Optimal control of a non-smooth semilinear elliptic equation
Constantin Christof, Christian Clason, Christian Meyer, Stephan, Walther
TL;DR
This paper investigates optimal control problems involving non-smooth semilinear elliptic equations, establishing differentiability properties, deriving optimality conditions, and demonstrating numerical solution methods.
Contribution
It characterizes the directional differentiability and Bouligand subdifferential of the control-to-state map, and develops numerical methods for solving the resulting optimality conditions.
Findings
Control-to-state map is directionally differentiable.
First-order optimality conditions are derived and interpreted.
Numerical examples show semi-smooth Newton method effectiveness.
Abstract
This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.
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Optimal control of a non-smooth semilinear elliptic equation
Constantin Christof TU Dortmund, Faculty of Mathematics, Vogelpothsweg 87, 44227 Dortmund, Germany, (, , )
Christian Clason University of Duisburg-Essen, Faculty of Mathematics, Thea-Leymann-Str. 9, 45127 Essen, Germany () [email protected]
Christian Meyer11footnotemark: 1 ,111Corresponding author
Stephan Walther11footnotemark: 1
(November 27, 2017)
Abstract
This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.
1 Introduction
In this paper, we consider the following non-smooth semilinear elliptic optimal control problem
[TABLE]
where , , is a bounded domain and is a smooth objective; for precise assumptions on the data, we refer to Section 1.1 below. The semilinear PDE in (P) models the deflection of a stretched thin membrane partially covered by water (see [18]); a similar equation arises in free boundary problems for a confined plasma; see, e.g., [35, 31, 18].
The salient feature of (P) is of course the occurrence of the non-smooth -function in the equality constraint in (P). This causes the associated control-to-state mapping to be non-smooth as well, and hence standard techniques for obtaining first-order necessary optimality conditions that are based on the adjoint of the Gâteaux-derivative of the control-to-state mapping cannot be applied. One remedy to cope with this challenge is to apply generalized differentiability concepts for the derivation of optimality conditions. Such concepts and the resulting conditions can be roughly grouped in two classes:
- (i)
(generalized) directional derivatives, leading to “purely primal” optimality conditions stating that the directional derivatives of the reduced objective in feasible directions are non-negative; 2. (ii)
various notions of subdifferentials, leading to abstract optimality conditions stating that zero is contained in a suitable subdifferential of the reduced objective.
For an introductory treatment of generalized derivatives and their relation in finite and/or infinite dimensions, we refer to, e.g., the textbooks [5, 32, 34, 29]; a more general, infinite-dimensional, treatment can be found in [25]. However, with the exception of the convex setting, concrete (and, in particular, numerically tractable) characterizations of these abstract conditions are only available in a restricted set of situations; see, e.g., [26, 11, 6, 20]. To be more precise, it is frequently unclear if these abstract optimality conditions – whether of type (i) or (ii) – are equivalent to optimality systems involving dual variables. (We recall that even in the convex setting, deriving such systems from conditions of type (ii) requires regularity conditions of, e.g., Slater type, which do not hold in all cases. Such regularity conditions are even more restrictive for Clarke and limiting subdifferentials.) For selected optimal control problems governed by variational inequalities (VIs), purely primal optimality conditions of type (i) have been transformed into optimality systems known as strong stationarity conditions. We refer to [23, 24, 27, 38, 39] for obstacle-type problems, to [12] for static elastoplasticity, to [21] for bilevel optimal control of ODEs, and to [8] for VIs of the second kind. In [22], the equivalence of strong stationarity to purely primal conditions of type (i) is shown for optimal control of non-smooth semilinear parabolic equations. Concerning the comparison of conditions of type (ii) with optimality systems involving dual variables for optimal control of non-smooth problems, the literature is comparatively scarce. Frequently, regularization and relaxation methods, respectively, are used to derive optimality systems, and there are numerous contributions in this field; we only refer to [1, 36, 2, 17, 13, 33] and the references therein. A limit analysis for vanishing regularization then yields an optimality system for the original non-smooth problem which is usually of intermediate strength and less rigorous compared to strong stationarity. A classification of the different optimality systems involving dual variables for the case of optimal control of the obstacle problem can be found in [14, 10]. In [27, 15, 40], optimality conditions for the case of VIs of the first kind are obtained by using limiting normal cones. As shown in [10, Thm. 5.7], this approach does in general not lead to optimality conditions which are more rigorous than what is obtained by regularization (even if the limiting normal cone in the spirit of Mordukhovich is used). On the other hand, it is known for the case of finite-dimensional mathematical programs with equilibrium constraints that under suitable assumptions, the optimality conditions obtained via regularization are equivalent to zero being in the Clarke subdifferential of the reduced objective; see, e.g., [4, Sec. 2.3.3]. The latter two results show that a comparison of an optimality system involving dual variables with conditions of type (ii) is in general far from evident, and we are not aware of any contributions in this direction for the case of optimal control of non-smooth PDEs. This is of particular interest, however, since optimality conditions of type (ii) may be satisfied by accumulation points of sequences generated by optimization algorithms; see [16].
The aim of our paper is to investigate this connection for the particular optimal control problem (P). For this purpose, we turn our attention to the Bouligand subdifferential of the control-to-state mapping, defined as the set of limits of Jacobians of smooth points in the spirit of, e.g., [28, Def. 2.12] or [19, Sec. 1.3]. Note that in infinite-dimensional spaces, one has to pay attention to the topology underlying these limit processes so that multiple notions of Bouligand subdifferentials arise, see Definition 3.1 below. We will precisely characterize these subdifferentials and use this result to interpret the optimality conditions arising in the regularization limit. We emphasize that the regularization and the associated limit analysis are not novel and fairly straightforward. The main contribution of our work is the characterization of the Bouligand subdifferential as a set of linear PDE solution operators; see Theorem 3.28. This characterization allows a comparison of the optimality conditions obtained by regularization with standard optimality conditions of type (ii) (specifically, involving Bouligand and Clarke subdifferentials), which shows that the former are surprisingly strong; cf. Theorem 4.12. On the other hand, it is well-known that one loses information in the regularization limit, and the same is observed in case of (P). In order to see this, we establish another optimality system, which is equivalent to a purely primal optimality condition of type (i). It will turn out that the optimality system derived in this way is indeed stronger than the one obtained via regularization since it contains an additional sign condition for the adjoint state. It is, however, not clear how to solve these strong stationarity conditions numerically. In contrast to this, the optimality system arising in the regularization limit allows a reformulation as a non-smooth equation that is amenable to solution by semi-smooth Newton methods. We emphasize that we do not employ the regularization procedure for numerical computations, but directly solve the limit system instead. Our work includes first steps into this direction, but the numerical results are preliminary and give rise to future research.
