π)

(α u− p + β λ) u d(µr × µϕ) ≤ 0

Lemma 2.2 implies ∫N×(0,2 π)

λ u d(µr × µϕ) ≥∫

N‖u(r, ·)‖L2(µϕ) dµr.

Hence, we have

0 ≥∫

N×(0,2 π)(α u− p + β λ) u d(µr × µϕ)

≥ α∫

N×(0,2 π)|u|2 d(µr × µϕ)− β

∫N‖u(r, ·)‖L2(µϕ) dµr + β

∫N‖u(r, ·)‖L2(µϕ) dµr.

This shows u(r, ·) ≡ 0 on N.

As expected, (2.7) implies that the optimal control u is sparse. Moreover, we infer theannular sparsity structure, since (2.7) implies that we have u(r, ϕ) ≡ 0 for all ϕ ∈ (0, 2 π)whenever ‖ p(r, ·)‖L2(µϕ) ≤ β holds.

In the unconstrained case, i.e., when Uad = L2r (Ω ) holds (or formally, −ua = ub =

∞), it is straightforward to show that (2.6b) implies α u − p + β λ = 0. In this case,we will exploit the following reformulation of this optimality system numerically, seeSection 3.3.

Lemma 2.5. Let (y, u) be the solution of (P) in the case Uad = L2r (Ω ). Then, there

exists a unique adjoint state p and a unique subgradient λ ∈ ∂‖·‖L1,2r (Ω )(u) such that

(2.6a), (2.6c) and

α u(r, ϕ) = max(

0, 1− β

‖ p(r, ·)‖L2(µϕ)

)p(r, ϕ) a.e. in Ω (2.8)

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are satisfied.

Proof. In view of Theorem 2.3, we only need to show that (2.8) is equivalent to (2.6b).This is done by a distinction of cases.

First, we consider the set N = r ∈ (0, R) : u(r, ·) ≡ 0. As already seen in the proof ofCorollary 2.4, (2.6b) is equivalent to ‖ p(r, ·)‖L2(µϕ) ≤ β on this set. It is easy to see thatalso (2.8) is equivalent to this condition.

On the complement set (0, R) \ N, (2.6b) is equivalent to

α u + βu

‖u(r, ·)‖L2(µϕ)= p (∗)

by Lemma 2.2. Similarly, (2.8) is equivalent to

α u = p− βp

‖ p(r, ·)‖L2(µϕ). (∗∗)

Both (∗) and (∗∗) imply

u‖u(r, ·)‖L2(µϕ)

=p

‖ p(r, ·)‖L2(µϕ).

Now, the equivalence of (2.6b) and (2.8) is easy to see via the equivalence of (∗) and(∗∗).

We remark that (2.8) implies the continuity of u P−1 in Ω if p P−1 is continuous.

We mention that the Newton differentiability in function space of the system (2.6a), (2.8)and (2.6c) can be shown analogously as in [Herzog et al., 2012, Section 3], which justifiesthe use of a semi-smooth Newton method in the case without control constraints. Webriefly sketch the ideas for this case. To this end, let us introduce the (affine) adjoint-state map

P : L2r (Ω ) 3 u 7→ p = P(u) ∈ p ∈ H1

0(Ω ),

which assigns to a given control the unique solution of the adjoint equation through(2.6c) and (2.6a). Then, in view of (2.8), the entire optimality system (2.6) can be formu-lated equivalently as the single condition F(u) = 0, where

F : L2r (Ω ) 3 u 7→ F(u) := u− α−1 G(P(u)) ∈ L2

r (Ω ),

G : L6,2r (Ω ) 3 p 7→ G(p) := max

(0, 1− β

‖p(r, ·)‖L2(µϕ)

)p ∈ L2

r (Ω ).

Here L6,2r (Ω ) denotes the Bochner-Lebesgue space analogous to (1.5). By the stan-

dard embedding H10(Ω ) → L6(Ω ) and a simple adaptation of Lemma 1.1 we get

H10(Ω ) → L6

r (Ω ), and clearly also H10(Ω ) → L6,2

r (Ω ) holds. This shows that F

12


has the mapping properties as claimed above. The Newton differentiability of G isshown in [Herzog et al., 2012, Lemma 3.2]. The Newton differentiability of F and thebounded invertibility of its generalized derivative with ‖F′(u)−1‖L(L2

r (Ω )) ≤ 1 followsas in [Herzog et al., 2012, Lemma 3.6].

