OPTIMAL CONTROL OF A FREE BOUNDARY PROBLEM WITH ...

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Feb 24, 2014 - George Mason University, Fairfax, VA 22030, USA. ([email protected]). ..... Given u ∈ Uad, the convex cone C (u) comprises all directions h ∈ L2 (0, 1) such that u + th ..... as m → ∞, where fm = − div(A∇wm). Since the unit ...
OPTIMAL CONTROL OF A FREE BOUNDARY PROBLEM WITH SURFACE TENSION EFFECTS: A PRIORI ERROR ANALYSIS ∗

arXiv:1402.5709v1 [math.OC] 24 Feb 2014

´§ HARBIR ANTIL† , RICARDO H. NOCHETTO‡ , AND PATRICK SODRE Abstract. We present a finite element method along with its analysis for the optimal control of a model free boundary problem with surface tension effects, formulated and studied in [1]. The state system couples the Laplace equation in the bulk with the Young-Laplace equation on the free boundary to account for surface tension. We first prove that the state and adjoint system have the requisite regularity for the error analysis (strong solutions). We discretize the state, adjoint and control variables via piecewise linear finite elements and show optimal O(h) error estimates for all variables, including the control. This entails using the second order sufficient optimality conditions of [1], and the first order necessary optimality conditions for both the continuous and discrete systems. We conclude with two numerical examples which examine the various error estimates. Key words. sharp interface model, free boundary, curvature, surface tension, pde constrained optimization, boundary control, finite element method, L2 projection, second order sufficient conditions, a priori error estimate. AMS subject classifications. 49J20, 35Q93, 35Q35, 35R35, 65N30.

1. Introduction. The purpose of this paper is to numerically analyze the optimal control problem we proposed in [1]. The underlying state system couples the Laplace equation in the bulk with the Young-Laplace equation on the free boundary to account for surface tension, as proposed by P. Saavedra and L. R. Scott in [11]. We first recall the continuous optimal control problem from [1, Section 2]. Γγ = (x1 , 1 + γ(x1 )) Ωγ Γ



Σ

Fig. 1.1. Ωγ denotes a physical domain with boundary ∂Ωγ = Σ∪Γγ . Here Σ includes the lateral and the bottom boundary and is assumed to be fixed. Furthermore, the top boundary Γγ (dotted line) ˚ 1 (0, 1) denotes a is “free” and is assumed to be a graph of the form (x1 , 1 + γ(x1 )), where γ ∈ W ∞ parametrization. The free boundary Γγ is further mapped to a fixed boundary Γ = (0, 1) × {1} and in 2 turn the physical domain Ωγ is mapped to a reference domain Ω = (0, 1) , where all computations are carried out. 1 ˚∞ Let γ ∈ W (0, 1) denote a parametrization of the top boundary (see Figure 1) ∗ This

research was supported in part by NSF grants DMS-0807811 and DMS-1109325. of Mathematical Sciences. George Mason University, Fairfax, VA 22030, USA ([email protected]). ‡ Department of Mathematics and Institute for Physical Science and Technology, University of Maryland College Park, MD 20742, USA ([email protected]). § Department of Mathematics, University of Maryland College Park, MD 20742, USA ([email protected]). † Department

1

of the physical domain Ωγ ⊂ Ω∗ ⊂ R2 with boundary ∂Ωγ := Γγ ∪ Σ, defined as Ω∗ := (0, 1) × (0, 2),  Ωγ := (x1 , x2 ) : 0 < x1 < 1, 0 < x2 < 1 + γ(x1 ) ,  Γγ := (x1 , x2 ) : 0 < x1 < 1, x2 = 1 + γ(x1 ) , Σ := ∂Ωγ \ Γγ ,  Γ := (x1 , x2 ) : 0 < x1 < 1, x2 = 1 . Here, Ω∗ and Σ are fixed while Ωγ and Γγ deform according to γ. We want to find an optimal control u ∈ Uad ⊂ L2 (0, 1) so that the solution pair (γ, y) to the free boundary problem (FBP) is the best least squares fit of desired boundary γd : (0, 1) → R and bulk yd : Ω∗ → R configurations. This amounts to solving the problem: minimize J (γ, y, u) :=

1 1 λ 2 2 2 kγ − γd kL2 (0,1) + ky − yd kL2 (Ωγ ) + kukL2 (0,1) , 2 2 2

subject to the state equations      

−∆y = 0

(1.2a)

in Ωγ

y = v on ∂Ωγ ,  −κH [γ] + ∂ y (·, 1 + γ) = u on (0, 1) ν     γ(0) = γ(1) = 0,

(1.2b)

the state constraints |dx1 γ| ≤ 1

a.e. in (0, 1) ,

(1.2c)

with dx1 being the total derivative with respect to x1 , and the control constraint u ∈ Uad

(1.2d)

dictated by Uad , a closed ball in L2 (0, 1), to be specified later in (2.6). Here λ > 0 is the stabilization parameter; v is given data which in principle could act as a Dirichlet boundary control;   dx1 γ   H [γ] := dx1  q  2 1 +|dx1 γ| is the curvature of γ; and κ > 0 plays the role of surface tension coefficient. We use a fixed domain approach to solve the optimal control free boundary problem (OC-FBP). In fact, we transform Ωγ to Ω = (0, 1)2 and Γγ to Γ = (0, 1)×{1} (see Figure 1), at the expense of having a governing PDE with rough coefficients. One of the challenges of an OC-FBP is dealing with state constraints which may allow or prevent topological changes of the domain. Our analysis in [1] yields the control constraint (1.2d), which always enforce the state constraint (1.2c) i.e., we can treat OC-FBP as a simpler control constrained problem. Moreover, we proved novel second order sufficient conditions for the optimal control problem for small data v. As a consequence we obtained that the above minimization has a (locally) unique solution. 2

In this paper we introduce the fully discrete optimization problem, using piecewise linear finite elements, and show that it converges with an optimal rate O(h) for all variables. In fact, the convergence analysis for the control requires a-priori error estimates for both the state and the adjoint equations. The state equations error estimates were developed by P. Saavedra and R. Scott in [11] under the assumption that the continuous state equations have suitable second order regularity (strong Sobolev solutions). Our first goal is to prove such a regularity for v ∈ Wp2 (Ω), p > 2, via a fixed-point argument, as well as to extend the analysis to the continuous adjoint equations. Our analysis of strong solutions for both the state and adjoint equations is novel in Sobolev spaces but not in H¨ older spaces [12]. We exploit this second order regularity to derive a-priori error estimates for the state and adjoint variables based on [11]; the former are a direct extension of [11] whereas the latter are new. An important difference with [11] is the presence of the control variable u, for which we obtain also a novel a-priori error estimate. There are two approaches for dealing with a discrete optimal control problem with PDE constraints. Both rely on an agnostic discretization of the state and adjoint equations, perhaps by the finite element method; they differ on whether or not the admissible set of controls is discretized as well. The first approach [2, 5, 10] follows a more physically appealing idea and also discretizes the admissible control set. The second approach [8] induces a discretization of the optimal control by projecting the discrete adjoint state into the admissible control set. From an implementation perspective this projection may lead to a control which is not discrete in the current mesh and thus requires an independent mesh. Its key advantage is obtaining an optimal quadratic rate of convergence [8, Theorem 2.4] for the control. We point that no such improvements can be inferred for our problem. This is due to the highly nonlinear nature of the state equations and the necessity to discretize the (rough) coefficients. We will provide more details on this topic in Section 6 below. We follow the first approach and discretize the entire optimization problem using piecewise linear finite elements. However, we exploit the structure of the admissible control set and show optimal convergence rates for the state, adjoint, and control variables. We largely base our error analysis of the state and adjoint equations on the work by P. Saavedra and L. R. Scott [11]. For analyzing the discrete control we use the second order sufficient condition we developed in [1, Theorem 5.6] together with the first order necessary conditions for the continuous and discrete systems. To prove well-posedness of the discrete state and adjoint systems we rely on the inf-sup theory and a fixed point argument. Therefore, the smallness assumption on the data v is still required as in the continuous case [1, Theorem 4.5]. However, with the aid of simulations we are able to explore the control problem beyond theory and test it for large data. The outline of this paper is as follows: In Section 2.1, we state the variational form for the OC-FBP. We summarize the first order necessary and second order sufficient conditions in Sections 2.2 and 2.3. For boundary data in Wp2 , p > 2, we show that the state and the adjoint systems have strong solutions in Section 3. We introduce a finite element discretization of the system in Section 4 and prove error estimates 1 in Wp1 × W∞ for the state variables and in Wq1 × W11 for the adjoint variables. We 2 derive an L -error estimate for the optimal control in Section 6. We conclude with two numerical examples in Section 7 which confirm our theoretical findings: the first example explores the unconstrained problem, whereas the second example deals with the constrained one. 3