Let us finally emphasize that our results and the underlying analysis are in no way limited to the PDE in (P). Instead, the arguments can easily be adapted to more general cases involving a piecewise -function rather than the -function and a (smooth) divergence-gradient-operator instead of the Laplacian. However, in order to keep the discussion as concise as possible and to be able to focus on the main arguments, we restrict the analysis to the specific PDE under consideration.
The outline of the paper is as follows: This introduction ends with a short subsection on our notation and the standing assumptions. We then turn to the control-to-state mapping, showing that it is globally Lipschitz and directionally differentiable and characterizing points where it is Gâteaux-differentiable. Section 3 is devoted to the characterization of the Bouligand subdifferentials. We first state necessary conditions that elements of the subdifferentials have to fulfill. Afterwards, we prove that these are also sufficient, which is by far more involved compared to showing their necessity. In Section 4, we first shortly address the regularization and the corresponding limit analysis. Then we compare the optimality conditions arising in the regularization limit with our findings from Section 3. The section ends with the derivation of the strong stationarity conditions. Section 5 deals with the numerical solution of the optimality system derived via regularization. The paper ends with an appendix containing some technical lemmas whose proofs are difficult to find in the literature.
1.1 Notation and standing assumptions
Let us shortly address the notation used throughout the paper. In what follows, always denotes a bounded domain. By we denote the characteristic function of a set . By we denote the -dimensional Lebesgue measure. Given a (Lebesgue-)measurable function , we abbreviate the set by ; the sets and are defined analogously. Note that in what follows, we always work with the notion of Lebesgue measurability (e.g., when talking about -spaces or representatives), although we could equivalently work with Borel measurability here. As usual, the Sobolev space is defined as the closure of with respect to the -norm. Moreover, we define the space
[TABLE]
Equipped with the scalar product
[TABLE]
becomes a Hilbert space. Here and in the remainder of this paper, denotes the distributional Laplacian. Note that is compactly embedded in since, for any sequence with in , we have
[TABLE]
by the compact embedding of into (cf. [9, Thm. 7.22]). Since is isometrically isomorphic to the subset
[TABLE]
of the separable space , it is separable as well. Note that the solution operator associated with the PDE in (P) is bijective as a function from to (cf. Proposition 2.1 below). The space is thus the natural choice for the image space of the control-to-state mapping appearing in problem (P). If the boundary possesses enough regularity (a -boundary would be sufficient here), then is isomorphic to by the classical regularity theory for the Laplace operator, cf. [9, Lem. 9.17].
With a little abuse of notation, in what follows we will denote the Nemytskii operator induced by the -function (with different domains and ranges) by the same symbol. In the same way, we will denote by the directional derivative of in the point in direction , both considered as a scalar function and as the corresponding Nemytskii operator.
Throughout the paper, we will make the following standing assumptions.
Assumption \thetheorem.
The set , , is a bounded domain. The objective functional in (P) is weakly lower semi-continuous and continuously Fréchet-differentiable.
Note that we do not impose any regularity assumptions on the boundary of .
2 Directional differentiability of the control-to-state mapping
We start the discussion of the optimal control problem (P) by investigating its PDE constraint, showing that it is uniquely solvable and that the associated solution operator is directionally differentiable.
Proposition 2.1**.**
For all , there exists a unique solution to
[TABLE]
Moreover, the solution operator associated with (PDE) is well-defined and globally Lipschitz continuous as a function from to .
Proof 2.2**.**
The arguments are standard. First of all, Browder and Minty’s theorem on monotone operators yields the existence of a unique solution in . If , then a simple bootstrapping argument implies . To prove the Lipschitz continuity of the solution mapping , we consider two arbitrary but fixed with associated solutions and . Using again the monotonicity, we obtain straightforwardly that holds with some absolute constant . From (PDE) and the global Lipschitz continuity of the -function, we now infer
[TABLE]
The above shows that is even globally Lipschitz as a function from to and completes the proof.
Theorem 2.3** (directional derivative of ).**
Let be arbitrary but fixed, set , and let be the unique solution to
[TABLE]
Then it holds
[TABLE]
In particular, the solution operator associated with (PDE) is Hadamard directionally differentiable with in all points in all directions .
Proof 2.4**.**
First observe that for every , (1) admits a unique solution by exactly the same arguments as in the proof of Proposition 2.1. Note moreover that (1) is equivalent to
[TABLE]
Now let be arbitrary but fixed and let and be sequences with and in . We abbreviate . Subtracting the equations for and from the one for yields
[TABLE]
Testing this equation with and using the monotonicity of the -operator, we obtain that there exists a constant independent of with
[TABLE]
Now the compactness of and the directional differentiability of (which directly follows from the directional differentiability of and Lebesgue’s dominated convergence theorem) give
[TABLE]
As is also Lipschitz continuous and thus Hadamard-differentiable, (3) implies
[TABLE]
Hence, (2) yields that the sequence is bounded in and thus (possibly after transition to a subsequence) converges weakly in . Because of (3), the weak limit is zero and therefore unique so that the whole sequence converges weakly to zero. This implies the first assertion. If now converges strongly to in , then (2), (3) and (4) yield in . From the definition of the norm , it now readily follows that in . This establishes the second claim.
Theorem 2.3 allows a precise characterization of points where is Gâteaux-differentiable. This will be of major importance for the study of the Bouligand subdifferentials in the next section.
Corollary 2.5** (characterization of Gâteaux-differentiable points).**
Let be arbitrary but fixed. Then the following are equivalent:
- (i)
* satisfies .* 2. (ii)
* is Gâteaux-differentiable in , i.e., .* 3. (iii)
* holds for all .*
If one of the above holds true, then the directional derivative in a direction is uniquely characterized as the solution to
[TABLE]
Proof 2.6**.**
In view of (1), it is clear that if , then is Gâteaux-differentiable in and the Gâteaux derivative is the solution operator for (5). Further, (ii) trivially implies (iii). It remains to prove that (iii) implies (i). To this end, we note that if for all , then (1) implies
[TABLE]
for all . Consider now a function with in and in , whose existence is ensured by Lemma A.1. Since was arbitrary, we are allowed to choose
[TABLE]
such that by virtue of (1). Consequently, we obtain from (6) that . Since in , this yields as claimed.