3 NUMERICAL REALIZATION AND RESULTS

We address in this section the discretization of the annular optimal control problem andits solution by a semi-smooth Newton method.

3.1 FINITE ELEMENT DISCRETIZATION OF THE FORWARD PROBLEM

We start with a triangular grid of the rectangular domain Ω = (0, R) × (0, 2π). Tosolve the variational formulation of the forward problem (1.9) in a conforming way, weneed to construct a discrete subspace of H1

0(Ω ). To this end, our starting point is aspace of piecewise linear functions vh on Ω with the standard nodal basis. In order toensure the conformity, i.e., vh P−1 ∈ H1

0(Ω ), we need to enforce the continuity of thepiecewise smooth function vh P−1, as well as the essential boundary conditions. Bothcan be achieved by appropriate conditions on the function vh, viz.

1. The Dirichlet conditions on vh P−1 are satisfied if and only if vh(R, ·) = 0 holds.2. Since the points (r, 0) and (r, 2π) are mapped by P to the same point in Ω ,

vh(r, 0) = vh(r, 2π) must necessarily hold for all r ∈ [0, R].3. Similarly, all points (0, ϕ) are mapped by P to the origin in Ω , vh(0, ϕ) must be

independent of ϕ ∈ [0, 2π].Conditions 1. and 3. can be easily formulated by appropriate restrictions on the degreesof freedom describing the function vh. To ensure condition 2. we impose the followingcondition on the mesh on Ω .

We require that the vertices on the lower boundary (r, 0) agree intheir r-coordinate with the vertices (r, 2π) on the upper boundary.

(M)

Then condition 2. is simply realized by a number of equality constraints for the degreesof freedom located in the relative interior of these boundaries. All these constraints aresketched in Figure 3.1. The resulting finite element space is termed Vh.Note that the requirement that the vertices on the lower and upper boundaries bealigned is important for the approximation properties of the ensuing subspaces. Whenthe vertices are not aligned, the continuity condition 2. would imply that r 7→ vh(r, 0) isglobally linear.We replace the variational forward problem (1.9) by its Galerkin approximation, i.e.,find yh ∈ Vh such that∫

Ω∇y>h

(r 00 r−1

)∇vh dr dϕ =

∫Ω

u vh r dr dϕ for all vh ∈ Vh. (3.1)

13


const 0= = = =

0 R

2π

Figure 3.1: Conditions on the discrete functions on Ω .

Let us briefly describe how the solution of (3.1) can be achieved in the MATLAB PDEtoolbox. As usual, we denote by [p,e,t] the mesh data. We denote by K the stiff-ness matrix associated with the left hand side of (3.1), and with all degrees of freedompresent and unconstrained, i.e.,

K = assema(p,t,char(’x’,’1./x’) ,0,0);

To incorporate the constraints of type 1.–3. one could set up a matrix N where each rowrepresents one of the constraints. The solution of the state equation in Vh is would thenbe achieved by solving the augmented system[

K N>

N 0

] [yz

]=

[b0

],

where b is the load vector generated by the right hand side in (3.1). Equivalently, withZ being a matrix whose columns span the nullspace of N, we could solve

Z>K Z y = Z>b

and then expand y := Z y. A suitable nullspace basis can be constructed easily. Recallthat each row in Z belongs to one of the degrees of freedom (nodes) in the mesh. Thereare three kinds of columns in Z:

1. each inner node generates a column with exactly one entry ’1’, at the row corre-sponding to its index,

2. each pair of nodes in the interior of the upper and lower boundaries of Ω gen-erates a column with exactly two entries ’1’, at the rows corresponding to theirindices,

3. the set of all nodes on the left boundary r = 0 generates one more column, withentries ’1’ in all rows pertaining to participating nodes.

Each coefficient vector in the range space of Z represents a piecewise linear function vhon Ω w.r.t. the standard nodal basis, and with the property that vh P−1 ∈ H1

0(Ω )holds.

