Second-Order Analysis and Numerical Approximation for Bang-Bang Bilinear Control Problems

We consider bilinear optimal control problems, whose objective functionals do not depend on the controls. Hence, bang-bang solutions will appear. We investigate sufficient second-order conditions for bang-bang controls, which guarantee local quadratic growth of the objective functional in $L^1$. In addition, we prove that for controls that are not bang-bang, no such growth can be expected. Finally, we study the finite-element discretization, and prove error estimates of bang-bang controls in $L^1$-norms.


Introduction.
In this article, we consider optimal control problems of the following type: Minimize the cost functional Here, Ω ⊂ R n is a bounded domain with Lipschitz boundary, L is a second-order elliptic operator, and b is a monotone nonlinearity. The presence of the nonlinear coupling χ ω uy motivates to call this problem 'bilinear', sometimes the term 'control affine problem' is used. In addition, this coupling complicates the analysis considerably. Since J does not depend explicitly on the control, it is expected that locally optimal controlsū are of bang-bang type, that isū(x) ∈ {α, β} for almost all x ∈ Ω.
We are interested in sufficient second-order optimality conditions and discretization error estimates for problem (1.1)- (1.3). To this end, we develop an abstract framework in Section 2. The analysis relies on a structural assumption on the behavior of the reduced gradient on almost inactive sets. This allows to prove a second-order condition, see Theorem 2.4. The abstract results are then applied in Section 3 to the bilinear control problem of elliptic equations.
In addition, we investigate the discretization of the original problems using finite elements. Here, we show that under the sufficient second-order condition we obtain an error estimate of the type ū −ū h L 1 ≤ c h, see Theorem 4.4. This extends earlier result for linear-quadratic bang-bang problems [12,29] and regularized nonlinear control problems [2,6].
Let us comment on the existing literature for bang-bang control problems. The present paper continues our research on bang-bang problems. It extends earlier works [7,9], which focused on problems with the control appearing linearly, to the bilinear case. In the literature on control problems governed by ordinary differential equations there are many contributions dealing with second-order conditions in the bang-bang case, e.g., [13,16,17,19,20,21,22]. In these contributions one typically assumes that the (differentiable) switching function σ : [0, T ] → R has finitely many zeros. Our structural assumption (2.5) can be considered as an extension to the distributed parameter case.
Bilinear control problems for time-dependent equations were studied, e.g., in [4,3], see also the references in these papers. By means of the Goh transform, the bilinear control problem is transferred into a problem, where the control appears linearly. It is an open problem, whether the idea of Goh transform can be applied to control of elliptic (thus time-independent) equations.
2. Abstract framework. Throughout this section we assume that (X, B, η) is a finite and complete measure space. We consider the abstract optimization problem Minimize J(u) subject to u ∈ U ad , where U ad = {u ∈ L ∞ (X) : α ≤ u(x) ≤ β a.e. in X} (2.1) with −∞ < α < β < +∞, and J : U ad → R is a given function.
In the sequel, we will denote the open ball with respect to the L p (X)-norm of radius r > 0 around v ∈ L p (X) by B p r (v).

2.1.
A negative result in the non-bang-bang case. In this section, we prove that we cannot expect any growth of the objective, if the optimal control is not of bang-bang type. Theorem 2.1 Let us assume that the measure space (X, B, η) is additionally separable and non-atomic. Suppose thatū is a local minimizer of (P) in the sense of L 1 (X), which is not bang-bang. Further, we assume that J is weak* sequentially continuous from L ∞ (X) to R. Then, there exists δ 0 > 0 such that for any δ ∈ (0, δ 0 ] and for any ε > 0, there exists u ∈ U ad with u −ū L 1 (X) = δ and J(u) ≤ J(ū) + ε. (2.2) Before proving the theorem we give some remarks and an auxiliary lemma. First, the theorem implies that a growth of type J(u) ≥ J(ū) + ν u −ū γ L p (X) ∀u ∈ U ad ∩ B 1 δ (ū) for some ν, δ, γ > 0 and p ∈ [1, ∞] is impossible. Indeed, let us argue by contradiction. Without loss of generality we can assume that the above growth holds for some δ < δ 0 . Then, according to the theorem, for every ε > 0 there exists u ε ∈ U ad such (2.2) holds. This implies with the assumed growth condition and Hölder's inequality that Finally, making ε → 0 we get a contradiction.