2. Continuous Optimal Control Problem. The purpose of this section is to recall the continuous optimal control problem in its variational form along with its first and second optimality conditions derived in [1]. We denote by . the inequality ≤ C with a constant independent of the quantities of interest. 2.1. Problem Formulation. We choose to present the formulation directly in its variational form on the reference domain Ω after having linearized the curvature H, and scaled the control u. These assumptions, not being crucial [1, Section 2, A1 A2 ], result in an optimal control problem subject to a nonlinear PDE constraint with (rough) coefficients depending on γ but without an explicit interface. We denote by 1 ˚∞ ˚ 1 (0, 1) → R and BΩ : W ˚p1 (Ω) × W ˚q1 (Ω) → R the bilinear forms BΓ : W (0, 1) × W 1 Z

1

dx1 γ(x1 )dx1 ξ(x1 ) dx1 , BΓ [γ, ξ] := κ Z 0   BΩ y, z; A [γ] := A [γ] ∇y · ∇z dx,

(2.1)



with surface tension constant κ. The coefficient matrix A [γ] arises from mapping the physical domain Ωγ to the reference domain Ω and is given by [1, Section 2] 

1 + γ(x1 )

A [γ] =  − dx1 γ(x1 )x2

 − dx1 γ(x1 )x2 2 . 1+(dx1 γ(x1 )x2 )

(2.2)

1+γ(x1 )

˚ 1 (0, 1) → Wq1 (Ω), 1 < q < 2, be a continuous linear extension operator, Let E : W 1 namely, Eξ|Γ = ξ,

Eξ|Σ = 0, |Eξ|Wq1 (Ω) ≤ CE |ξ|W 1 (0,1) , 1

(2.3)

where CE is the stability constant. It is convenient to introduce the product space: ˚1 ˚1 W1,1 t,s := Wt (0, 1) × Ws (Ω)

1 ≤ t, s ≤ ∞.

(2.4)

Given v ∈ Wp1 (Ω), a lifting of the boundary data to Ω, yd ∈ L2 (Ω∗ ), γd ∈ L2 (0, 1), let δy := y + v − yd , δγ := γ − γd . The optimal control problem is to minimize J (γ, y, u) :=

1 2 kδγkL2 (0,1) + 2

2 1 λ

p

2 + kukL2 (0,1) ,

δy 1 + γ 2 2 2 L (Ω)

(2.5a)

with state variable (γ, y) ∈ W1,1 ∞,p , p > 2 satisfying the state equations   −1 BΓ [γ, ξ] + BΩ y + v, z + Eξ; A [γ] = hu, ξiW∞ ˚ 1 (0,1) (0,1)×W 1

1,1 , (2.5b) ∀ (ξ, z) ∈ W1,q

the state constraint dx1 γ(x1 ) ≤ 1

a.e. x1 ∈ (0, 1) ,

(2.5c)

with dx1 being the total derivative with respect to x1 , and the control constraint u ∈ Uad . 4

(2.5d)

The set Uad of admissible controls is a closed ball in L2 (0, 1) defined as follows: if θ1 ∈ (0, 1) is chosen as in [1, Lemma 4.3], and α is the inf-sup constant for BΓ [1, Proposition 4.1], then n o Uad := u ∈ L2 (0, 1) : kukL2 (0,1) ≤ θ1 /2α . (2.6) n o We also need the open ball U := u ∈ L2 (0, 1) : kukL2 (0,1) < θ1 /α , so that Uad ⊂ U. −1 In view of the regularity of u and ξ, the duality pairing hu, ξiW∞ ˚ 1 (0,1) reduces (0,1)×W 1 R1 to 0 uζ. We refer to [1, Equations 4.5-4.6] for details. Since Uad is not open, we need to define a proper set of admissible directions to compute derivatives with respect to u. Given u ∈ Uad , the convex cone C (u) comprises all directions h ∈ L2 (0, 1) such that u + th ∈ Uad , t > 0, i.e., n o C (u) := h ∈ L2 (0, 1) : u + th ∈ Uad , t > 0 .

The proof of existence and local uniqueness of a minimizer of (2.5) involves multiple steps. The first step [1, Subsection 4.1.1] is to impose a smallness condition on the data v and restrict the radius of the L2 (0, 1) ball Uad to solve the nonlinear system (2.5b)-(2.5c). The second step [1, Subsection 4.1.2] is to improve the regularity of γ 2−1/p (0, 1). This addifrom Lipschitz continuous to the fractional Sobolev space Wp tional regularity is in turn used to prove the existence of a minimizer in [1, Subsection 5.1]. The last two steps [1, Subsection 5.2 and 5.3] consist of computing the first and second order optimality conditions. The optimality conditions are essential tools for this paper. From the simulation perspective, the first order condition yields a way to compute a minimizing sequence. From a numerical analysis perspective, the second order condition is the starting point for proving an a-priori error estimate. We recall now these conditions and prove a new result, Lemma 2.1, which is instrumental for implementing the adjoint system. 2.2. First-order Optimality Condition. The purpose of this section is to state the first order necessary optimality conditions using the reduced cost functional approach; a formal Lagrange multiplier approach is also presented in [1, Section 3]. Given u ∈ U, there exists a unique solution (γ, y) ∈ W1,1 ∞,p of (2.5b)-(2.5c). This induces the so-called control-to-state map Gv : U → W1,1 ∞,p , where Gv (u) =  γ(u), y(u) , and we can write the cost functional J in (2.5a) in the reduced form as J (u) := J1 (Gv (u)) + J2 (u)

(2.7)

with J1 (γ, y) :=

1 2 kδγkL2 (0,1) + 2

2 1

p

δy 1 + γ 2 , 2 L (Ω)

J2 (u) =

λ 2 kukL2 (0,1) . 2

Therefore, if u ¯ ∈ Uad is a minimizer of J , then u ¯ satisfies the variational inequality

0 J (¯ u), u − u ¯ L2 (0,1)×L2 (0,1) ≥ 0 ∀u ∈ Uad . (2.8) Deriving an expression for J 0 (¯ u) is one of the main difficulties of an optimization problem. It turns out that J 0 (¯ u) = λ¯ u + s¯ [1, Sections 3 and 5.2], whence hλ¯ u + s¯, u − u ¯iL2 (0,1)×L2 (0,1) ≥ 0 5

∀u ∈ Uad ,

(2.9)

where (¯ s, r¯ − E¯ s) ∈ W1,1 s, r¯) satisfies the adjoint equations in variational form 1,q and (¯   0 BΓ [ξ, s¯] + DΩ (ξ, z), r¯; γ¯ , y¯ = J1 (¯ γ , y¯), (ξ, z) * + Z

(2.10) 1 1 2 |δ y¯| dx2 + z, δ y¯ (1 + γ¯ ) , = ξ, δ¯ γ+ 2 0  for all (ξ, z) in W1,1 γ , y¯) := γ(¯ u), y(¯ u) , where DΩ is the parametrized form ∞,p with (¯       DΩ (ξ, z), r¯; γ¯ , y¯ := BΩ z, r¯; A [¯ γ ] + BΩ y¯ + v, r¯; DA [¯ γ ] hξi (2.11) with derivative of A with respect to γ given by DA [¯ γ ] hhi = A1 [¯ γ ] h + A2 [¯ γ ] dx1 h [1, (2.4)]. Moreover, the duality pairings on the right hand side of (2.10) are reduced to standard integrals due to the L2 -regularity imposed by the cost functional. The assembly of (2.10) is nontrivial. In view of (2.1) and (2.11) we have ! Z 1 Z 1   BΩ y¯ + v, r¯; DA [¯ γ ] hξi = A1 [¯ γ ] ∇(¯ y + v) · ∇¯ r dx2 ξ dx1 (2.12) 0

0 1

Z

Z

!

1

A2 [¯ γ ] ∇(¯ y + v) · ∇¯ r dx2

+ 0

ξx1 dx1 ,

(2.13)

0

and the computation of the inner integrals in the variable x2 alone might seem like a daunting task. However we can circumvent this issue altogether with the following simple observation. The control-to-state map Gv : U → W1,1 echet ∞,p admits a Fr´ derivative (γ, y) = G0v (¯ u)h ∈ W1,1 which satisfies the linear variational system ∞,p Z 1   1,1 BΓ [γ, ζ] + DΩ (γ, y) , z + Eζ; γ¯ , y¯ = hζ ∀ (ζ, z) ∈ W1,q , (2.14) 0

for every u ¯ ∈ U and h ∈ L2 (0, 1). This formal differentiation of (2.5b) is rigorously justified in [1, Theorem 4.12]. The system (2.14) consists of the two equations ˚q1 (Ω) ∀z ∈ W ˚11 (0, 1). BΩ [y, Eζ; A [¯ γ ]] + BΩ [¯ y + v, Eζ; DA [¯ γ ] hγi] + BΓ [γ, ζ] = hh, ζi ∀ζ ∈ W BΩ [y, z, A [¯ γ ]] + BΩ [¯ y + v, z; DA [¯ γ ] hγi] = 0

The formal adjoint of this system, obtained upon regarding (γ, y) ∈ W1,1 ∞,p as test 1,1 functions and (ζ, z) ∈ W1,q as unknowns, reads as follows: BΩ [y, z, A [¯ γ ]] + BΩ [y, Eζ; A [¯ γ ]] = hf, yi BΩ [¯ y + v, z; DA [¯ γ ] hγi] + BΩ [¯ y + v, Eζ; DA [¯ γ ] hγi] + BΓ [γ, ζ] = hg, γi.