3 Bouligand subdifferentials of the control-to-state mapping
This section is devoted to the main result of our work, namely the precise characterization of the Bouligand subdifferentials of the PDE solution operator from Proposition 2.1.
3.1 Definitions and basic properties
We start with the rigorous definition of the Bouligand subdifferential. In the spirit of [28, Def. 2.12], it is defined as the set of limits of Jacobians of differentiable points. However, in infinite dimensions, we have of course to distinguish between different topologies underlying this limit process, as already mentioned in the introduction. This gives rise to the following
Definition 3.1** (Bouligand subdifferentials of ).**
Let be given. Denote the set of smooth points of by
[TABLE]
In what follows, we will frequently call points in Gâteaux points.
- (i)
The weak-weak Bouligand subdifferential of in is defined by
[TABLE]
- (ii)
The weak-strong Bouligand subdifferential of in is defined by
[TABLE]
- (iii)
The strong-weak Bouligand subdifferential of in is defined by
[TABLE]
- (iv)
The strong-strong Bouligand subdifferential of in is defined by
[TABLE]
Remark 3.2**.**
Based on the generalization of Rademacher’s theorem to Hilbert spaces (see [23, Thm. 1.2]) and the generalization of Alaoglu’s theorem to the weak operator topology, one can show that and are non-empty for every ; see also [7]. In contrast to this, it is not clear a priori if and are non-empty, too. However, Theorem 3.28 at the end of this section will imply this as a byproduct.
From the definitions, we obtain the following useful properties.
Lemma 3.3**.**
- (i)
For all it holds
[TABLE] 2. (ii)
If is Gâteaux-differentiable in , then it holds . 3. (iii)
For all and all , it holds
[TABLE]
where is the Lipschitz constant of .
Proof 3.4**.**
Parts (i) and (ii) immediately follow from the definition of the Bouligand subdifferentials (to see (ii), just choose for all ). In order to prove part (iii), observe that the definition of implies the existence of a sequence of Gâteaux points such that in and in for all . For each , the global Lipschitz continuity of according to Proposition 2.1 immediately gives . Consequently, the weak lower semi-continuity of the norm implies
[TABLE]
This yields the claim.
Remark 3.5**.**
The Bouligand subdifferentials and do not change if the condition “ in for all ” in Definition 3.1(i) and (iii) is replaced with either “ in for all ” or “ in for all ”, where is a normed linear space such that is compactly embedded into , e.g., or . This can be seen as follows: Suppose that a sequence is given such that holds in for all . Then, by Lemma 3.3(iii), we can find for every a subsequence such that converges weakly in . From the weak convergence in , we obtain that the weak limit has to be equal to independently of the chosen subsequence. Consequently, in for all , and we arrive at our original condition. If, conversely, we know that holds in for all , then, by the compactness of the embedding , it trivially holds in for all . This yields the claim. The case of in proceeds analogously.
Next, we show closedness properties of the two strong subdifferentials.
Proposition 3.6** (strong-strong-closedness of ).**
Let be arbitrary but fixed. Suppose that
- (i)
* and for all ,* 2. (ii)
* in ,* 3. (iii)
, 4. (iv)
* in for all .*
Then is an element of .
Proof 3.7**.**
The definition of implies that for all , one can find a sequence of Gâteaux points with associated derivatives such that in as and
[TABLE]
Since is separable, there exists a countable set that is dense in . Because of the convergences derived above, it moreover follows that for all , there exists an with
[TABLE]
Consider now a fixed but arbitrary , and define
[TABLE]
Then the density property of implies in as , and we may estimate
[TABLE]
where the boundedness of follows from Lemma 3.3(iii). The above proves that for all , we have in . Since was arbitrary and the Gâteaux points satisfy in as by (7) and our assumptions, the claim follows from the definition of .
Proposition 3.8** (strong-weak-closedness of ).**
Let be arbitrary but fixed. Assume that:
- (i)
* and for all ,* 2. (ii)
* in ,* 3. (iii)
, 4. (iv)
* in for all .*
Then is an element of .
Proof 3.9**.**
As in the proof before, for all the definition of implies the existence of a sequence of Gâteaux points with associated derivatives such that in as and
[TABLE]
Now the compact embedding of in gives that in as , and we can argue exactly as in the proof of Proposition 3.6 to show that there is a diagonal sequence of Gâteaux points such that in and
[TABLE]
On the other hand, by Lemma 3.3(iii), the operators are uniformly bounded in . Therefore, for an arbitrary but fixed , the sequence is bounded in , so that a subsequence converges weakly to some . Because of (8), and the uniqueness of the weak limit implies the weak convergence of the whole sequence in . As was arbitrary, this implies the assertion.
3.2 Precise characterization of the Bouligand subdifferentials
This section is devoted to an explicit characterization of the different subdifferentials in Definition 3.1 without the representation as (weak) limits of Jacobians of sequences of Gâteaux points. We start with the following lemma, which will be useful in the sequel.
Lemma 3.10**.**
Assume that
- (i)
* is monotonically increasing and globally Lipschitz continuous,* 2. (ii)
* is a sequence with ,* 3. (iii)
* is a sequence satisfying a.e. in for all and in for some ,* 4. (iv)
* is the unique solution to*
[TABLE] 5. (v)
* is the unique solution to*
[TABLE]
Then it holds that in , and if we additionally assume that pointwise a.e. and strongly in , then we even have strongly in .
Proof 3.11**.**
First note that due to the monotonicity and the global Lipschitz continuity of , the equations (9) and (10), respectively, admit unique solutions in by the same arguments as in the proof of Proposition 2.1. Moreover, due to the weak and weak- convergence, the sequences and are bounded in and , respectively, so that is bounded in . Hence there exists a weakly converging subsequence – w.l.o.g. denoted by the same symbol – such that in and, by the compact embedding , strongly in . Together with the weak convergence of , this allows passing to the limit in (9) to deduce that satisfies
[TABLE]
As the solution to this equation is unique, we obtain . The uniqueness of the weak limit now gives convergence of the whole sequence, i.e., in .