Furthermore, even a growth of type f ( u −ū L p (X) ) cannot be satisfied, as long as f is a non-decreasing function and f (t) > 0 for t > 0.
Recall that the measure space is non-atomic, if for all holds. It is easy to check that this is equivalent to the separability of L p (X) for all p ∈ [1, ∞). In particular, all regular Borel measures are separable measures.
Before proving the theorem, we need to state a lemma. Lemma 2.2 Let the measure space (X, B, η) be as in Theorem 2.1. Let a measurable set B ⊂ X be given.
Proof. We define the set Then, according to [23,Proposition 6.4.19], we have where F w is the closure of F w.r.t. the weak topology of L 2 (B). The space L 2 (B) is reflexive and separable, since (X, B, η) is assumed to be separable. Hence, the weak topology is metrizable on the bounded set F w . Thus, there is a sequence {v k } ⊂ L 2 (B) Finally, the result follows if v k is extended by 0 to X. Now we are in the position to prove Theorem 2.1.

2.2.
Second-order analysis. In this section, we consider the second-order analysis of problem (P). To this end, letū ∈ U ad be a fixed control. We make the following assumptions on J andū.
(H1) The functional J can be extended to an L ∞ (X)-neighborhood A of U ad . It is twice continuously Fréchet differentiable w.r.t. L ∞ (X) in this neighborhood. Moreover, we assume thatū satisfies the first-order condition J ′ (ū)(u−ū) ≥ 0 for all u ∈ U ad .
Under the previous assumptions we can prove some sufficient second-order optimality conditions forū. To this end we introduce the following cone of critical directions: for every τ > 0 we define Before establishing the second-order conditions we state the following result, whose proof can be found in [9, Proposition 2.7]. Theorem 2.3 Let us assume that (H1), (H4) and (H5) hold, then The next theorem provides a second-order condition which allows us to prove a quadratic growth of the objective J in the neighborhood ofū. In particular,ū is a strict local solution under this assumption. Note that condition (2.9) is slightly weaker than the corresponding results [9, Theorems 2.8 and 3.3], which required κ ′ < κ in (2.9). This improvement has been possible by some slightly more refined estimates in the proof. Theorem 2.4 Suppose that the above assumptions (H1)-(H5) are satisfied. Let κ be as in Theorem 2.3. Further, we assume that Then, there exist ν > 0 and δ > 0 such that (2.10) The following lemma will be used to prove this theorem. Lemma 2.5 Suppose that the above assumptions (H1)-(H5) are satisfied. Let κ be as in Theorem 2.3. Further, we assume that there exist τ > 0 and κ ′ ≥ 0 such that Then, for every γ ∈ (0, 3κ), there is a δ > 0 such that where u θ =ū + θ(u −ū) and 0 ≤ θ ≤ 1 is arbitrary.
Now we are in the position to prove Theorem 2.4.

Approximation results.
The rest of this section is dedicated to the numerical approximation of the optimization problem (P). To this end we make the following assumptions. First, we fix an approximation of the underlying set X.
Associated with the approximation X h of X, we define the following two notions of convergence. For a sequence u h ∈ L 1 (X h ) and u ∈ L 1 (X), we say that Next, we state assumptions to define the approximation of our problem (P).
that are weakly lower semicontinuous with respect to the L 2 (X) topology.
(D4) The following properties hold for sequences u h ∈ U ad,h and u ∈ U ad is a neighborhood of U ad,h . Moreover, for all u h ∈ U ad,h and for all u ∈ U ad , J ′ h (u h ) and J ′ (u) are linear and continuous forms on L 1 (X h ) and L 1 (X), respectively. Hence, there exist elements ψ h ∈ L ∞ (X h ), ψ ∈ L ∞ (X) such that the following identifications hold: First, we state a lemma which provides a partial converse to (D2). Lemma 2.6 Let us assume that (D1) and Proof. We argue by contradiction. Assume that u ≤ β is not satisfied a.e. on X. Then, there is a measurable set B ⊂ X with η(B) > 0 and ε > 0 such that It remains to check the second assertion. By extending u h withū on X \ X h , we get u h * ⇀ u in L ∞ (X), in particular, u h ⇀ u in L 1 (X). Now, the assertion follows from the weak lower semicontinuity of the norm of L 1 (X).