(2.15)

Lemma 2.1 (relation between (2.10) and (2.15)). The linear system (2.10) coinR1 cides with (2.15) provided f = δ y¯(1 + γ¯ ) and g = δ¯ γ + 12 0 |δ y¯|2 dx2 . Proof. It suffices to check that (ζ, z+Eζ) satisfies (2.10) and invoke the uniqueness of (2.10), the latter being a consequence of [1, Lemma 5.6]. Lemma 2.1 is instrumental for the implementation of this control problem. Later in Section 7, we choose Newton’s method instead of a fixed point iteration to solve the state equations, because it is locally a second order method. Secondly, computing a Newton direction (γ, y) requires solving a linear system of type (2.14). By transposition, the same matrix can be used to solve for the adjoint variables thereby making the seemingly complicated coupling DΩ rather simple to deal with. 6

2.3. Second-order Sufficient Conditions. It is well known that the first order necessary optimality conditions are also sufficient for the well-posedness of a convex optimization problem with linear constraints. Unfortunately, we cannot assert the convexity of our problem due to the highly nonlinear nature of (2.5a)-(2.5b). A large portion of our previous work [1, Theorem 5.7] was devoted to proving a second-order sufficient condition. We restrict ourselves to merely restating that result. Theorem 2.2 (second-order sufficient conditions). If |v|W 1 (Ω) is small enough p and u ¯ in Uad is an optimal control, then there exists a neighborhood of u ¯ such that 2

J 00 (¯ u) (u − u ¯) ≥

λ 2 ku − u ¯kL2 (0,1) 2

∀u ∈ u ¯ + C (¯ u).

(2.16)

Furthermore, the following two conditions hold: local quadratic growth J (u) ≥ J (¯ u) +

λ 2 ku − u ¯kL2 (0,1) 8

∀u ∈ u ¯ + C (¯ u),

(2.17)

and local convexity

J 0 (u) − J 0 (¯ u), u − u ¯

L2 (0,1)×L2 (0,1)



λ 2 ku − u ¯kL2 (0,1) 4

∀u ∈ u ¯ + C (¯ u).

(2.18)

3. Strong Solutions: Second-order Regularity. The goal of this section is to prove the existence of strong solutions to the state and adjoint equations. The underlying second-order Sobolev regularity is crucial for the a-priori error estimates of Section 5, and thus an important contribution of this paper. 3.1. Second-order Regularity in the Square. We start with an auxiliary regularity result for the square Ω. This type of results are well known for C 1,1 domains [7, Theorem 9.15 and Lemma 9.17]. Lemma 3.1 (second-order regularity in the square). Let Ω = (0, 1)2 and A = 1 (aij )2ij=1 ∈ W∞ (Ω). If f ∈ Lp (Ω), with 1 < p < ∞, then there exists a unique solution 2 ˚p1 (Ω) to w ∈ Wp (Ω) ∩ W − div(A∇w) = f

in Ω,

(3.1)

1 (Ω) and p such that and a constant C# depending on kAkW∞

kwkW 2 (Ω) ≤ C# kf kLp (Ω) . p

(3.2)

Proof. We proceed in several steps. First we prove (3.2) assuming that there is a solution in Wp2 (Ω), and next we show the existence of such a solution. e of 1. Reflection: We introduce an odd reflection fe of f and an even reflection A e A to the adjacent unit squares of Ω so that the extended domain Ω is a square with 1 e e and A e ∈ W∞ vertices (−1, −1), (2, −1), (2, 2), (−1, 2). We observe that fe ∈ Lp (Ω) (Ω). e e Since kf kH −1 (Ω) . k f k by Sobolev embedding in two dimensions, there exists a p e e L (Ω) 1 e unique solution w e ∈ H (Ω) to the extended problem 0

e w) e − div(A∇ e = fe in Ω. 7

(3.3)

e of its restriction w to Ω, whence w has a vanishing Such w e is an odd reflection to Ω trace on ∂Ω. Moreover, kwk e Lp (Ω) e H 1 (Ω) e . kwk e because of Sobolev embedding and 0

kwk e Lp (Ω) e . kwkLp (Ω) .

(3.4)

2 e ∩ Lp (Ω) e is a solution of (3.3), then 2. A priori Wp2 (Ω)-estimate: If w e ∈ Wp,loc (Ω) we write (3.3) in nondivergence form

e : D2 w e · ∇w −A e − div A e = fe and apply the interior Wp2 estimates of [7, Theorem 9.11] to write e kwkWp2 (Ω) = kwk e Wp2 (Ω) . kwk e Lp (Ω) e + kf kLp (Ω) e . kwkLp (Ω) + kf kLp (Ω)

(3.5)

˚ 1 (Ω). To show that this estimate implies (3.2), we argue by contradiction and w ∈ W p ˚ 1 (Ω) be a sequence satisfying as in [7, Lemma 9.17]. Let {wm } ⊂ Wp2 (Ω) ∩ W p kwm kWp2 (Ω) = 1,

kfm kLp (Ω) → 0

as m → ∞, where fm = − div(A∇wm ). Since the unit ball in Wp2 (Ω) is weakly compact for 1 < p < ∞, there exists a subsequence, still labeled wm , that converges weakly ˚p1 (Ω). Therefore in Wp2 (Ω) and strongly in Wp1 (Ω) to a function w ∈ Wp2 (Ω) ∩ W Z

Z vfm = −







2

Z

v A : D wm + div A · ∇wm → − Ω

  v A : D2 w + div A · ∇w = 0



for all v ∈ Lq (Ω), whence − div(A∇w) = 0 and w = 0 because of uniqueness. On the other hand, (3.5) yields 1 . kwkLp (Ω) , which is a contradiction. This thus shows the validity of (3.2). 2 e of (3.3). If 3. Existence. It remains to show that there is a solution Wp,loc (Ω) 2 e and e then the unique solution w e of (3.3) belongs to Hloc (Ω) fe ∈ L2 (Ω), e ∈ H01 (Ω) e kwk e H 2 (Ω0 ) . kwk e H 1 (Ω) e + kf kL2 (Ω) e . kf kL2 (Ω) , 0

e [7, Theorem 8.8]. We first let p > 2 and lift the for all Ω0 compactly contained in Ω 2 e upon applying [7, Lemma 9.16] which gives the estimate regularity of w e to Wp,loc (Ω) e kwk e Wp2 (Ω0 ) . kwk e Lp (Ω) e + kf kLp (Ω) e .

(3.6)

e by a sequence {fem } ⊂ L2 (Ω). e Since If 1 < p < 2 instead, we approximate fe ∈ Lp (Ω)  2 2 2 e e w em ∈ Hloc (Ω) ⊂ Wp,loc Ω , we can apply the interior Wp estimates of [7, Theorem 9.11] to deduce (3.6) again for w em . Moreover, (3.6) shows that {w em } is a Cauchy sequence in Wp2 (Ω0 ) for any Ω0 , whence (3.6) remains valid for the limit w. This finishes the proof. 3.2. State Equations. We resort to the fixed point argument in [11, Section 2] and [1, Section 4.1.1]. It consists of three steps: defining a convex set which acts as domain for the fixed point iterator, linearizing the free boundary problem by freezing one variable, and identifying conditions to guarantee a contraction on the convex set. 8

To deal with second-order regularity we need to introduce, besides the space W1,1 ∞,p , the second order Banach subspace product  1,1 2 2 W2,2 ∞,p := W∞,p ∩ W∞ (0, 1) × Wp (Ω) , 2,2 and endow both W1,1 ∞,p and W∞,p with the norms

(γ, y) 1,1 := (1 + βCA )|v| 1 |γ| 1 W (Ω) W (0,1) +|y|W 1 (Ω) , W ∞,p

p



p

and



(γ, y) 2,2 := 1 + 2CA C# kvk 2 kγk 2 Wp (Ω) W∞ (0,1) +kykWp2 (Ω) , W ∞,p

respectively; hereafter CA is a bound in L∞ (Ω) on the operator A and its first and second order derivatives [1, Proposition 2.1], and C# is the constant in (3.2). To guarantee that the assumptions for the first-order regularity results in [1, Section 4.1.1] hold, we must iterate on a subset of [1, Eq. (4.12)] n o B1 := (γ, y) ∈ W1,1 , 1 (0,1) ≤ 1 ∞,p : |y|Wp1 (Ω) ≤ βCA |v|Wp1 (Ω) , |γ|W∞ where β is the inf-sup constant for BΩ in Wp1 (Ω) [1, Proposition 4.1]. For the purpose of finding a strong solution, we further restrict B1 as follows: n o : B2 := (γ, y) ∈ B1 ∩ W2,2 kyk ≤ 2C C kvk , |γ| ≤ 1 . 2 2 2 A # ∞,p Wp (Ω) Wp (Ω) W∞ (0,1) We linearize the free boundary problem by considering the following operator 2,2 T : B2 → W∞,p defined as  T (γ, y) := T1 (γ, y), T2 (γ, y) = (e γ , ye)