To prove the strong convergence under the additional assumptions, note that the difference satisfies
[TABLE]
For the first term on the right-hand side of (11), we have in by assumption. The second term in (11) is estimated by
[TABLE]
The first term in (12) converges to zero due to in and the compact embedding, while the convergence of the second term follows from the pointwise convergence of in combination with Lebesgue’s dominated convergence theorem. The global Lipschitz continuity of and the strong convergence of in finally also give in . Therefore, the right-hand side in (11) converges to zero in . As induces the norm on , we thus obtain the desired strong convergence.
By setting and , we obtain as a direct consequence of the preceding lemma the following weak continuity of .
Corollary 3.12**.**
The solution operator is weakly continuous, i.e.,
[TABLE]
We will see in the following that all elements of the subdifferentials in Definition 3.1 have a similar structure. To be precise, they are solution operators of linear elliptic PDEs of a particular form.
Definition 3.13** (linear solution operator ).**
Given a function with , we define the operator to be the solution operator of the linear equation
[TABLE]
We first address necessary conditions for an operator in to be an element of the Bouligand subdifferentials. Afterwards we will show that these conditions are also sufficient, which is more involved compared to their necessity.
Proposition 3.14** (necessary condition for ).**
Let be arbitrary but fixed and set . Then for every there exists a unique satisfying
[TABLE]
such that .
Proof 3.15**.**
If is arbitrary but fixed, then there exists a sequence of Gâteaux points such that in and in for all . Now, let be arbitrary but fixed and abbreviate , , and . Then we know from Corollary 3.12 that in and from Corollary 2.5 that . Moreover, from the Banach–Alaoglu Theorem it follows that – after transition to a subsequence (which may be done independently of ) – it holds that in . From the weak- closedness of the set we obtain that holds a.e. in . Further, the definition of and the convergences in and in yield
[TABLE]
Due to the sign of the integrand, the above yields a.e. in , and this entails a.e. in and a.e. in . This shows that satisfies (14). From Lemma 3.10, we may deduce that in . We already know, however, that in . Consequently, since was arbitrary, , and the existence claim is proven. It remains to show that is unique. To this end, assume that there are two different functions with . If we then consider a function with in and in (whose existence is ensured by Lemma A.1) and define , then we obtain , which gives rise to
[TABLE]
Subtraction now yields a.e. in and, since in , this yields .
Proposition 3.16** (necessary condition for ).**
Let be arbitrary but fixed with . Then for every there exists a unique function satisfying
[TABLE]
such that .
Proof 3.17**.**
Let be fixed but arbitrary. Since , the preceding proposition yields that there is a unique function satisfying (14) such that . It remains to prove that only takes values in . To this end, first observe that the definition of implies the existence of a sequence of Gâteaux points such that in and in for every , where, according to Corollary 2.5, with . As in the proof of Proposition 3.14, we choose the special direction , where is again a function with in and in . Then , and the strong convergence of to in allows passing to a subsequence to obtain and pointwise a.e. in . From the latter, it follows that for almost all there exists an (depending on ) with for all , and consequently
[TABLE]
But, as takes only the values [math] and for all , pointwise convergence almost everywhere is only possible if a.e. in . This proves the claim.
As an immediate consequence of the last two results, we obtain:
Corollary 3.18**.**
If is Gâteaux-differentiable in a point , then it holds
[TABLE]
Proof 3.19**.**
The inclusion was already proved in Lemma 3.3. The reverse inclusion follows immediately from Propositions 3.14 and 3.16, and the fact that in a Gâteaux point there necessarily holds (see Corollary 2.5).
Remark 3.20**.**
Note that even in finite dimensions, the Bouligand and the Clarke subdifferential can contain operators other than the Gâteaux derivative in a Gâteaux point; see, e.g., [5, Ex. 2.2.3]. Thus, Corollary 3.18 shows that, in spite of its non-differentiability, the solution operator is comparatively well-behaved.
Remark 3.21**.**
Similarly to Theorem 2.3, where the directional derivative of the -function appears, Propositions 3.14 and 3.16 show that elements of and are characterized by PDEs which involve a pointwise measurable selection of the set-valued a.e.-defined functions and , respectively. Here, and denote the Bouligand and the convex subdifferential of the function , respectively.
Now that we have found necessary conditions that elements of the subdifferentials and have to fulfill, we turn to sufficient conditions which guarantee that a certain linear operator is an element of these subdifferentials. Here we focus on the subdifferentials and . It will turn out that a linear operator is an element of these subdifferentials if it is of the form with as in (14) and (15), respectively. Thanks to Lemma 3.3(i) and the necessary conditions in Propositions 3.14 and 3.16, this will finally give a sharp characterization of all Bouligand subdifferentials in Definition 3.1; see Theorem 3.28 below. We start with the following preliminary result.
Lemma 3.22**.**
Let be arbitrary but fixed and write . Assume that is a function satisfying
[TABLE]
and define via
[TABLE]
Then as in Definition 3.13 is an element of the strong-strong Bouligand subdifferential .
Proof 3.23**.**
We have to construct sequences of Gâteaux points converging strongly to such that also the corresponding Gâteaux derivatives in an arbitrary direction converge strongly in to . For this purpose, set , , and . Then we obtain , in and in as . Choose now arbitrary but fixed representatives of and and define . Then for all and all , we have
[TABLE]
i.e., the sets in the collection are disjoint (and obviously Lebesgue measurable). Furthermore, the underlying measure space is finite. Thus, we may apply Lemma B.1 to obtain a -zero set such that
[TABLE]
for all . According to Corollary 2.5, this implies that is Gâteaux-differentiable in for all with where . Consider now an arbitrary but fixed sequence with as and fix for the time being a direction . Then satisfies
[TABLE]
and it holds that
[TABLE]
where is as defined in (17). Note that according to assumption (16), the case is negligible here. Using Lemma 3.10, we now obtain that converges strongly in to . Since was arbitrary, this proves the claim.
In the following, we successively sharpen the assertion of Lemma 3.22 by means of Lemma 3.10 and the approximation results for characteristic functions proven in Appendix B.
Proposition 3.24** (sufficient condition for ).**
Let be arbitrary but fixed and set . If satisfies
[TABLE]
then .
Proof 3.25**.**
Since takes only the values [math] and , there exists a Lebesgue measurable set with . We now proceed in two steps:
- (i)
If is open, then we know from Lemma B.3(i)* that there exists a sequence of functions satisfying (16) such that*
[TABLE]
Let us abbreviate . Then, thanks to (18) and (19), we arrive at
[TABLE]
Moreover, due to Lemma 3.22, we know that for all and, from Lemma 3.10 and (20), we obtain strongly in for all . Proposition 3.6 now gives as claimed.