The following theorem proves that (P h ) realizes a convergent approximation of (P). Theorem 2.7 Let us assume that (D1)-(D4) hold. Then for every h, the problem thenũ is a global solution of (P). Conversely, ifū is a bang-bang strict local minimum of (P) in the L 1 (X) sense, then there exists a sequence {ū h } h of local minimizers of problems (P h ) in the sense of L 1 (X h ) such thatū h →ū in L 1 (X).
Proof. The existence of a global solutionū h of (P h ) follows from the boundedness, convexity and closedness of U ad,h and the weak lower semicontinuity of J h ; see assumptions (D2) and (D3). Now, consider a subsequence, denoted in the same way, such thatū h * ⇀ũ in L ∞ (X). Sinceū h ∈ U ad,h for every h, the inclusionũ ∈ U ad holds by Lemma 2.6. Furthermore, given an element u ∈ U ad , according to assumption (D2) we can take a sequence {u h } h with u h ∈ U ad,h such that u h → u in L 1 (X). Then, using (D4) and the global optimality of everyū h , we infer Hence,ũ is a solution of (P).
Conversely, we assume thatū is a bang-bang strict local minimum of (P). Then, there exists δ > 0 such that Then, we consider the problems . Therefore the feasible set of (P δ,h ) is not empty for every h small enough, and arguing as before we have that (P δ,h ) has a solutionū h for every h small enough. Moreover, the sequence {ū h } is bounded in L ∞ (X). Thus, there exists a weak* converging subsequence. Additionally, for any subsequence converging toũ in L ∞ (X) weak*, we get thatũ ∈ U ad ∩B 1 δ (ū) by Lemma 2.6, and as above J(ũ) ≤ J(ū). The strict local optimality ofū in U ad ∩ B 1 δ (ū) implies thatũ =ū. Moreover, we conclude that the whole sequence {ū h } h converges toū in L ∞ (X) weak*. In addition, by using the bang-bang property ofū, we get From here we get that ū h −ū L 1 (X h ) < δ for all h small enough. Hence,ū h is a local minimum of (P h ) for every small h.
We finish this section by proving an estimate ofū h −ū in terms of the order of the approximations ofū by elements of U ad,h and J ′ by J ′ h . Theorem 2.8 Let us assume that (H1)-(H5) and (D1)-(D5) hold. Additionally, we suppose thatū satisfies the second-order condition (2.9) with κ ′ ∈ (0, κ). Let {ū h } h be a sequence of local solutions of problems (P h ) converging toū in L 1 (X). Then, for γ = (κ − κ ′ )/2 we obtain that the estimate holds for all h small enough, whereû h andû h denote the extensions ofū h and u h bȳ u to X, respectively.
This specific extension of the elements u h is quite convenient for the derivation of the error estimate. We will also see in Section 4 below, that this will not impede the applicability of our abstract framework to derive discretization error estimates for optimal control problems. Let us observe that for every u h ∈ U ad,h , its extensionû to X by settingû(x) =ū(x) in X \ X h belongs to U ad , henceû ∈ A as well.
Proof. Let u h ∈ U ad,h , and denote byû h its extension to X byū. Sinceū h is a local minimum of (P h ), Due to (D5) this inequality can be written in the form Note that our choice of extension is crucial for the above rearrangement. Next, we rewrite the left-hand side, and by the mean value theorem and by denoting . This estimate is now used in (2.20). After applying Young's inequality we obtain From this inequality we deduce Since u h is an arbitrary element of U ad,h , this inequality implies (2.19).
In Section 4 we will provide precise estimates for the right hand side of (2.19) for some distributed optimal control problems, including bilinear controls.
3. Second-order analysis for bilinear control problems. In this section, we apply the second-order analysis results proved in the abstract framework in Section 2 to the study of some optimal control problems. The first part of this section will be devoted to the analysis of a bilinear distributed control problem associated with a semilinear elliptic equation. In the second part, we will consider a bilinear Neumann control problem.
In what follows, Ω denotes a bounded open subset of R n , 1 ≤ n ≤ 3, with a Lipschitz boundary Γ. In Ω we consider the elliptic partial differential operator where a ij , a 0 ∈ L ∞ (Ω) and a 0 ≥ 0 in Ω. Associated with this operator we define the usual bilinear form a : Let Γ D be a closed subset of Γ, possibly empty, and set Γ N = Γ \ Γ D . We define the space equipped with the usual norm of H 1 (Ω) and the operator L : Ly, z = a(y, z) ∀y, z ∈ V, and we assume its coercivity.