14


3.2 CONVERGENCE OF THE DISCRETIZATION

This subsection is new.In order to show the convergence of the above scheme for the forward problem, we canapply Céa’s Lemma in H1

0(Ω ). To this end, we have to show

infvh∈Vh‖v− vh‖H1

0 (Ω ) → 0 as h→ 0 (3.2)

for all v ∈ H10(Ω ). Let us introduce

C∞0 (Ω ) = v : Ω → R such that v P−1 ∈ C∞

0 (Ω ).

By the definition (1.10) of the norm in H10(Ω ) and the density of C∞

0 (Ω ) in H10(Ω ),

it is sufficient to verify (3.2) for all v ∈ C∞0 (Ω ). Due to the smoothness of P, it is easy

to show that v ∈ C∞0 (Ω ) implies

• v ∈ C∞(Ω ),• v is constant along the line r = 0,• the values of v and its derivatives at (r, 0) coincide with the values at (r, 2 π), for

all r ∈ [0, R],• v is zero in a neighborhood of the line r = R,

see also Figure 3.1. Due to these conditions, the Lagrange interpolation of functions inC∞

0 (Ω ) is well defined for meshes satisfying condition (M).

Lemma 3.1. Let Th be a quasi-uniform family of geometrically conforming triangu-lar meshes on Ω which satisfy condition (M). Let us denote by vh the Lagrange inter-polant of v ∈ C∞

0 (Ω ). Then we have

‖v− vh‖2H1

0 (Ω )≤ C B2 h2 (1 + |ln h|),

where C depends only on c1, c2 and R, and B is the ∞-norm of the second derivativesof v.

Proof. First, we consider an arbitrary triangle with vertices (ri, ϕi), i = 1, 2, 3, such that,w.l.o.g., r1 ≤ r2 ≤ r3. We set

J = det(

r2 − r1 r3 − r1ϕ2 − ϕ1 ϕ3 − ϕ1

)and denote by

λ2(r, ϕ) =1J[(ϕ3 − ϕ1) (r− r1) + (r1 − r3) (ϕ− ϕ1)

]λ3(r, ϕ) =

1J[(ϕ1 − ϕ2) (r− r1) + (r2 − r1) (ϕ− ϕ1)

]λ1(r, ϕ) = 1− λ2(r, ϕ)− λ3(r, ϕ)

15


the barycentric coordinates. Then, vh is given by

vh = λ1 v(r1, ϕ1) + λ2 v(r2, ϕ2) + λ3 v(r3, ϕ3)

and it is easy to verify that

∂vh

∂ϕ=

r3 − r2

Jv(r1, ϕ1) +

r1 − r3

Jv(r2, ϕ2) +

r2 − r1

Jv(r3, ϕ3) (3.3a)

=∂v∂ϕ

(r1, ϕ1) + C B h. (3.3b)

We remark that the quasi-uniformity of the mesh family implies that there exist c1, c2 >0 such that the edge lengths of all triangles in Th belong to [c1 h, c2 h]. In order to es-timate the interpolation error, we distinguish three different cases. We denote by Th,ithe set of all triangles of a given mesh which satisfy the conditions of case #i. In whatfollows, C denotes a generic constant which depends only on the quantities c1, c2, R andwhich may change from line to line.

Case 1: 0 = r1 = r2Referring to (3.3a) and using v(0, ϕ) = 0, we obtain ∂vh/∂ϕ = 0. Moreover, we get∣∣∣∣ ∂v

∂ϕ(r, ϕ)

∣∣∣∣ ≤ ∣∣∣∣ ∂v∂ϕ

(r1, ϕ)

∣∣∣∣+ ∥∥∥ ∂2v∂r ∂ϕ

∥∥∥∞|r− r1| ≤ B r

by a Taylor estimate. This yields for any triangle4 ∈ Th,1∫4

1r

∣∣∣ ∂v∂ϕ

(r, ϕ)− ∂vh

∂ϕ

∣∣∣2 dr dϕ =∫4

1r

∣∣∣ ∂v∂ϕ

(r, ϕ)∣∣∣2 dr dϕ

≤ B2∫4

r dr dϕ ≤ C B2 h3.