∀ (γ, y) ∈ B2 ,

(3.7)

1 2 ˚∞ (0, 1) is the unique solution to (0, 1) ∩ W where γ e = T1 (γ, y) ∈ W∞

− κ d2x1 γ e = A [γ] ∇(y + v) · ν + u in (0, 1) ,

(3.8)

˚p1 (Ω) is the unique solution to and ye = T2 (γ, y) ∈ Wp2 (Ω) ∩ W       − div A T1 (γ, y) ∇e y = div A T1 (γ, y) ∇v in Ω, (3.9)   where div A T1 (γ, y) is computed row-wise. The operator T maps B1 into itself provided v and u are restricted to verify |v|Wp1 (Ω) ≤

1 − θ1 , αCE CA (1 + βCA )

kukL2 (0,1) ≤

θ1 , α

(3.10)

for some θ1 ∈ (βCA /(1 + βCA ), 1) [1, Lemma 4.3]. We now investigate additional conditions for T to map B2 into itself. Lemma 3.2 (range of T ). Let CS be the Sobolev embedding constant between 1 Wp2 (Ω) and W∞ (Ω). The operator T maps B2 to B2 if, in addition to (3.10), the following relation holds  CA CS 1 + 2CA C# kvkWp2 (Ω) +kukL∞ (0,1) ≤ κ. (3.11) 9

Proof. In view of (3.10) we have T (B2 ) ⊂ B1 . Given (γ, y) ∈ B2 , we readily get from (3.8)





≤ A [γ] L∞ (0,1) ∇(y + v) L∞ (Ω) +kukL∞ (0,1) . e κ d2x1 γ ∞ L

(0,1)

As A [γ] L∞ (0,1) ≤ CA , by definition of CA , and for p > 2, Wp2 (Ω) is continuously 1 embedded in W∞ (Ω) with constant CS , we deduce



≤ CA CS ky + vkW 2 (Ω) +kukL∞ (0,1) ≤ κ e κ d2x1 γ ∞ p L

(0,1)

because of (3.11). This implies |e γ |W∞ 2 (0,1) ≤ 1, which is consistent with B2 .  To deal with (3.9), we invoke the a priori estimate (3.2) with f = div A[e γ ]∇v  

γ ] L∞ (Ω) |v|Wp1 (Ω) . ke y kWp2 (Ω) ≤ C# kA[e γ ]kL∞ (Ω) kvkWp2 (Ω) + div A [e This gives ke y kW 2 (Ω) ≤ 2CA C# kvkW 2 (Ω) , p

p

and, together with the previous bound on γ e, yields (e γ , ye) ∈ B2 , as asserted. Remark 3.3 (boundedness of u ¯). We point out that, within the context of the optimal control problem, the L∞ -estimate requirement on u ¯ in (3.11) can be satisfied. The reason is that the variational inequality (2.9) implies that ( if λ¯ u + s¯ = 0 − λs¯ , u ¯= − 2αk¯skθ1 s¯2 , if λ¯ u + s¯ 6= 0 L (0,1)

i.e. the optimal control u ¯ is proportional to the adjoint function s¯ which in turn is ˚ 1 (0, 1) ⊂ L∞ (0, 1). absolutely continuous, i.e s¯ ∈ W 1 Theorem 3.4 (second-order regularity of the state variables). Let CS be the 1 Sobolev embedding constant between Wp2 (Ω) and W∞ (Ω), and 2 Λ = κ−1 CA CS 1 + 2CA C# . If, in addition to (3.10) and (3.11), the function v further satisfies kvkWp2 (Ω) ≤ (1 − θ2 )Λ−1 ,

(3.12)

for some θ2 ∈ (0, 1), then the map T defined in (3.7) is a contraction on B2 with constant 1 − θ2 for all u ∈ Uad . Proof. Let (γ1 , y1 ), (γ2 , y2 ) ∈ B2 with (γ1 , y1 ) 6= (γ2 , y2 ), and set δγ := γ1 − γ2 , δy := y1 − y2 . Combining (3.7) and (3.8) we get an equation for δe γ := γ e1 − γ e2  −κ d2x1 δe γ = A [γ1 ] − A [γ2 ] ∇(y1 + v) · ν + A [γ2 ] ∇δy · ν. Since δγ(0) = δγ(1) = 0 we infer that δγ 0 (x1 ) = 0 for some x1 ∈ (0, 1) and 1 (0,1) ≤ |δγ|W 2 (0,1) , whence |δγ|W∞ ∞

−1 2 (0,1) = |δe kδe γ kW∞ γ |W∞ CA CS (δγ, δy) W2,2 . (3.13) 2 (0,1) ≤ κ ∞,p

10

We next estimate δe y := ye1 − ye2 . In view of (3.9) we see that   − div A [e γ1 ] ∇δe y = div (A [e γ1 ] − A [e γ2 ])∇(e y2 + v) . Invoking the a priori estimate (3.2), we deduce

kδe y kWp2 (Ω) ≤ C# A [e γ1 ] − A [e γ2 ] L∞ (0,1) |e y2 + v|Wp2 (Ω)



|e y2 + v|Wp1 (Ω) . γ1 ] − A [e γ2 ] + C# div A [e ∞ L

(0,1)

Since ke y2 kWp2 (Ω) ≤ 2CA C# kvkWp2 (Ω) , we infer that  kδe y kWp2 (Ω) ≤ 2CA C# 1 + 2CA C# kvkWp2 (Ω) kδe γ k W∞ 2 (0,1) , and, applying (3.13), we obtain 

 2 kδe y kW 2 (Ω) ≤ 2κ−1 CA CS C# 1 + 2CA C# kvkW 2 (Ω) (δγ, δy) W2,2 . p

∞,p

p

(3.14)

The definition of W2,2 ∞,p norm, together with (3.13) and (3.14), leads to



2

(δe γ , δe y ) W2,2 ≤ κ−1 CA CS 1 + 2CA C# kvkW 2 (Ω) (δγ, δy) W2,2 , ∞,p

p

∞,p



and (3.12) gives (δe γ , δe y ) W2,2 ≤ θ2 (δγ, δy) W2,2 , which is the assertion. ∞,p

∞,p

3.3. Adjoint Equations. We begin by assuming that (¯ γ , y¯) belongs to B2 and rewriting the adjoint equations (2.10) in strong divergence form in Ω  − div A [¯ γ ] ∇¯ r = δ y¯ (1 + γ¯ ) , (3.15a) and in (0, 1) −κ d2x1 s¯ = δ¯ γ+ Z −

1 2

Z

1

2

|δ y¯| dx2 0

1

Z A1 [¯ γ ] ∇ (¯ y + v) · ∇¯ r dx2 + dx1

0

(3.15b)

1

A2 [¯ γ ] ∇ (¯ y + v) · ∇¯ r dx2 0

together with the boundary conditions r¯ = 0 on Σ, r¯ = s¯ on Γ, and s¯(0) = s¯(1) = 0. Theorem 3.5 (second-order regularity of adjoint variables). The solution (¯ s, r¯) along with the following a-priori estimates to (3.15) satisfies (¯ s, r¯ − E¯ s) ∈ W2,2 1,q 2

k¯ skW 2 (0,1) +k¯ rkWq2 (Ω) . kδ¯ γ kL1 (0,1) +kδ y¯kL2 (Ω) +kδ y¯kLq (Ω) , 1

provided the function v satisfies   4CA CE (1 + 2C# ) 2(1 + 2CA C# ) + α(1 + βCA ) kvkWp2 (Ω) ≤ 1.