- (ii)
Assume now that for some Lebesgue measurable set . Then Lemma B.3(ii)* implies the existence of a sequence of open sets such that*
[TABLE]
Similarly to above, we abbreviate so that (21) again yields the convergence in (20). Since from (i) we know that for all , we can argue completely analogously to (i) to prove in the general case.
Proposition 3.26** (sufficient condition for ).**
Let be arbitrary but fixed and . If satisfies
[TABLE]
then .
Proof 3.27**.**
We again proceed in two steps:
- (i)
If is a simple function of the form with for all , , and Lebesgue measurable and mutually disjoint, then we know from Lemma B.3(iv)* that there exists a sequence of Lebesgue measurable sets such that in . In view of (22), this yields*
[TABLE]
so that, by Lemma 3.10, we obtain in for all . Moreover, from Proposition 3.24, we already know that for all . Therefore Proposition 3.8 gives as claimed.
- (ii)
For an arbitrary but fixed satisfying (22), measurability implies the existence of a sequence of simple functions satisfying (22) and converging pointwise a.e. to . (Note that the pointwise projection of a simple function onto the set of functions satisfying (22) remains simple as is fixed and measurable.) Since pointwise a.e. convergence and a uniform bound in imply weak- convergence in , we can now apply (i) and again Lemma 3.10 and Proposition 3.8 to obtain the claim.
Thanks to Lemma 3.3(i), the necessary conditions in Propositions 3.14 and 3.16, respectively, in combination with the sufficient conditions in Propositions 3.26 and 3.24, respectively, immediately imply the following sharp characterization of the Bouligand subdifferentials of .
Theorem 3.28** (precise characterization of the subdifferentials of ).**
Let be arbitrary but fixed and set . Then:
- (i)
It holds . Moreover, if and only if there exists a function satisfying
[TABLE]
such that . Furthermore, for each the associated is unique.
- (ii)
It holds . Moreover, if and only if there exists a function satisfying
[TABLE]
such that . Furthermore, for each the associated is unique.
Remark 3.29**.**
Theorem 3.28* shows that it does not matter whether we use the weak or the strong topology for the approximating sequence in the definition of the subdifferential; only the choice of the operator topology makes a difference. We further see that the elements of the strong resp. weak Bouligand subdifferential are precisely those operators generated by a function that is obtained from a pointwise measurable selection of the Bouligand resp. convex subdifferential of the -function, cf. Remark 3.21. Note that for all with , a.e. in , all , and all , it holds*
[TABLE]
This implies that the set is convex and contains the convex hull of the set . We point out that the convex combination of two elements of, e.g., is typically not an element of (due to the bilinear term in the definition of ). The above “convexification effect” appears only when we consider the inverse operators.
4 First-order optimality conditions
In this section we turn our attention back to the optimal control problem (P), where we are mainly interested in the derivation of first-order necessary optimality conditions involving dual variables. Due to the non-smoothness of the control-to-state mapping caused by the -function in (PDE), the standard procedure based on the adjoint of the Gâteaux derivative of cannot be applied. Instead, regularization and relaxation methods are frequently used to derive optimality conditions as in, e.g., [1]. We will follow the same approach and derive an optimality system in this way in the next subsection. Since the arguments are rather standard, we keep the discussion concise; the main issue here is to carry out the passage to the limit in the topology of . We again emphasize that the optimality conditions themselves are not remarkable at all. However, in Section 4.2, we will give a new interpretation of the optimality system arising through regularization by means of the Bouligand subdifferentials from Section 3 (cf. Theorem 4.12), which is the main result of this section.
4.1 Regularization and passage to the limit
For the rest of this section, let be an arbitrary local minimizer for (P). We follow a widely used approach (see, e.g., [24]) and define our regularized optimal control problem as
[TABLE]
with a regularized version of the -function satisfying the following assumptions.
Assumption 4.1**.**
The family of functions satisfies
* for all ;* 2.
there is a constant such that for all ; 3.
for all and all , there holds ; 4.
for every , the sequence converges uniformly to on and uniformly to [math] on as .
There are numerous possibilities to construct families of functions satisfying Assumption 4.1; we only refer to the regularized -functions used in [24, 33]. As for the -function, we will denote the Nemytskii operator associated with by the same symbol.
Lemma 4.2**.**
For every , there exists a unique solution of the PDE in (Pε). The associated solution operator is weakly continuous and Fréchet-differentiable. Its derivative at in direction is given by the unique solution to
[TABLE]
where .
Proof 4.3**.**
The arguments are standard. The monotonicity of by Assumption 4.1 yields the existence of a unique solution, and bootstrapping implies that this is an element of . The weak continuity of follows from Lemma 3.10 in exactly the same way as Corollary 3.12. Due to Assumption 4.1 and , the Nemytskii operator associated with is continuously Fréchet-differentiable from to and, owing to the non-negativity of , the linearized equation in (23) admits a unique solution for every . The implicit function theorem then yields the claimed differentiability.
Lemma 4.4**.**
There exists a constant such that for all , there holds
[TABLE]
Moreover, for every sequence with in and every sequence , there exists a subsequence and an operator such that
[TABLE]
Proof 4.5**.**
Given , let us set and . Then it holds that
[TABLE]
Testing this equation with and employing the monotonicity of the -operator and Assumption 4.1 gives . Then, thanks to the Lipschitz continuity of the -function and again Assumption 4.1, a bootstrapping argument completely analogous to that in the proof of Proposition 2.1 yields (24), cf. (25).
To obtain the second part of the lemma, let and be sequences with in and . Then (24) and Proposition 2.1 imply
[TABLE]
as , i.e., in . Now, given an arbitrary but fixed direction , we know that the derivative is characterized by
[TABLE]
Then, due to pointwise a.e. in (at least for a subsequence) and Assumption 4.1 and , there is a subsequence (not relabeled for simplicity) such that
[TABLE]
with
[TABLE]
Note that the transition to a subsequence above is independent of . Using Lemma 3.10 and Theorem 3.28 then yields the second claim.
Theorem 4.6** (optimality system after passing to the limit).**
Let be locally optimal for (P) with associated state . Then there exist a multiplier and an adjoint state such that
[TABLE]
where denotes the convex subdifferential of the -function. The solution operator is thus an element of .