Since |ϕ1 − ϕ2| is bounded from below by c1 h, there are at most 2 π/(c1 h) = C/htriangles in Th,1. This shows

∑4∈Th,1

∫4

1r

∣∣∣ ∂v∂ϕ

(r, ϕ)− ∂vh

∂ϕ

∣∣∣2 dr dϕ ≤ C B2 h2.

Case 2: 0 = r1 < r2By (3.3b) we obtain ∣∣∣∣∂vh

∂ϕ

∣∣∣∣ ≤ ∣∣∣∣ ∂v∂ϕ

(r1, ϕ1)

∣∣∣∣+ C B h ≤ C B h

and as in Case 1 ∣∣∣∣ ∂v∂ϕ

(r, ϕ)

∣∣∣∣ ≤ B r ≤ C B h.

16


Let us denote by w(s) the length of the intersection of the triangle with some line r = s.We have∫4

1r

∣∣∣ ∂v∂ϕ

(r, ϕ)− ∂vh

∂ϕ

∣∣∣2 dr dϕ ≤ C B2 h2∫4

1r

dr dϕ = C B2 h2∫ r3

0w(r)

1r

dr ≤ C B2 h3.

Here we used w(r) ≤ C r and r3 ≤ c2 h, which follow from the assumptions on themesh family. Since all triangles in Th,2 lie in the strip [0, c2 h]× [0, 2 π], there are at most(c2 h 2 π)/(β h2) = C/h triangles in this case, where β > 0 is chosen such that the areaof any triangle is bounded from below by β h2. This yields

∑4∈Th,2

∫4

1r

∣∣∣ ∂v∂ϕ

(r, ϕ)− ∂vh

∂ϕ

∣∣∣2 dr dϕ ≤ C B2 h2.

Case 3: 0 < r1Using (3.3b), we get∫

4

1r

∣∣∣ ∂v∂ϕ

(r, ϕ)− ∂vh

∂ϕ

∣∣∣2 dr dϕ ≤ Cr1

∫4

∣∣∣ ∂v∂ϕ

(r, ϕ)− ∂vh

∂ϕ

∣∣∣2 dr dϕ ≤ C B2 h4

r1.

For each triangle in Th,3, there must be another triangle in the mesh which has (r1, ϕ1)as a vertex and lies to the left of (r1, ϕ1). More precisely, this triangle contains a non-trivial portion of the line segment connecting (r1, ϕ1) with the boundary point (0, ϕ1).Due to the assumptions on the mesh regularity, this implies r1 ≥ α h for some α > 0which depends only on c1, c2.Now, the set Th,3 is further divided into subsets, based on the value of r1. In particular,we set

T nh,3 =

4 ∈ Th,3 : r1 ∈ [α n h, α (n + 1) h)

for n = 1, . . . , b R

α hc. Since b Rα hc > R

α h − 1, we get Th,3 =⋃b R

α h cn=1 T n

h,3. Moreover, sinceall triangles in T n

h,3 lie in the strip [α n h, α (n + 1) h + c2 h] × [0, 2 π], there are at most(α + c2) h 2 π/(β h2) = C/h triangles in T n

h,3. We obtain

∑4∈T n

h,3

∫4

1r

∣∣∣ ∂v∂ϕ

(r, ϕ)− ∂vh

∂ϕ

∣∣∣2 dr dϕ ≤ Ch· C B2 h4

α n h≤ C B2 h2

n.

By using an upper bound for the harmonic series

b Rα h c

∑n=1

1n≤ 1 + lnb R

α hc ≤ C + |ln h|

we obtain

∑4∈Th,3

∫4

1r

∣∣∣ ∂v∂ϕ

(r, ϕ)− ∂vh

∂ϕ

∣∣∣2 dr dϕ ≤b R

α h c

∑n=1

∑4∈T n

h,3

∫4

1r

∣∣∣ ∂v∂ϕ

(r, ϕ)− ∂vh

∂ϕ

∣∣∣2 dr dϕ

≤ C B2 h2b R

α h c

∑n=1

1n≤ C B2 h2 (1 + |ln h|).