(3.16)

˚ 1 (0, 1) → Wq2 (Ω) is the Proof. Setting r¯ = r¯0 + E¯ s, where E : W12 (0, 1) ∩ W 1 extension operator for q < 2, we can rewrite (3.15a) as   − div A [¯ γ ] ∇¯ r0 = div A [¯ γ ] ∇E¯ s + δ y¯ (1 + γ¯ ) , 11

with r¯0 |∂Ω = 0. We apply (3.2) to obtain r¯0 ∈ Wq2 (Ω) and k¯ r0 kW 2 (Ω) ≤ 2C# CE k¯ skW 2 (0,1) + 2C# kδ y¯kLq (Ω) , q

1

and similarly for s¯ 1 2 |¯ s|W 2 (0,1) ≤ kδ¯ γ kL1 (0,1) + kδ y¯kL2 (Ω) + 4CA k¯ y + vkWp2 (Ω) k¯ rkWq2 (Ω) . 1 2 Using the fact y¯ ∈ B2 we deduce  1 2 |¯ s|W 2 (0,1) ≤ kδ¯ γ kL1 (0,1) + kδ y¯kL2 (Ω) + 4CA 1 + 2CA C# kvkWp2 (Ω) k¯ rkWq2 (Ω) . 1 2 Recalling the estimate for |¯ s|W 1 (0,1) from [1, Lemma 5.5] 1

|¯ s|W 1 (0,1) ≤ αkδ¯ γ kL1 (0,1) + 1

 α 2 kδ y¯kL2 (Ω) + αCA 1 + βCA |v|W 1 (Ω) |¯ r|W 1 (Ω) , p q 2

and using that k¯ skL1 (0,1) ≤ |¯ s|W 1 (0,1) , we end up with 1

1 2 rkWq2 (Ω) , k¯ skW 2 (0,1) ≤ (1 + 2α)kδ¯ γ kL1 (0,1) + (1 + 2α)kδ y¯kL2 (Ω) + λkvkWp2 (Ω) k¯ 1 2  where λ = 2CA 2(1 + 2CA C# ) + α(1 + βCA ) . Inserting the estimate for k¯ r0 kWq2 (Ω) into this estimate, we get 1 2 k¯ skW 2 (0,1) ≤ (1 + 2α)kδ¯ γ kL1 (0,1) + (1 + 2α)kδ y¯kL2 (Ω) 1 2 + λCE (1 + 2C# )kvkWp2 (Ω) k¯ skW 2 (Ω) + 2λC# kvkWp2 (Ω) kδ y¯kLq (Ω) 1

and the desired estimate for k¯ skW 2 (0,1) follows from (3.16). This, and the relation 1 r¯ = r¯0 + E¯ s yields the remaining estimate for k¯ rkWq2 (Ω) , and concludes the proof. 4. Discrete Optimal Control Problem. The goal of this section is to introduce the discrete counterpart of the optimization problem (2.5). The discretization uses the finite element method and is classical. Let T denote a geometrically conforming rectangular quasi-uniform triangulation of the fixed domain Ω such that Ω = ∪K∈T K and h ≈ hK be the meshsize of T . Additionally, let 0 = ζ0 < ζ1 < . . . < ζM +1 = 1 be a partition of [0, 1] with nodes ζi compatible with T . Consider the following finite dimensional spaces, where the capital letters stand for discrete objects: n o ¯ : Y |K ∈ P 1 (K), K ∈ T , Vh := Y ∈ C 0 (Ω) (4.1a) ˚ ˚p1 (Ω), Vh := Vh ∩ W n o Sh := G ∈ C 0 ([0, 1]) : G|[ζi ,ζi+1 ] ∈ P 1 ([ζi , ζi+1 ]), 0 ≤ i ≤ M , 1 ˚ ˚∞ Sh := Sh ∩ W (0, 1), Uad := Sh ∩ Uad ,

(4.1b) (4.1c) (4.1d) (4.1e)

and P 1 (D) stands for bilinear polynomials on an element D = K ∈ T or linear polynomials on an interval D = [ζi , ζi+1 ]. The spaces ˚ Vh , ˚ Sh and Uad in (4.1) will be 12

used to approximate the continuous solutions (y, γ, u) of (2.5). This discretization is classical [4, Chapter 3], except perhaps for the L2 constraint in Uad , which we enforce by scaling the functions with their L2 -norm; for more details we refer to Section 7. Next we present a discrete analog of the continuous extension (2.3), namely ∀G ∈ ˚ Sh .

Eh G := (Sh ◦ E)(G),

The caveat is that functions in Wq1 (Ω) are not necessarily continuous. This issue is addressed by utilizing the Scott-Zhang interpolation operator Sh : Wq1 (Ω) → Vh . This operator satisfies the optimal estimate [4], |w − Sh w|Wq1 (Ω) . h|w|Wq2 (Ω) , ∀w ∈ Wq2 (Ω), 1 ≤ q ≤ ∞.

(4.2)

1 For functions in Wp1 (Ω) with p > 2, W∞ (0, 1) and W11 (0, 1) we will use the standard Lagrange interpolation operator Ih . This is justified by the Sobolev embedding theorems, i.e. we can identify functions in those spaces with their continuous equivalents. Moreover, the following optimal interpolation estimates hold,

|y − Ih y|Wp1 (Ω) . h|y|Wp2 (Ω) , ∀y ∈ Wp2 (Ω), |γ − Ih γ|Wp1 (0,1) . h|γ|Wp2 (0,1) , ∀γ ∈

Wp2

2 < p ≤ ∞,

(0, 1),

1 ≤ p ≤ ∞.

(4.3a) (4.3b)

Next we state the discrete counterpart of the optimal control problem (2.5a) in its variational form: if δG := G − γd , δY := Y + v − yd , then minimize

2 1 λ 1

√ 2 2 + kU kL2 (0,1) , (4.4a) Jh (G, Y, U ) := kδGkL2 (0,1) + δY 1 + G 2 2 2 L2 (Ω) subject to the discrete state equation (G, Y ) ∈ ˚ Sh × ˚ Vh Z 1   BΓ [G, Ξ] + BΩ Y + v, Z + Eh Ξ; A [G] = U Ξ ∀(Ξ, Z) ∈ ˚ Sh × ˚ Vh ,

(4.4b)

0

the state constraints 0 G ≤ 1

on (ζi , ζi+1 ) , i = 0, ..., M − 1,

(4.4c)

and the control constraints U ∈ Uad . We point out that Y |∂Ω = 0 in (4.4b). This is not the standard approach in finite element literature because it requires knowing an extension of v to Ω; we adopt this approach to simplify the exposition. We must include the following mild regularity assumptions on data in order to obtain an order of convergence: (A3 ) The given data satisfy v ∈ Wp2 (Ω), γd ∈ L2 (0, 1) and yd ∈ L2 (Ω∗ ). ¯ denote the optimal control to (4.4a), whose existence will be shown in Now let U  ¯ Y¯ be the optimal state, which satisfy discrete state equations Theorem 4.2, and G, in variational form (4.4b). The discrete adjoint equations in variational form read: ¯ R) ¯ such that (S, ¯ R ¯ − Eh S) ¯ ∈˚ find (S, Sh × ˚ Vh and for every (Ξ, Z) ∈ ˚ Sh × ˚ Vh , Z 1 E D 2     D E ¯ . (4.5) ¯ G, ¯ Y¯ = Ξ, δ G ¯+ 1 δ Y¯ dx2 + Z, δ Y¯ 1 + G BΓ Ξ, S¯ + DΩ Ξ, Z, R; 2 0 13

¯ satisfies the variational inequality Finally, the optimal control U

0 ¯ ), U − U ¯ 2 ≥ 0 ∀U ∈ Uad , Jh ( U L (0,1)×L2 (0,1)

(4.6)

¯ ) = S¯ + λU ¯ . Therefore (4.6) reads where Jh0 (U

¯, U − U ¯ 2 ≥0 S¯ + λU L (0,1)×L2 (0,1)

(4.7)

∀U ∈ Uad .

The following discrete estimates mimic the continuous inf-sup [1, Proposition 4.1]. Proposition 4.1 (discrete inf-sup). The following two statements hold: (i) There exists constant 0 < α < ∞ independent of h such that BΓ [G, Ξ] , |Ξ| W 1 (0,1) 06=Ξ∈˚ Sh

(4.8a)

BΓ [Ξ, S] . |Ξ| 1 (0,1) W∞ 06=Ξ∈˚ Sh

(4.8b)

|G|W∞ sup 1 (0,1) ≤ α

1

|S|W 1 (0,1) ≤ α sup 1

(ii) There exists constant 0 < β < ∞ independent of h and constants Q < 2 < P , h0 > 0, such that for p ∈ [Q, P ] and 0 < h ≤ h0   BΩ Y, Z; A [G] . (4.9) |Y |Wp1 (Ω) ≤ β sup |Z|W 1 (Ω) 06=Z∈˚ Vh q