Proof 4.7**.**
Based on the previous results, the proof follows standard arguments, which we briefly sketch for the convenience of the reader. We introduce the reduced objective functional associated with (Pε) as
[TABLE]
and consider the following auxiliary optimal control problem
[TABLE]
where is the radius of local optimality of . Thanks to the weak continuity of by Lemma 4.2 and the weak lower semi-continuity of by Section 1.1, the direct method of the calculus of variations immediately implies the existence of a global minimizer of (Pε,r). Note that due to the continuous Fréchet-differentiability of , the global Lipschitz continuity of , and (24), there exists (after possibly reducing the radius ) a constant such that for all sufficiently small , it holds
[TABLE]
As a consequence, we obtain (with the same constant)
[TABLE]
and
[TABLE]
and therefore
[TABLE]
Thus, for every sufficiently small, any global solution of (Pε,r) must necessarily satisfy
[TABLE]
In particular, for small enough, is in the interior of the -ball around and, as a global solution of (Pε,r), is also a local one of (Pε). It therefore satisfies the first-order necessary optimality conditions of the latter, which, thanks to the chain rule and Lemma 4.2, read
[TABLE]
From Lemma 4.4 we obtain that there exists a sequence and an operator such that
[TABLE]
Further, we deduce from (27), the global Lipschitz continuity of , and (24), that in . Combining all of the above and using our assumptions on , we can pass to the limit in (28) to obtain
[TABLE]
By setting , this together with Theorem 3.28 and Remark 3.29 finally proves the claim.
Corollary 4.8**.**
Assume that is continuously Fréchet-differentiable from to . If is locally optimal for (P) with associated state , then there exists a multiplier and an adjoint state such that
[TABLE]
If is even Fréchet-differentiable from to , then .
Proof 4.9**.**
According to Definition 3.13, is the solution operator of (13), which is formally self-adjoint. Thus we can argue as in [37, Sec. 4.6] to deduce (29a) and the -regularity of . The -regularity is again obtained by bootstrapping.
4.2 Interpretation of the optimality conditions in the limit
In classical non-smooth optimization, optimality conditions of the form , where denotes one of the various subdifferentials of , frequently appear when a function is minimized over a normed linear space ; we only refer to [34, Secs. 7 and 9] and the references therein. With the help of the results of Section 3 (in particular Theorem 3.28), we are now in the position to interpret the optimality system in (26) in this spirit. To this end, we first consider the reduced objective and establish the following result for its Bouligand subdifferential.
Proposition 4.10** (chain rule).**
Let be arbitrary but fixed and let be the reduced objective for (P) defined by . Moreover, set . Then it holds
[TABLE]
Proof 4.11**.**
Let and be arbitrary but fixed. Then the definition of guarantees the existence of a sequence of Gâteaux points with in and in for all . Since is Fréchet- and thus Hadamard-differentiable and so is by Theorem 2.3, we may employ the chain rule to deduce that is Gâteaux-differentiable in the points with derivative
[TABLE]
for . As in by Proposition 2.1 and is continuously Fréchet-differentiable by Section 1.1, we obtain for every that
[TABLE]
Since was arbitrary, this proves .
With the above result, we can now relate the optimality conditions obtained via regularization to the Bouligand subdifferential of the reduced objective, and in this way rate the strength of the optimality system in (26).
Theorem 4.12** (limit optimality system implies Bouligand-stationarity).**
It holds:
[TABLE]
Here, denotes the Clarke subdifferential as defined in [5, Sec. 2.1].
Proof 4.13**.**
The first two implications immediately follow from Theorem 4.6 and Proposition 4.10. For the third implication, observe that the weak closedness of (see [5, Prop. 2.1.5b]) and in all Gâteaux points (cf. [5, Prop. 2.2.2]) result in .
Remark 4.14**.**
The above theorem is remarkable for several reasons:
- (i)
Theorem 4.12* shows that is a necessary optimality condition for the optimal control problem (P). This is in general not true even in finite dimensions, as the minimization of the absolute value function shows.*
- (ii)
The above shows that the necessary optimality condition in Theorem 4.6, which is obtained by regularization, is comparatively strong. It is stronger than Clarke-stationarity and even stronger than Bouligand-stationarity (which is so strong that it does not even make sense in the majority of problems).
Remark 4.15**.**
It is easily seen that the limit analysis in Section 4.1 readily carries over to control constrained problems involving an additional constraint of the form for a closed and convex . The optimality system arising in this way is identical to (26) except for (26c), which is replaced by the variational inequality
[TABLE]
The interpretation of the optimality system arising in this way in the spirit of Theorem 4.12 is, however, all but straightforward, as it is not even clear how to define the Bouligand subdifferential of the reduced objective in the presence of control constraints. Intuitively, one would choose the approximating sequences in the definition of from the feasible set , but then the arising subdifferential could well be empty. This gives rise to future research.
4.3 Strong stationarity
Although comparatively strong, the optimality conditions in Theorem 4.6 are not the most rigorous ones, as we will see in the sequel. To this end, we apply a method of proof which was developed in [22] for optimal control problems governed by non-smooth semilinear parabolic PDEs and inspired by the analysis in [23, 24]. We begin with an optimality condition without dual variables.
Proposition 4.16** (purely primal optimality conditions).**
Let be locally optimal for (P) with associated state . Then there holds
[TABLE]
Proof 4.17**.**
As already argued above, and are Hadamard-differentiable, so that the reduced objective is Hadamard-differentiable with directional derivative by the chain rule for Hadamard-differentiable mappings. Thus by classical arguments, the local optimality of implies for all .
Lemma 4.18**.**
Let fulfill (26a), i.e., with some , . Then for every there holds
[TABLE]
Proof 4.19**.**
Let be arbitrary and define by so that . Then implies
[TABLE]
as claimed.
Theorem 4.20** (strong stationarity).**
Let be locally optimal for (P) with associated state . Then there exist a multiplier and an adjoint state such that
[TABLE]
Proof 4.21**.**
From Theorem 4.6, we know that there exist and such that (26) is valid, which already gives (32a), (32b), and (32d). It remains to show (32c). To this end, let be arbitrary and define
[TABLE]
so that by Theorem 2.3. By testing (33) with and using (26c) and (30) from Proposition 4.16, we arrive at
[TABLE]
On the other hand, we know from Lemma 4.18 that and satisfy (31). Subtracting this equation from (34) and using the density of and the global Lipschitz continuity of yields
[TABLE]
Note that due to (32b), the bracket in (35) vanishes a.e. in . Thus, we obtain
[TABLE]
for all . The above implies that and holds a.e. in for all , and this in turn yields by addition that holds a.e. in for all . Inequality (32c) now follows immediately.