(A1) We have that Moreover, we consider a Carathéodory function b : Ω × R −→ R of class C 2 with respect to the second variable, such that the following assumptions are satisfied. 3.1. A bilinear distributed control problem. In this section, we consider the following state equation where ω is an open subset of Ω, and u and f satisfy the following assumptions.
(A4) We assume that u ∈ A, where the open set A ⊂ L ∞ (ω) is given by where Λ was introduced in (A1).
In the next theorem, we analyze the equation (3.4). Theorem 3.1 The following statements hold.
(1) For any u ∈ A there exists a unique solution y u ∈ Y := V ∩ L ∞ (Ω) of the state equation (3.4). Moreover, there exists a constant C such that

6)
and given Proof. For the proof of existence and uniqueness of a solution of (3.4) in Y , first we observe that the linear operator L + χ ω u is coercive in V for all u ∈ A due to the fact that u ≥ − Λ 2 and assumption (A1). Then, the arguments are standard; see, for instance, [28, §4.1]. We recall that the boundedness of y needed in this proof is a consequence of Stampacchia's result [27,Theorem 4.2]. To prove the differentiability of the mapping G we use the implicit function theorem as follows. We define which is a Banach space when it is endowed with the graph norm. Now, we consider the mapping L : Yp × A −→ W 1,p ′ (Ω) * given by L(y, u) = Ly + b(·, y) + χ ω uy − f.
From assumption (A2) we get that L is of class C 2 and ∂L ∂y (y u , u)z = Lz + b ′ (·, y u )z + χ ω uz defines an isomorphism between Yp and W 1,p ′ (Ω) * for all u ∈ A. Indeed, it is obvious that ∂L ∂y (y u , u) : Yp −→ W 1,p ′ (Ω) * is a continuous linear mapping. The bijectivity is a consequence of the Lax-Milgram theorem and, once again, [27,Theorem 4.2]. Hence, a straightforward application of the implicit function theorem implies that G is of class C 2 and (3.6) and (3.7) hold.
Associated with the state equation (3.4) we introduce the following bilinear distributed control problem where This problem is included in the abstract framework considered in Section 2 by taking X = ω and η equal to the Lebesgue measure.
The next theorem is an immediate consequence of Theorem 3.1 and the chain rule. Theorem 3.2 The reduced objective J : A → R is twice Fréchet differentiable and the first and second derivatives are given by where ϕ u ∈ Y is the unique solution of
Using Theorems 3.1 and 3.2 we infer the next result by standard arguments. whereȳ andφ are the state and adjoint state, respectively, corresponding toū.
In the rest of this section,ū will denote a fixed element of U ad satisfying (3.12). We are going to apply the results obtained in the abstract framework in Section 2. To this end, we observe that (H1) obviously holds with X = ω and (H4) is fulfilled withψ = −(φȳ)| ω . Assumption (H5) is formulated in our setting as follows: there exists a constant K such that where | · | denotes the Lebesgue measure in ω. Then, (2.8) holds.
For the second-order analysis we introduce the cone C τ u as in (2.6). The rest of this section is devoted to prove that the quadratic growth condition (2.10) holds under the second-order condition (2.9). For that, we apply Theorem 2.4. Therefore, we only need to verify that assumptions (H2) and (H3) hold. The following lemma will be used for this verification. Lemma 3.4 Given c ∈ L ∞ (Ω) with c ≥ 0, we consider the equation (3.14) Then, the following statements hold y L 6 (Ω) ≤ C L f L 6/5 (Ω) ∀f ∈ L 6/5 (Ω), where y ∈ V denotes the unique solution of (3.14).

Now, we have
This implies y L p (Ω) ≤ C p ′ f L 1 (Ω) .
Of course, better estimates can be obtained in the previous lemma for dimensions n < 3, but we do not need them here. Remark 3.5 Let us observe that the solution z v of (3.6) satisfies the estimates (3.15)-(3.17) for f = −χ ω vy u . It is enough to take c(x) = b ′ (x, y u (x)) + χ ω (x)u(x). Moreover, using (3.5), we get that {y u } u∈U ad is uniformly bounded in L ∞ (Ω). Hence, the mentioned estimates for z v can be written in terms of the norm of v in ω.