Proof. We refer to [11, Proposition 3.2] for a proof of (4.8a) and to [4, Proposition 8.6.2] for a proof of (4.9). The technique of [11] extends to (4.8b). Existence and uniqueness of solutions to the state and adjoint equations can be shown similarly to the continuous case [1, Corollary 4.6, and Theorem 5.6] provided ¯ U ∈ Uad and |v|Wp1 (Ω) is small. We will next prove existence of an optimal control U solving (4.4a). Theorem 4.2 (existence of optimal control). There exists a discrete optimal ¯ ∈ Uad which solves (4.4a). control U Proof. The proof follows by using a minimizing sequence argument similar to the continuous proof [1, Theorem 5.1]. However, weak convergence of a minimizing sequence {Un } yields strong convergence in finite dimensional spaces. Following [1, Theorem 4.8] it is routine to show that the discrete control-to-state map is Lipschitz continuous. Together with this Lipschitz continuity and the strong convergence of Un , we also obtain strong convergence of the associated state sequence {(Gn , Yn )}. 5. A-priori Error Estimates: State and Adjoint Variables. The goal of this section is to derive a-priori error estimates between the continuous and discrete solutions of the state and adjoint equations for given functions u ∈ Uad and U ∈ Uad . This is the content of Lemmas 5.1 through 5.6. These estimates are the stepping ¯ in Theorem 6.1. stone for the L2 estimate of u ¯−U Lemma 5.1 (preliminary error estimate for γ). Given u ∈ Uad and U ∈ Uad , let (γ, y) and (G, Y ) solve (2.5b) and (4.4b) respectively for v ∈ Wp2 (Ω) with |v|W 1 (Ω) p small. Then the following error estimate for γ − G holds |γ − G|W∞ 1 (0,1) . h|γ|W 2 (0,1) +|y − Y |W 1 (Ω) +ku − U kL2 (0,1) . ∞ p 14

Proof. We use the discrete inf-sup (4.8a) to infer that |Ih γ − G|W 1 (0,1) . sup ∞

06=Ξ∈˚ Sh

BΓ [Ih γ − G, Ξ] . |Ξ|W 1 (0,1) 1

Next, we rewrite BΓ [Ih γ − G, Ξ] = BΓ [Ih γ − γ, Ξ] + BΓ [γ − G, Ξ], and estimate the first term using H¨ older’s inequality and (4.3b). For the second term we set w = y + v and W = Y + v, use that γ and G satisfy (2.5b) and (4.4b) respectively, and the fact that |y|Wp1 (Ω) . |v|Wp1 (Ω) , to obtain     BΓ [γ − G, Ξ] = −BΩ w, Eh Ξ; A [γ] + BΩ W, Eh Ξ; A [G] + hu − U, Ξi     = BΩ w, Eh Ξ; −A [γ] + A [G] + BΩ W − w, Eh Ξ; A [G] + hu − U, Ξi   . |γ − G|W∞ 1 (0,1) |v|W 1 (Ω) +|y − Y |W 1 (Ω) +ku − U kL2 (0,1) |Ξ|W 1 (0,1) , p p 1

where hu − U, Ξi = hu − U, ΞiL2 (0,1)×L2 (0,1) . Combining the above two estimates with the triangle inequality and (4.3b), we end up with |γ − G|W∞ 1 (0,1) . h|γ|W 2 (0,1) ∞ +|γ − G|W∞ 1 (0,1) |v|W 1 (Ω) +|y − Y |W 1 (Ω) +ku − U kL2 (0,1) . p p Using that |v|Wp1 (Ω) is small finally yields the desired result. Lemma 5.2 (error estimate for y). Given u ∈ Uad and U ∈ Uad , let (γ, y) and (G, Y ) solve (2.5b) and (4.4b) respectively with |v|W 1 (Ω) small. Then the following p estimate for y − Y holds   |y − Y |Wp1 (Ω) . h |γ|W∞ +|v|Wp1 (Ω) ku − U kL2 (0,1) . 2 (0,1) +|y|W 2 (Ω) p Proof. We proceed as in Lemma 5.1. We use the discrete inf-sup followed by the interpolation estimate (4.3a), together with the state constraint |G|W∞ 1 (0,1) ≤ 1, to obtain   BΩ Ih y − Y, Z; A [G] |Ih y − Y |Wp1 (Ω) . sup |Z|Wq1 (Ω) 06=Z∈˚ Vh   BΩ y − Y, Z; A [G] . h|y|W 2 (Ω) + sup . p |Z|Wq1 (Ω) 06=Z∈˚ Vh We handle the last term by using that y and Y are solutions to (2.5b) and (4.4b), i.e.       BΩ y − Y, Z; A [G] = BΩ y + v, Z; A [G] − BΩ Y + v, Z; A [G]   = BΩ y + v, Z; A [G] − A [γ] , followed by the bound |y|Wp1 (Ω) . |v|Wp1 (Ω) in the definition of B1 to yield   BΩ y − Y, Z; A [G] . |γ − G|W∞ 1 (0,1) |v|W 1 (Ω) |Z|W 1 (Ω) . p q 15

Combining the above estimates with Lemma 5.1, and using the triangle inequality in conjunction with (4.3a), we obtain |y − Y |Wp1 (Ω) . h|y|Wp2 (Ω) +|γ − G|W∞ 1 (0,1) |v|W 1 (Ω) p   . h |γ|W∞ +ku − U kL2 (0,1) |v|Wp1 (Ω) 2 (0,1) |v|W 1 (Ω) +|y|W 2 (Ω) p p +|y − Y |W 1 (Ω) |v|W 1 (Ω) . p

p

The desired estimate is a consequence of the smallness assumption on |v|Wp1 (Ω) . Lemma 5.3 (error estimate for γ). Given u ∈ Uad and U ∈ Uad , let (γ, y) and (G, Y ) solve (2.5b) and (4.4b) respectively with |v|W 1 (Ω) small. Then the following p error estimate for γ − G holds   |γ − G|W∞ +ku − U kL2 (0,1) . 1 (0,1) . h |γ|W 2 (0,1) +|y|W 2 (Ω) ∞ p Proof. The assertion follows by combining Lemma 5.2 with Lemma 5.1. Lemma 5.4 (preliminary error estimate for s). Given u ∈ Uad and U ∈ Uad , ˚ 1 (0, 1) × W ˚q1 (Ω) satisfy the continuous adjoint system (2.10), and let (s, r − Es) ∈ W 1 (S, R − Eh S) ∈ ˚ Sh × ˚ Vh satisfy the discrete counterpart (4.5). Then the following error estimate for s − S is valid   |s − S|W 1 (0,1) . h|s|W 2 (0,1) + 1 +|r|W 1 (Ω) |v|W 1 (Ω) |γ − G|W 1 (0,1) ∞ q p 1 1   + kδykL2 (Ω) +|y − Y |W 1 (Ω) +|r|W 1 (Ω) |y − Y |W 1 (Ω) +|r − R|W 1 (Ω) |v|W 1 (Ω) . p

p

q

q

p

Proof. We again employ the discrete inf-sup (4.8b), now taking the form |Ih s − S|W 1 (0,1) . sup 1

06=Ξ∈˚ Sh

BΓ [Ξ, Ih s − S] |Ξ|W 1 (0,1) ∞

. h|s|W 2 (0,1) + sup 1

06=Ξ∈˚ Sh

BΓ [Ξ, s − S] , |Ξ|W∞ 1 (0,1)

where the last inequality follows by adding and subtracting s, the continuity of BΓ , and the interpolation estimate (4.3b) for s − Ih s. It remains to control the last term. We use that s and S satisfy equations (2.10) and (4.5) to obtain BΓ [Ξ, s − S] = hΞ, γ − γd i − hΞ, G − γd i D 1Z 1 E D 1Z 1 E 2 2 |y + v − yd | dx2 − Ξ, |Y + v − yd | dx2 + Ξ, 2 2 0  0    − BΩ y + v, r; DA [γ] hΞi + BΩ Y + v, R; DA [G] hΞi D 1Z 1 E = hΞ, γ − Gi + Ξ, (y − Y ) (y + Y + 2v − 2yd ) dx2 2 0 h i    − BΩ y − Y, r; DA [γ] hΞi − BΩ Y + v, r; DA [γ] − DA [G] hΞi   − BΩ Y + v, r − R; DA [G] hΞi . 16