Proposition 4.22**.**
The strong stationarity conditions are equivalent to the purely primal optimality conditions, i.e., together with its state and a multiplier and an adjoint state satisfies (32) if and only if they also fulfill the variational inequality (30).
Proof 4.23**.**
Let be arbitrary and define . Then the gradient equation in (32d) and Theorem 2.3 give
[TABLE]
For the last term, (32b) and the sign condition in (32c) yield
[TABLE]
Together with Lemma 4.18, this implies that (36) results in
[TABLE]
which is (30).
The other direction follows analogously to [22, proof of Thm. 5.3]. Assume that with satisfies (30). Defining
[TABLE]
and , we have
[TABLE]
From (30) we thus obtain that
[TABLE]
Let now . The surjectivity of then yields
[TABLE]
By the Hahn–Banach extension theorem, we can extend from to , i.e., we obtain a that satisfies this variational inequality for all . Pointwise inspection then shows that
[TABLE]
Hence,
[TABLE]
satisfies a.e. as well as
[TABLE]
Together with (37) and the third relation of (38), this yields (32).
Remark 4.24**.**
As in case of the optimality system (26), the regularity of the adjoint state in Theorem 4.20 is again only limited by the mapping and differentiability properties of the objective functional. Thus, arguing as in Corollary 4.8, one shows that if is differentiable from or to , the adjoint state satisfying (29a) is an element of or , respectively.
Remark 4.25**.**
Although the optimality system (26) is comparatively strong by Theorem 4.12, it provides less information compared to the strong stationarity conditions in (32) since it lacks the sign condition (32c) for the adjoint state. The conditions (32) can be seen as the most rigorous dual-multiplier based optimality conditions, as by Proposition 4.22 they are equivalent to the purely primal condition. We point out, however, that the method of proof of Theorem 4.20 can in general not be transferred to the case with additional control constraints (e.g., for a closed and convex set ), since it requires the set to be dense in . In contrast to this, the adaptation of the limit analysis in Section 4.1 to the case with additional control constraints is straightforward as mentioned in Remark 4.15.
5 Algorithms and numerical experiments
One particular advantage of the optimality system in (26) is that it seems amenable to numerical solution as we will demonstrate in the following. We point out, however, that we do not present a comprehensive convergence analysis for our algorithm to compute stationary points satisfying (26) but only a feasibility study. For the sake of presentation, we consider here an tracking objective of the form
[TABLE]
with a given desired state and a Tikhonov parameter .
5.1 Discretization and semi-smooth Newton method
Let us start with a short description of our discretization scheme, where we restrict ourselves from now on to the case . For the discretization of the state and the control variable, we use standard continuous piecewise linear finite elements (FE), cf., e.g., [3]. Let us denote by the associated FE space spanned by the standard nodal basis functions . The nodes of the underlying triangulation belonging to the interior of the domain are denoted by . We then discretize the state equation in (P) by employing a mass lumping scheme for the non-smooth nonlinearity. Specifically, we consider the discrete state equation
[TABLE]
where denote the FE-approximations of and . With a slight abuse of notation, we from now on denote the coefficient vectors and by . The discrete state equation can then be written as the nonlinear algebraic equation
[TABLE]
where and denote stiffness and mass matrix, is the componentwise -function, and
[TABLE]
with denoting the lumped mass matrix. Due to the monotonicity of the -operator, one easily shows that (40) and (41) admit a unique solution for every control vector . The objective functional is discretized by means of a suitable interpolation operator (e.g., the Clément interpolator or, if , the Lagrange interpolator). If – again by the abuse of notation – we denote the coefficient vector of with respect to the nodal basis by , we end up with the discretized objective
[TABLE]
Again regularizing the -function in (41), a limit analysis analogous to Section 4.1 yields the following discrete counterpart to (26) with vectors as necessary optimality conditions for the discretized optimal control problem:
[TABLE]
Here, denotes the Hadamard product, and we have eliminated the control by means of the gradient equation .
Next, we reformulate (42) as a non-smooth system of equations. To this end, let us introduce for a given the proximal point mapping of by
[TABLE]
Since the proximal point mapping of coincides with the resolvent of its convex subdifferential, it is straightforward to show that if and only if . Thus, for every , (42c) is equivalent to the non-smooth equation
[TABLE]
Since is Lipschitz continuous and piecewise continuously differentiable by (43), it is semi-smooth as a function from . As the same holds for the -function, it seems reasonable to apply a semi-smooth Newton method to numerically solve the system consisting of (42a), (42b), and (42c*′*). However, the application of a Newton-like scheme to (42) is a delicate issue. This can already be seen by observing that is not unique in points where and vanish at the same time. Moreover, the Newton matrix may well be singular. For a clearer notation, let us introduce the index sets and , and denote by and the characteristic vectors of these index sets. The Newton matrix associated with (42) is given by
[TABLE]
where we have multiplied the last row corresponding to (42c*′*) by the lumped mass matrix to ensure a uniform scaling of the Newton equations. As already noted, the matrix becomes singular if there is an index such that and . To resolve this problem, we follow an active set-type strategy where we remove the components of corresponding to these indices from the Newton equation and leave them unchanged in the Newton update. This is motivated by the fact that components that are fixed in this way will likely either be changed in a further iteration (and hence no longer be singular) or correspond to underdetermined components in the optimality conditions for which we thus take the (feasible) starting value as a final value. A similar strategy was also followed in [30] (see Chap. 3.4.1 and Rem. 3.9 ii)). This procedure worked well in our numerical tests at least for small values of (where the influence of the critical set on the Newton matrix is smaller), as we will demonstrate below.
Remark 5.1**.**
It is not clear how to numerically solve the strong stationarity conditions in (32) since the system may well become overdetermined by the sign condition in (32c).
5.2 Numerical results
We now present two different examples with a constructed exact solution to (26) in order to demonstrate convergence of the proposed algorithm and to illustrate the dependence on the parameters and as well as on the mesh size . In both examples, the state vanishes in parts of the domain so that the non-smoothness of the -function becomes apparent. For this purpose, we introduce an additional inhomogeneity in the state equation, i.e., we replace the PDE in (P) by
[TABLE]
with a given function . It is easy to see that this modification does not influence the analysis in the preceding sections. The domain is chosen as the unit square , which is discretized by means of Friedrich–Keller triangulations. In all cases, we take as a starting guess for the Newton iteration , and terminate the iteration if either the combined norm of the residuals in (42a), (42b), and (42c*′*) becomes less than or if the maximum number of Newton iterations is reached. The Newton system in each iteration is solved by matlab’s sparse direct solver.