17


In all cases, a Taylor estimate yields∫4

r∣∣∣∂v

∂r(r, ϕ)− ∂vh

∂r

∣∣∣2 dr dϕ ≤ C B2 h4

Summing over all triangles yields the claim.

As already discussed above, this result implies the convergence of the discretizationscheme of the forward problem as h → 0. Based on this, the convergence of an opti-mal solution in a discretize–then optimize setting can be deduced as well using standardarguments. In the sequel, we will however follow an optimize–then discretize approachfor which the discussion of convergence is more involved and beyond the scope of thispaper.

3.3 DISCRETIZATION AND SOLUTION OF THE UNCONSTRAINEDOPTIMALITY SYSTEM

To complete the discretization of (P), we follow an optimize–then discretize approach. Inthis section, we consider the case without control constraints. We employ piecewiselinear controls on the same triangular grid which we use for the state. Therefore, theright hand side in the discrete state equation (3.1) can be realized by the term M u,where M is mass matrix

[∼,M] = assema(p,t,0,char(’x’) ,0);

We do not impose additional conditions on the discrete control (as we did for the state)because its continuity will follow automatically from the discrete optimality systemgiven below.

To preserve the iterated structure of the term ‖ p(r, ·)‖L2(µϕ) in (2.8), we choose a partic-ular quadrature formula and impose further structural conditions on the mesh. Fromnow on, the mesh vertices are supposed to form a rectangular lattice. This supersedesthe conditions set forth in Section 3.1. For simplicity, we elaborate on the case of con-stant mesh widths hr and hϕ. It will be convenient to address the components of thecoefficient vector u by a double index. The value of the control at (r, ϕ) = (i hr, j hϕ) isdenoted by uij with 1 ≤ i ≤ nr and 1 ≤ j ≤ nϕ.

The discrete state and adjoint state are represented by vectors y and p, which are ex-panded to the full nodal basis by multiplication with Z. Our discrete optimality systemnow consists of the discrete state equation,

Z>K Z y = ZM u, y = Z y, (3.4)

the discrete adjoint equation,

Z>K Z p = −ZM (y− yd), p = Z p, (3.5)

18


and the following discretization of (2.8),

α uij = max

(0, 1− β( nϕ

∑k=1

ωk p2ik

)1/2

)pij =:

[G(p)

]ij pij =

[G(p) p

]ij. (3.6)

Here the weights ωk are equal to hϕ/2 for k ∈ 1, nϕ and ωk = hϕ otherwise. Thevector yd represents the coefficient of a nodal interpolation of the desired state yd. Inthe last term in (3.6), G(p) denotes a vector and is the pointwise product betweenvectors of the same size.

Note that (3.6) implies that the discrete control inherits the continuity of the discreteadjoint state, which in turn follows from the fact that its coefficient vector p is in thenull space of the constraint matrix N, and that the factor in parentheses depends onlyon the index i in radial direction.

We solve (3.4)–(3.6) by a semismooth Newton method. We mention that the matrix inthe resulting Newton system for the update stepZ>M Z 0 Z>K Z

0 α I diag(p)G′(p)Z + diag(G(p))ZZ>K Z −Z>M 0

δyδuδp

= −R(y, u, p) (3.7)

turns out to be non-symmetric, and not obviously symmetrizable. Here G′(p) denotesa Newton derivative of G, and the right hand side is the negative residual of (3.5), (3.6)and (3.4) in this order. Numerical experiments revealed that globalization efforts werenot required for convergence.

3.4 CASE OF POINTWISE CONTROL CONSTRAINTS

The numerical treatment of (P) is more involved in the presence of pointwise inequalityconstraints for the control. The reason is that these constraints are not compatible withthe sparsity structure induced by the iterated norm in the objective. This was observedalready in Herzog et al. [2012]. Following Lemma 4.1 in that paper, we reformulate theoptimality system (2.6) as a non-smooth equation. One can show that the following twostatements are equivalent:

1. λ ∈ ∂‖·‖L1,2r (Ω )(u) and (2.6) holds.