Consequently, after normalization |Ξ|W∞ 1 (0,1) = 1, we infer that   BΓ [Ξ, s − S] . kγ − Gk 1 L (0,1) +ky − Y kL2 (Ω) 2ky + v − yd k +kY − ykL2 (Ω)   +|y − Y |Wp1 (Ω) |r|Wq1 (Ω) +|v|Wp1 (Ω) |r|Wq1 (Ω) |γ − G|W∞ . 1 (0,1) +|r − R|W 1 (Ω) q The desired result for|Ih s − S|W 1 (0,1) follows after combining the above estimate with 1 kγ − GkL1 (0,1) . |γ − G|W 1 (0,1) and ky − Y kL2 (Ω) . |y − Y |W 1 (Ω) . Finally, applying p ∞ the triangle inequality in conjunction with (4.3b) we deduce the asserted estimate for |s − S|W 1 (0,1) . 1 Lemma 5.5 (error estimate for r). Given u ∈ Uad and U ∈ Uad , let (s, r − Es) ∈ ˚ 1 (0, 1) × W ˚ 1 (Ω) satisfy the continuous adjoint system (2.10), and (S, R − Eh S) ∈ W q 1 ˚ Sh × ˚ Vh satisfy the discrete counterpart (4.5). Then the following a-priori error estimate for r − R holds   |r − R|Wq1 (Ω) . h |s|W 2 (0,1) +|r|Wq2 (Ω) 1     + 1 + 1 +|v|Wp1 (Ω) |r|Wq1 (Ω) +kδykLq (Ω) |γ − G|W∞ 1 (0,1)   + 1 +kδykL2 (Ω) +|y − Y |Wp1 (Ω) +|r|Wq1 (Ω) |y − Y |Wp1 (Ω) . Proof. Since the discrete inf-sup (4.9) is for functions in ˚ Vh , we write r = r0 + Es, ˚q1 (Ω) and R0 ∈ ˚ Vh , to obtain and R = R0 + Eh S, with r0 ∈ W |r − R|Wq1 (Ω) ≤ |r0 − Sh r0 |Wq1 (Ω) +|Sh r0 − R0 |Wq1 (Ω) +|Es − Eh S|Wq1 (Ω) . Consequently, applying (4.9) |Sh r0 − R0 |Wq1 (Ω) . h|r0 |Wq2 (Ω) +

sup 06=Z∈˚ Vh

  BΩ Z, r0 − R0 ; A [G] , |Z|Wp1 (Ω)

where we have added and subtracted r0 . Moreover, we handle the last term as before, i.e.       BΩ Z, r0 − R0 ; A [G] = BΩ Z, r0 + Es; A [γ] − BΩ Z, R0 + Eh S; A [G]     + BΩ Z, r0 + Es; A [G] − A [γ] + BΩ Z, Eh S − Es; A [G] . Invoking the adjoint equations (2.10) and (4.5), we see that     BΩ Z, r0 + Es; A [γ] − BΩ Z, R0 + Eh S; A [G]



= (y + v − yd ) (1 + γ) , Z − (Y + v − yd ) (1 + G) , Z

= hy − Y, Zi + hyγ − Y G, Zi + (v − yd ) (γ − G) , Z . Since yγ − Y G = y (γ − G) − (Y − y) G, after normalization |Z|Wp1 (Ω) = 1 and using (4.4c), we obtain     BΩ Z, r0 − R0 ; A [G] . ky − Y kLq (Ω) +|γ − G|W 1 (0,1) |r|W 1 (Ω) +kδykLq (Ω) ∞

+|Es − Eh S|Wq1 (Ω) . 17

q

Combining this together with |Es − Eh s|Wq1 (Ω) . h|s|W 2 (0,1) and |Eh s − Eh S|Wq1 (Ω) . 1 |s − S|W 1 (0,1) , we end up with 1

  |r − R|W 1 (Ω) . h |s|W 2 (0,1) +|r|W 2 (Ω) q

q

1

  +ky − Y kLq (Ω) +|γ − G|W∞ 1 (0,1) |r|W 1 (Ω) +kδykLq (Ω) q

+|s − S|W 1 (0,1) . 1

Finally, under the smallness assumption on |v|Wp1 (Ω) and ky − Y kLq (Ω) . |y − Y |Wp1 (Ω) , Lemma 5.4 yields the desired result. Lemma 5.6 (error estimate for s). The following a-priori estimate for s−S holds   |s − S|W 1 (0,1) . h |γ|W 2 (0,1) +|y|W 2 (Ω) +|s|W 2 (0,1) +|r|W 2 (Ω) +ku − U kL2 (0,1) . 1



p

q

1

Proof. We use Lemmas 5.4 and 5.5, to obtain    |s − S|W 1 (0,1) . h |s|W 2 (0,1) +|v|Wp1 (Ω) |s|W 2 (0,1) +|r|Wq2 (Ω) 1 1 1   + c1 +|v|Wp1 (Ω) (c1 + c3 ) |γ − G|W∞ 1 (0,1)    + c2 +|y − Y |Wp1 (Ω) +|v|Wp1 (Ω) 1 + c2 +|y − Y |Wp1 (Ω) |y − Y |Wp1 (Ω) where c1 = 1 +|r|W 1 (Ω) |v|W 1 (Ω) , q

p

c2 = |r|Wq1 (Ω) +kδykL2 (Ω) , c3 = |r|W 1 (Ω) +kδykLq (Ω) . q

The assertion follows by applying Lemmas 5.2 and 5.3, together with |v|Wq1 (Ω) ≤ 1. 6. A-priori Error Estimates: Optimal Control. Next we derive the a-priori ¯. error estimate between u ¯ and U Theorem 6.1 (error estimate for u). Let both h0 and |v|Wp1 (Ω) be sufficiently small. If h ≤ h0 , then

4 ¯ ¯ 2 ¯ ) 2

u ¯−U ≤ s(U ) − S(U , L (0,1) L (0,1) λ

(6.1)

 ¯ ) is the solution of the continuous adjoint equation (2.10) with γ(U ¯ ), y(U ¯) where s(U ¯ , and S(U ¯ ) is the solution of the solutions of the state equation (2.5b) with control U discrete adjoint equation (4.5). Proof. The proof relies primarily on the continuous quadratic growth condition (2.18) and on the continuous and discrete first-order optimality conditions (2.9) and ¯ ∈ Uad is admissible, according to (4.1e), replacing u by U ¯ in (2.18) we (4.6). Since U get

2

λ ¯ −u ¯ ) − J 0 (¯ ¯ −u

U ¯ L2 (0,1) ≤ J 0 (U u), U ¯ L2 (0,1)×L2 (0,1) . 4 18

¯ ) gives Adding and subtracting Jh0 (U

2

λ ¯ ) − J 0 (U ¯ ), U ¯ −u ¯ −u

U ¯ L2 (0,1)×L2 (0,1) ¯ L2 (0,1) ≤ J 0 (U h 4



¯ ), U ¯ −u ¯ 2 + Jh0 (U . ¯ L2 (0,1)×L2 (0,1) + J 0 (¯ u), u ¯−U L (0,1)×L2 (0,1)

¯ 2 ≤ 0, according to (2.8), we deduce Since J 0 (¯ u), u ¯−U L (0,1)×L2 (0,1)

2

λ ¯ ) − J 0 (U ¯ ), U ¯ −u ¯ −u

U ¯ L2 (0,1)×L2 (0,1) ¯ L2 (0,1) ≤ J 0 (U h 4

¯ ), U ¯ −u + Jh0 (U ¯ L2 (0,1)×L2 (0,1) . Add and subtract Ph u ¯, the L2 orthogonal projection of u ¯ onto Uad , to get

λ ¯ ¯ ) − Jh0 (U ¯ ), U ¯ −u kU − u ¯k2L2 (0,1) ≤ J 0 (U ¯ L2 (0,1)×L2 (0,1) 4



¯ ), Ph u ¯ ), U ¯ − Ph u + Jh0 (U ¯−u ¯ L2 (0,1)×L2 (0,1) + Jh0 (U ¯ L2 (0,1)×L2 (0,1) . ¯ ) ∈ Sh the middle term vanishes. In view of (4.6) and the fact that Since Jh0 (U

¯ ), U ¯ − Ph u Ph u ¯ ∈ Uad , we deduce Jh0 (U ¯ L2 (0,1)×L2 (0,1) ≤ 0 and

2

λ ¯ ) − Jh0 (U ¯ ), U ¯ −u ¯ −u

U ¯ L2 (0,1)×L2 (0,1) , ¯ L2 (0,1) ≤ J 0 (U 4 ¯ ) = λU ¯ + s(U ¯ ) and J 0 (U ¯ ) = λU ¯ + S(U ¯ ) yield The explicit expressions J 0 (U h

2

λ ¯ −u ¯ ) − S(U ¯ ), U ¯ −u

U ¯ L2 (0,1) ≤ s(U ¯ L2 (0,1)×L2 (0,1) , 4 which imply the desired estimate (6.1). Corollary 6.2 (rate of convergence). Let both h0 and |v|W 1 (Ω) be sufficiently p ¯ ), r(U ¯ )) be the solutions of the continuous adjoint equasmall. Furthermore, let (s(U  ¯ ), y(U ¯ ) solutions for the continuous state equation (2.5b) with tion (2.10) with γ(U  ¯ ¯ ¯ ) solve the discrete adjoint equation (4.5) with control U .  Let S(U ), R(U ¯ ), Y (U ¯ ) solutions for the discrete state equation (4.4b) with control U ¯ . If h ≤ G(U h0 , then there is a constant C0 ≥ 1, depending on kγkW∞ 2 (0,1) , kykW 2 (Ω) , kskW 2 (0,1) , p 1 krkWq2 (Ω) , kγd kL2 (0,1) , kyd kL2 (Ω) , such that ¯ ) − G(U ¯ ) 1 ¯ ) − Y (U ¯ ) 1 γ(U + y(U W∞ (0,1) Wp (Ω)