Then, we apply the convenient inequality of Lemma 3.4 to estimate y u2 − y u1 L r (Ω) in terms of u 2 − u 1 L p (ω) .
As a further preparation, we provide an estimate for the difference e = z θ −z. By taking the difference of the corresponding equations (3.6), we find that e solves the equation Owing to Lemma 3.4, we can estimate e L 6 (Ω) by the L 6/5 (Ω) norm of the right-hand side. Together with Hölder's inequality, we obtain the estimate Now, we can use (A2) and Remark 3.5, and we arrive at (3.18) for δ ≤ 1 the above estimates becomes Now, we are in position to estimate the above integrals. For the first integral, we have where we used Remark 3.5 and (3.19). Next, where again Remark 3.5 and (3.18) have been utilized. For the next integral, we remark that b ′′ (·,ȳ) − b ′′ (·, y θ ) L ∞ (Ω) can be estimated by any small positive number if ȳ − y θ L ∞ (Ω) is small enough, cf. (A2). For this, it is sufficient that δ is small enough, since u θ ∈ B 1 δ (ū) ∩ U ad , see again Remark 3.5. This along with (3.17) leads to the estimate Finally, we obtain by using similar arguments the estimates and where we used additionally (3.19). Putting these inequalities together, we obtain the desired estimate if δ > 0 is chosen small enough. Hence, we verified (H3) in our current setting.
Application of Theorem 2.4. We have verified that the assumptions (H1)-(H4) are satisfied in the setting of the bilinear distributed control problem (BDP). Thus, we can apply Theorem 2.4 and we obtain the following sufficient second-order condition. Theorem 3.6 Let us assume that (A1)-(A5) are satisfied. Moreover, we suppose that there is a constant K > 0, such that (3.13) holds and that there exist τ > 0 and κ ′ < 2 κ such that

A bilinear boundary control problem.
In this section we assume that n = 2. We outline the main steps which are necessary to transfer the analysis of Section 3.1 to a bilinear boundary control problem. We follow the notation introduced in Section 3 and assume that (A1)-(A3) hold. Further, we take ω = Γ N equipped with the surface measure. We define the operator S ω : where we are denoting the trace of z on ω by z as well. It is well known that there exist a constant C ω depending on Ω such that (3.21) Now, we consider the state equation with u ∈ A. Here, A is defined as follows where Λ was introduced in (A1). From the assumptions (A1) and (A3) along with (3.21) we get Then, Theorem 3.1 holds with the obvious modifications. In particular, the equations (3.6) and (3.7) are modified as follows and Associated with the state equation (3.4) we introduce the bilinear boundary control problem where with 0 ≤ α < β < ∞. We suppose that y d satisfies the assumption (A5). Then, Theorem 3.2 holds, we only need to change the adjoint state equation (3.11) by We also have that Theorem 3.3 holds. To get the sufficient second-order conditions we assume that (3.13) is fulfilled. Then, to check that Theorems 2.3 and 2.4 hold we need to check that assumptions (H1)-(H5) are satisfied. As in Section 3.1, it is enough to verify (H2) and (H3). To this end we will use the following lemma. Lemma 3.7 Let c ∈ L ∞ (Ω) be nonnegative and u ∈ A. For (f, g) ∈ L 2 (Ω) × L 2 (ω) let y ∈ V be the solution of the equation Then, for every p ∈ [1, ∞) and q > 1 there exist constants C p and M q independent of (f, g), c and u such that Proof. Since L 1 (Ω) and L 1 (ω) are subspaces of the space of real and regular Borel measures in Ω and ω, respectively, we can apply the well known results for measures to deduce that the solution y of (3.26) satisfies for every s ∈ [1, n n−1 ) and some constant C s independent of (f, g), c and u; see, for instance, [1], [5], or [18].
Hence, though simpler estimates can be used, the estimates used in Section 3.1 are valid to verify (H2) and (H3). As a consequence, we obtain a second-order sufficient condition analogously to Theorem 3.6 in the distributed case.
We finally mention that the same technique cannot be used to address the case n > 2. The verification of (H2) and (H3) for bilinear boundary control problems in more than two spatial dimensions remains an open problem.