In the first example, the optimal state and adjoint state are set to
[TABLE]
and the data and are constructed such that (26) is fulfilled, i.e., the optimal control is . We note that there is a subset where and vanish at the same time, but it is only of measure zero. Table 1 presents the numerical results for different values of the mesh size , the Tikhonov parameter in the objective in (39), and the parameter in the proximal point mapping. For the state , we report the relative error of the computed approximation with respect to the ( projection of the) constructed optimal state in the continuous norm. For this choice of the adjoint state , the relative error is of course not appropriate, and we thus report here the absolute error in the continuous norm. The error for is given in the discrete norm, , where the are the interior nodes of the triangulation. (Note that is everywhere constant except in , and the triangulation is chosen such that .)
First, we remark that for almost all combinations of parameter values, only a few Newton iterations are needed to reach a residual norm below . Regarding the dependence on the mesh size, we can observe quadratic convergence of the state and the adjoint state . Since in this case, the active set satisfies for all , the approximation only depends on the sign of and is hence independent of the mesh size (for the considered values of ). Turning to the behavior with respect to , we first note that the Newton iteration failed to converge for larger values. This can be explained by the fact that for larger values of , the critical set where and hence becomes larger so that we expect the local convergence of Newton’s method to become an issue. For smaller values of , the Newton iteration converges quickly, and the approximation of and is independent of . This is not the case for , where the approximation becomes worse. Here we point out that while (42c*′*) is equivalent to (42c) for any value of , this only holds for exact solutions. A simple pointwise inspection shows that if (42c*′*) does not hold exactly but only up to a residual of , then . Finally, we see that the Newton method is robust with respect to , and the approximation of and even improves for smaller . However, this seems to be a particularity of this example, since the Tikhonov parameter enters the data through the construction of this exact solution.
For the second example, we choose
[TABLE]
Note that and are twice continuously differentiable and vanish on the right half of the unit square. Therefore, the non-smoothness of the -function occurs on a set of positive measure in this example. Moreover, as and vanish at the same time, is not unique in this set. This example can thus be seen as a worst-case scenario. Nevertheless, our algorithm is able to produce reasonable results as Table 2 demonstrates. (Note that it does not make sense to list the discrete error for , since is not unique as mentioned above. In contrast, we can now report relative errors for .)
The algorithm shows a similar behavior as in the first example. Again, we observe quadratic convergence with respect to mesh refinement and that the Newton method does not converge if is chosen too large. (Note that in this example, the optimal state is scaled differently, which influences the effect of in (42c*′*).) In contrast to the first example, smaller values of lead to worse numerical approximation and slower convergence of the Newton method (and even non-convergence for ). This is a typical observation which is also made in case of smooth optimal control problems.
In summary, one can conclude that our semi-smooth Newton-type method seems to be able to solve the discrete optimality system (42) for a certain range of parameters, even in genuinely non-smooth cases. However, a comprehensive convergence analysis is still lacking, and the choice of the parameter appears to be a delicate issue. Moreover, as already mentioned in Remark 5.1, it is completely unclear how to incorporate the sign condition in (32c) into the algorithmic framework. This is the subject of future research.
Appendix A Smooth “characteristic” functions of open sets
Lemma A.1**.**
For every open set there exists a function such that everywhere in and in .
Proof A.2**.**
Since the collection of all open balls with rational radii and rational centers forms a base of the Euclidean topology on , given an arbitrary but fixed open set we find (non-empty) open balls , , such that
[TABLE]
For every ball , there is a smooth rotational symmetric bump function with in and in . Defining
[TABLE]
it holds that in , in , and for all . Sobolev’s embedding theorem then yields the claim.
Appendix B Approximation of functions in and
Lemma B.1**.**
If is a finite measure space and if , , is a collection of measurable disjoint sets , then there exists a countable set such that for all .
Proof B.2**.**
Define , . Then every with is contained in at least one . Suppose are contained in . Then their disjointness implies
[TABLE]
and thus . Consequently, there can only be finitely many in each such that the set is countable.
Lemma B.3**.**
- (i)
If is open, then there exists a sequence with such that
[TABLE] 2. (ii)
If is Lebesgue measurable, then there exists a sequence of open sets such that
[TABLE] 3. (iii)
If is Lebesgue measurable and if is arbitrary but fixed, then there exists a sequence of Lebesgue measurable sets such that
[TABLE] 4. (iv)
If is a simple function of the form
[TABLE]
with , , and Lebesgue measurable and mutually disjoint, then there exists a sequence of Lebesgue measurable sets such that
[TABLE]
Proof B.4**.**
Ad (i): Let be an arbitrary but fixed open set. By Lemma A.1, there exist functions with
[TABLE]
So, if we define , then for all and for all as (here and in the following, we always use continuous representatives). Moreover, the sign conditions on and imply that
[TABLE]
Consider now some with . Then it holds that
[TABLE]
showing that the collection is disjoint. Analogously to the proof of Lemma 3.22, we now obtain by means of Lemma B.1 that there exists a sequence with as such that
[TABLE]
Together with (44), this establishes the assertion of part (i).
Ad (ii): The outer regularity of the Lebesgue measure implies the existence of open sets such that
[TABLE]
Let us define
[TABLE]
Then is open for all , and it holds that
[TABLE]
The above implies that
[TABLE]
Since the exceptional set appearing above has measure zero, this proves (ii).
Ad (iii): We apply a homogenization argument. Given , let us define
[TABLE]
The sequence is bounded in , and we may extract a subsequence (not relabeled) such that in for some . Consider now an arbitrary but fixed . Then it holds that
[TABLE]
Using standard density arguments and the uniqueness of the weak- limit, we deduce from the above that in for the whole original sequence . But now for all , it holds that
[TABLE]
i.e., in . This gives the claim with .
Ad (iv): According to part (iii), we can find for every a sequence of Lebesgue measurable sets such that
[TABLE]
Defining , the claim follows immediately by superposition.
Acknowledgments
C. Clason was supported by the DFG under grant CL 487/2-1, and C. Christof and C. Meyer were supported by the DFG under grant ME 3281/7-1, all within the priority programme SPP 1962 “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization”.
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