2. (2.6a) and (2.6c) and

− p + α u + β λ + µ = 0 (3.8a)

max(1, ‖λ(r, ·) + c1 u(r, ·)‖L2(µϕ)

)λ− (λ + c1 u) = 0 (3.8b)

µ−max(0, µ + c2 (u− ub)

)−min

(0, µ + c2 (u− ua)

)= 0 (3.8c)

hold for any choice of positive constants c1, c2.

19


We mention that (3.8b) is equivalent to λ ∈ ∂‖·‖L1,2r (Ω )(u), and (3.8c) is equivalent to µ

being an element of the normal cone of Uad at u. This can be shown by a straightforwarddistinction of cases.Unfortunately, both versions of the optimality system have their drawbacks. The firstversion is not directly amenable to numerical treatment due to the variational inequal-ity in (2.6b). For the second version, the viability of the semismooth Newton methodin function space is unknown. Nevertheless, a semismooth Newton type iteration canbe successfully applied in the discrete setting. To this end, we discretize the optimalitysystem in a similar way as before in Section 3.3. The additional variables λ and µ areof the same dimension as the control u in the discrete setting. The term involving the‖·‖L2(µϕ) in (3.8b) is treated in the same way as in (2.8).The derivation of the discrete semismooth Newton system proceeds in the same way asin the unconstrained case, and we refer to [Herzog et al., 2012, Section 4] for details. Aswas observed already there, two modifications should be applied to the Newton systemin order to ensure its well-posedness in each step as well as the global convergence ofthe method in practice.Firstly, note that (3.8b) implies u = 0 on the subset of [0, R] where the ’max’ attainsthe value one. On the other hand, (3.8c) implies u ∈ ua, ub on some subsets of Ω ,which are termed active sets. This may lead to contradictory conditions at intermediateiterations, and a singular Newton matrix ensues. Therefore, we give preference to (3.8b)and modify the determination of the active sets in (3.8c) so that they can only be subsetsof (r, ϕ) : ‖λ(r, ·) + c1 u(r, ·)‖L2(µϕ) < 1.The second modification concerns the linearization of (3.8b), to which a damping isapplied. We refer to [Herzog et al., 2012, eq. (4.10)] for details. We point out that bothmodifications vanish in the limit and they do not impair fast local convergence.In Figure 3.2 we show the optimal control for the sectorial problem (with and withoutcontrol constraints) obtained for the following setting,

α = 0.01, β = 0.15, yd(x, y) = e2 x sin(y π),ua = −1, ub = 1.

(3.9)

In the complementarity formulation (3.8), we used the parameters

c1 =α

βand c2 = 100.

4 SECTORIAL FORMULATION

In this section we point out which changes to (P) are necessary to obtain optimal con-trols with sectorial sparsity patterns. As can be expected, we have to modify the sparsitypromoting term ‖u‖L1,2

r (Ω ) in the objective. To this end, we define the space

L1,2r (Ω )

def= L1(µϕ; L2(µr)).

20


Figure 3.2: Optimal control of the annular sparsity problem (P) without (left plot) andwith control constraints (right plot). The parameters are given in (3.9).

Compared with L1,2r (Ω ) = L1(µr; L2(µϕ)) we have changed the order of integration.

The norm in the space L1,2r (Ω ) is given by

‖u‖L1,2r (Ω ) =

∫ 2π

0

(∫ R

0|u(r, ϕ)|2 dµr

)1/2dµϕ =

∫ 2π

0

(∫ R

0|u(r, ϕ)|2 r dµ

)1/2dµϕ.

The optimal control problem now reads

Minimize12‖y− yd‖2

L2r (Ω ) +

α

2‖u‖2

L2r (Ω ) + β ‖u‖L1,2

r (Ω )

such that (y, u) satisfy (1.9)and ua ≤ u ≤ ub a.e. on Ω .

(P)

Following the analysis in Section 2, we have to identify the dual space of L1,2r (Ω ) and

the subdifferential of its norm in L2r (Ω ). Standard arguments yield that the dual space

is L∞,2r (Ω )

def= L∞(µϕ; L2(µr)), and using Lemma 2.1, we have

∂‖·‖L1,2r (Ω )(v) =

w ∈ L2

r (Ω ) : ‖w‖L∞,2r (Ω ) ≤ 1 and

∫Ω

v w d(µr × µϕ) = ‖v‖L1,2r (Ω )

.