4. Numerical approximation of distributed control problems. In this section, we consider the following boundary value problem where A is given by (3.1) with coefficients a ij ∈ C 0,1 (Ω) satisfying the ellipticity condition n i,j=1 a ij (x)ξ i ξ j ≥ Λ|ξ| 2 ∀x ∈ Ω and ∀ξ ∈ R n .
We also introduce the adjoint state equation associated to the control u Now, we consider the control problem (BDP) associated to the equation (4.1). Here we suppose that y d ∈ Lp(Ω)∩L 2 (Ω). We also assume thatω ⊂ Ω. Let us observe that if this condition does not hold, then the assumption (3.13) can be fulfilled only in some extreme cases. This is due to the fact thatȳ andφ vanish on Γ and, hence, the {x ∈ Ω : |ȳ(x)φ(x)| ≤ ε} contains a strip along the boundary with a measure of order √ ε. The situation is different for Neumann boundary problems.
Since assumptions (A1)-(A5) are satisfied, Theorems 3.2 and 3.3 are valid for the the control problem (BDP) associated to the state equation (4.1). In what follows, u will denote a local solution of (BDP) satisfying the regularity condition (3.13). Therefore, Theorem 3.6 holds as well.
The goal of this section is to prove error estimates for the numerical approximation of (BDP) based on a finite element discretization. To this end, we assume that Ω is convex and Γ is of class C 1,1 . Therefore, we have additional regularity for the states y u and adjoint states ϕ u for every u ∈ A, namely y u , ϕ u ∈ W 2,p (Ω) ∩ W 1,p 0 (Ω); see [14,Chapter 2]. Sincep > n, we have that W 2,p (Ω) ⊂ C 1 (Ω). If n = 2, this regularity holds for a convex and polygonal domain Ω assuming that the coefficients a ij are of class C 1 inΩ. In dimension n = 3, the regularity result is valid for rectangular parallelepipeds under the same C 1 regularity of the coefficients; see [14,Chapter 4], [11,Corollary 3.14].
Let {T h } h>0 be a quasi-uniform family of triangulations ofΩ; see [10]. We set Ω h = ∪ T ∈T h T with Ω h and Γ h being its interior and boundary, respectively. We assume that the vertices of T h placed on the boundary Γ h are also points of Γ and there exists a constant C Γ > 0 such that dist(x, Γ) ≤ C Γ h 2 for every x ∈ Γ h . This always holds if Γ is a C 2 boundary and n = 2. From this assumption we know [25,Section 5.2] that where | · | denotes the Lebesgue measure. Let us denote by T ω,h the family of all elements T ∈ T h such that T ⊂ω. We setω h = T ∈T ω,h T and ω h is its interior. We also assume that |ω \ ω h | ≤ C ω h pω with p ω > n/2.
Associated with this triangulation we define the spaces where P k (T ) denotes the polynomial of degree k in T with k = 0, 1. Now, for every u ∈ A we consider the discrete system of nonlinear equations where the bilinear form a was defined in (3.2). Using our assumptions on b and the ellipticity of the operator y → Ay + uy, the existence and uniqueness of a solution of (4.3) follows by standard arguments. This solution will be denoted by y h (u). We also consider the discrete adjoint state equation The solution of this adjoint equation is denoted by ϕ h (u).
The following approximation results are needed for the numerical analysis of the discrete control problem. Lemma 4.1 Let u ∈ A fulfill u L ∞ (ω) ≤ M , and let y, y h , ϕ and ϕ h be the solutions of (4.1), (4.4), (4.2) and (4.5), respectively. Then, for some constant C depending on M we have Proof. Let us denote u h = χ| ω h u and y u h its continuous associated state. From Lemma 3.4 and Remark 3.5, and using the classical L ∞ -estimates for finite element approximations, see [2,8] and [24,26], we get where we have used that |ω \ ω h | ≤ C ω h pω . From this estimate we deduce the corresponding estimate for ϕ − ϕ h by using similar arguments.
Finally, we define the discrete control problem Moreover, we included a Tikhonov parameter α h ≥ 0 and require α h → 0 as h → 0. This regularization term is beneficial for the numerical solution of (BDP h ) and we will prove that the choice α h = c h yields the same order of convergence as α h = 0, see (4.9) below.
Let us check that these approximations of (BDP) fit into the framework described in Section 2.3. To this end we have to check the assumptions (D1)-(D5). First, we observe that taking X = ω, X h = ω h and η = Lebesgue measure in ω, (D1) follows from our assumption |ω \ ω h | → 0 as h → 0.
Assumption (D2) is immediate. Indeed, it is enough to observe that given u ∈ U ad we can take u h as the projection of u on U h : where χ T denotes the characteristic function of T . It is well known that u h → u strongly in L p (ω) under the assumption u ∈ L p (ω); see [15]. Now, (D3) is obvious. (D4) is a straightforward consequence of the following lemma. Lemma 4.2 If u h ⇀ u weakly in L 1 (ω) with u h ∈ A ∩ U h and u ∈ A, and there exists a constant M > 0 such that Proof. Let us extend every u h to ω by setting u h (x) = 0 ∀x ∈ ω \ ω h . From (4.6) we get Now, we prove that y u − y u h L ∞ (Ω) → 0 as h → 0. Since u h L ∞ (ω) ≤ M ∀h > 0, then {y u h } h is bounded in W 2,p (Ω). Using the compactness of the embedding W 2,p (Ω) ⊂ L ∞ (Ω), we deduce easily the convergence y u − y u h L ∞ (Ω) → 0 as h → 0. The convergence J h (u h ) → J(u) follows easily by using α h → 0.
To check (D5) we take It is easy to prove that J h : A h −→ R is of class C 2 and its first derivative is given by u v dx ∀u ∈ A h and ∀v ∈ L ∞ (ω h ), (4.8) where y h (u) and ϕ h (u) are the solutions of (4.4) and (4.5), respectively. Hence, it is enough to take ψ h = −(ϕ h (u)y h (u))| ω h + α h u. Concerning the function J : A → R, we already know that it is of class C 2 (Theorem 3.2), and according to (3.8) we can take ψ = −(ϕ u y u )| ω .
Therefore, Theorems 2.7 and 2.8 hold. Observe that Theorem 2.7 is formulated as follows. Theorem 4.3 Assume that (A1)-(A5) hold. For every h, the problem (BDP h ) has at least a global solutionū h . If {ū h } h is a sequence of global solutions of (BDP h ) and u h * ⇀ũ in L ∞ (ω), thenũ is a global solution of (BDP). Conversely, ifū is a bangbang strict local minimum of (BDP) in the L 1 (ω) sense, then there exists a sequence {ū h } h of local minimizers of problems (BDP h ) with respect to the same topology such thatū h →ū in L 1 (ω). Now, we apply Theorem 2.8 to get the following result. Theorem 4.4 Assume that (A1)-(A5) hold. Additionally, we suppose that (3.13) is fulfilled andū satisfies the second-order condition (3.20) with κ ′ ∈ (0, κ). Let {ū h } h be a sequence of local solutions of problems (BDP h ) converging toū in L 1 (ω). Then, there exists a constant C independent of h such that ū −ū h L 1 (ω h ) ≤ C (h + α h ). (4.9) Proof. To prove this theorem we will estimate the three terms in the right hand side of (2.19). First, we observe that whereȳ h andφ h are the discrete state and adjoint state associated withū h , and yû h and ϕû h are the continuous state and adjoint state corresponding toû h , which is the extension ofū h to ω byū. Now using Lemma 4.1 we obtain Now, we estimate the second term of (2.19). To this end, we take u h as the projection ofū on U h ; see (4.7). Sinceū is bang-bang by assumption, it holdsū = u h on all elements, whereū is constant. It remains to estimate |u h −ū| on elements T , whereū takes the values α and β on some points of T . Let us denote the family of such elements by T h,ū . Let us take T ∈ T h,ū . This means thatφȳ changes the sign in T . Sinceφȳ is continuous inΩ, there exists a point ξ T ∈ T such thatφ(ξ T )ȳ(ξ T ) = 0. Sinceφȳ ∈ W 2,p (Ω) ⊂ C 1 (Ω), we get the existence of constantL such that |φ(x)ȳ(x)| = |φ(x)ȳ(x) −φ(ξ T )ȳ(ξ T )| ≤L|x − ξ T | ≤Lh ∀x ∈ T.

This inequality implies that
This along with (3.13) leads to T ∈T h,ū |T | ≤ KLh.