(4.1)Analogously to the computations in the proof of Lemma 2.2, we find that, for any givenu ∈ L2

r (Ω ), the function λ ∈ L2r (Ω ) belongs to the subdifferential ∂‖·‖L1,2

r (Ω )(u) ifand only if

‖λ(·, ϕ)‖L2(µr) ≤ 1 where u(·, ϕ) ≡ 0,

λ(·, ϕ) =u(·, ϕ)

‖u(·, ϕ)‖L2(µr)elsewhere

holds for almost all ϕ ∈ (0, 2π).

The same arguments as used in the proof of Theorem 2.3 show that the optimality sys-tem (2.6) is the same as in the annular formulation, except that now λ ∈ ∂‖·‖L1,2

r (Ω )(u)

21


holds. Following the calculations leading to (2.7), we also obtain the equivalence

u(·, ϕ) ≡ 0 ⇔ ‖ p(·, ϕ)‖L2(µr) ≤ β (4.2)

for almost all ϕ ∈ (0, 2π), which confirms the occurrence of sectorial sparsity patterns.The analog of formula (2.8) in the absence of control constraints reads

α u(r, ϕ) = max(

0, 1− β

‖ p(·, ϕ)‖L2(µr)

)p(r, ϕ) a.e. in Ω . (4.3)

However, (4.3) does not imply the continuity of the (transformed) optimal control u P−1 at the origin, even if p P−1 is continuous. To see this, note that the continuity ofp P−1 implies that p(0, ϕ) is constant, but (4.3) does not imply that u(0, ϕ) is constantw.r.t. ϕ since the max(. . .) term depends on ϕ.

The numerical treatment of the sectorial problem requires only a few changes comparedto the annular variant. The restrictions we impose on the mesh and the discrete stateand adjoint states, as well as the formulation of the discrete state equation are the sameas in Section 3.1. Concerning the numerical algorithm, we simply need to replace (2.8)by (4.3) in the unconstrained case and proceed as before. In the constrained case, wemerely need to replace (3.8b) by

max(1, ‖λ(·, ϕ) + c1 u(·, ϕ)‖L2(µr)

)λ− (λ + c1 u) = 0

and continue as in Section 3.4.

In Figure 4.1 we show the optimal solution for the following data,

α = 0.05, β = 0.02, yd(x, y) = e2 x sin(y π),ua = −1, ub = 1.

(4.4)

In the complementarity formulation, we used the parameters

c1 =α

βand c2 = 10.

5 CONCLUSIONS AND OUTLOOK

We introduced and analyzed convex optimal control problems whose optimal controlsfeature particular spatial sparsity patterns. Optimality systems were derived, whichled to semismooth Newton type methods for the numerical solution. We elaborated onthe cases of annular and sectorial sparsity patterns on circular domains by way of thepolar coordinate transform and an iterated norm in the objective.

The extension to domains which are described by rectangles (R1, R2)× (ϕ1, ϕ2) in polarcoordinates (annular sectors) is straightforward. The conditions set forth in Section 3.1

22


Figure 4.1: Optimal control of the sectorial sparsity problem (P) without (left plot) andwith control constraints (right plot). The parameters are given in (4.4).

to achieve conformity need to be adjusted. In fact, there is no need to use only rectanglesas Ω . And finally, the polar coordinate transformation could be replaced by a generaldiffeomorphism to accommodate domains and sparsity patterns of rather arbitraryshape.

In three dimensions, the variety of choices by which the three coordinates can be groupedinto outer (sparse) and inner coordinates increases from two to six. For example, whenwe use spherical coordinates and place the radial coordinate outside and the two angu-lar coordinates inside, the support of an optimal control will consist of spherical shells.

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Documents

SECTORIAL PARSITY IN O CONTROL OF ELLIPTIC EQUATIONS · Annular and Sectorial Sparsity in Optimal Control Herzog, Obermeier, Wachsmuth Jacobians of these mappings as J P (r, j) =