¯ ¯ ¯ ¯ ) 1 ¯ 2

u + s(U ) − S(U ) W 1 (0,1) + r(U ) − R(U + λ ¯ − U ≤ C0 h. W (Ω) L (0,1) q

1

Proof. We combine the estimate

¯ ) − S(U ¯ ) 2

s(U

L (0,1)

¯ ) − S(U ¯ ) 1 ≤ s(U , W (0,1) 1

¯ with Lemma 5.6 for u = U = U

¯ ) − S(U ¯ ) 2

s(U ≤ C1 h L (0,1) 19

(6.2)

where the constant C1 has the

same dependencies as C0 . This together with (6.1), ¯ 2 implies the error estimate for u ¯−U in (6.2). For the remaining estimates in L (0,1) ¯ ¯ (6.2) set u = U , and U = U in Lemmas 5.2, 5.3, 5.5 and 5.6 to complete the proof. Remark 6.3 (linear rate). The first-order convergence rate of (6.2) is optimal for a piecewise-linear finite element discretization of (γ, y, s, r). For a control u in L2 , one might expect an increased rate of convergence. For example, it would be possible to use the standard Aubin-Nitsche duality argument if we were in a traditional linear setting to obtain

¯ ) − S(U ¯ ) 1 ¯ ) − S(U ¯ ) 2

s(U , ≤ h1/2 s(U W (0,1) L (0,1) 1

¯ in the proof which in turn would yield an optimal rate of convergence h3/2 for u ¯−U of Corollary 6.2. Unfortunately, the duality method fails in our setting because the right-hand-side of the adjoint equations is discretized as well. Thus we are left merely with the the Sobolev embedding

¯ ) − S(U ¯ ) 2 ¯ ) − S(U ¯ ) 1

s(U . s(U . L (0,1) W (0,1) 1

¯. In turn, this yields the linear rate of convergence for u ¯−U Remark 6.4 (dependence on κ). Since κ is the ellipticity constant for s, then the estimate of u is inversely proportional to the surface tension coefficient κ in view of (6.1). 7. Simulations. In our computations we assume the cost functional J in (2.5a) to be independent of y and yd ; we thus have J (γ, y, u) :=

1 λ 2 2 kγ − γd kL2 (0,1) + kukL2 (0,1) . 2 2

(7.1)

Our goal is to compute an approximation to the optimization problem presented in §2.5 with the cost functional in (7.1), the Dirichlet data v = x2 (1 − x2 )(1 − 2x1 ) applied to the entire boundary of Ω, and the desired configuration γd set to be an inverted hat function (see Figure 7.1). Moreover, we recall that γ satisfies the state 2 (0, 1) (see Theorem 3.4), equations (1.2b), and the second-order regularity γ ∈ W∞ whence the profile γd is not achievable. We also remark that the curvature is not linearized as was done for the analysis of (2.5b) and (2.1). In view of the control constraint u ∈ Uad (2.6), we need kukL2 (0,1) ≤ θ1 /2α. Since α ∼ 1/κ and θ1 < 1, we have kukL2 (0,1) ∼ κ. In our computations we have κ ≤ 1, this motivates us to consider the following set for the admissible controls. n o Uad = u ∈ L2 (0, 1) : kukL2 (0,1) ≤ 3 . We discretize the state variables (γ, y), the adjoint variables (s, r) and the control u using piecewise bi-linear finite elements. We remark that in our case the first optimize then discretize approach is equivalent to first discretize then optimize (see [9, p. 160-164]). To solve the state equations we use an affine invariant Newton strategy from [6, NLEQ-ERR, p. 148-149] because of its local quadratic convergence. The weak adjoint equations (2.10), or (3.15) in strong form, involve the coupling between the 2d bulk and 1d interface. This seemingly complicated coupling might entail an unusual assembly procedure and geometric mesh restrictions to evaluate integrals in (2.12). This is fortunately not the case because the matrix of the adjoint 20

Fig. 7.1. The inverted hat function indicates the desired state γd and the colors indicate the the 2 (0, 1). state y corresponding to the configuration γd . This profile γd is not achievable because γ ∈ W∞

system happens to be the transpose of the Jacobian of the state equations, according to Lemma 2.1, which is available to us from the Newton method. The assembly can thus be done with ease. We work on the platform provided by the deal.II finite element library [3] and use a direct (built-in) solver to invert the Jacobian at every Newton iteration, as well as the linear adjoint algebraic system. Consequently we can compute the derivative J 0 of the cost functional. We use a gradient based minimization algorithm to solve the minimization problem in Matlab. In particular, we use the built-in Matlab functions fmincon (constrained case), and fminunc (unconstrained case). Stopping criterion: the optimization algorithm stops when the gradient of the cost function is less than or equal to λ · 1e-4, or if the difference between two consecutive values of the cost function are less than or equal to λ · 1e-4. We present two examples to illustrate our theoretical a priori estimate for the control in Corollary 6.2. In particular, we study the behavior of the solution as the regularization parameter λ goes to zero; they differ on whether or not the control is a constrained quantity. For each of these examples we collect the following metrics • The cost function value J (¯ u). • The smallest eigenvalue of J 00 (¯ u), representing the constant δ in the 2nd 2 order sufficient condition J 00 (¯ u)h2 ≥ δkhkL2 (0,1) . This metric is obtained in Matlab through the approximated Hessian provided by the fmincon or fminunc functions.  ¯T MU ¯ 1/2 , where • The discrete L2 norm of the optimal control u ¯ is equal to U M denotes the mass matrix corresponding to 1d problem in the interval (0, 1). • The “self-convergence” rate of the optimal control as we uniformly refine the finite element mesh. We first solve the problem on a very fine mesh, 8 uniform refinement cycles, and use it in place of a closed form solution. Deriving a closed form solution to a nonlinear optimization problem is rather complicated and thus impractical. λ J (¯ u) J 00 (¯ u) k¯ ukL2 (Γ) rate

1 7.32e-2 7.56e-2 0.05 1.1610

1e-1 6.27e-2 8.10e-3 0.45 2.0202

1e-2 2.77e-2 1.30e-3 1.71 1.1224

1e-3 7.80e-3 6.24e-4 3.00 1.8402

1e-4 1.60e-3 5.57e-4 5.00 1.70

1e-5 2.52e-4 5.50e-4 6.30 1.5019

1e-6 4.20e-5 5.49e-4 8.06 1.2117

Table 7.1 Example 1 (Unconstrained case): the values of the cost function J (¯ u), the smallest eigenvalue of J 00 (¯ u), the L2 -norm of u ¯ and the convergence rate of optimal control as λ varies from 1 to 1e-6. 21

7.1. Example 1: Unconstrained control. We begin with the nominal case u ∈ L2 (0, 1) and κ = 1, i.e. the control is unconstrained and the surface tension coefficient is fixed. We are interested in the metrics J (¯ u), J 00 (¯ u), k¯ ukL2 (Γ) and convergence rate as the control regularization parameter λ approaches zero; see Table 7.1. Recall that we used a fixed point argument to prove the existence and uniqueness of a solution for the state equations which required u ¯ ∈ Uad . For λ = 1e-6, we have k¯ ukL2 (0,1) = 8, i.e. u ¯ 6∈ Uad . Nevertheless, we can still solve the state equations. This indicates that our choice of Uad is not sharp and we can solve the state equations even for larger u ¯. The smallest eigenvalue of the approximated Hessian J 00 (¯ u) for λ = 1e-6 is 5.5e4 i.e. the control u ¯ is also locally unique. The last row in Table 7.1 justifies the theoretical findings in Corollary 6.2. (¯ γ , y¯) u ¯ λ = 1e − 3, u ¯ ∈ (−5.38, 0.93)

(¯ γ , y¯) u ¯ λ = ∞, u ¯ ∈ [0, 0]

λ = 1, u ¯ ∈ (−0.0784971, 0]

λ = 1e − 4, u ¯ ∈ (−9.78, 4.96)

λ = 1e − 1, u ¯ ∈ (−0.675277, 0]

λ = 1e − 5, u ¯ ∈ (−17.63, 6.07)

λ = 1e − 2, u ¯ ∈ (−2.65675, 0]

λ = 1e − 6, u ¯ ∈ (−33.1146, 6.73)

Fig. 7.2. Example 1 (Unconstrained case): The optimal state solution (¯ γ , y¯), the applied control u ¯ in solid blue, and the previous control in dashed red for comparison. Each picture displays the corresponding value of λ from λ = ∞ to λ = 1e-6.

The first column in Figure 7.2 shows the optimal state (¯ γ , y¯) as λ approaches zero. The second column shows the control applied (solid blue); for reference we also 22

plot the previous control (dotted red). For λ = 1 to λ = 1e-2 one can see that the control acts at the center and tries to move γ towards γd . For λ = 1e-3 the control needs to push γ in the right-half up, and in the left-half down and therefore it adjusts accordingly. For λ = 1e-6 the control again mostly acts at the center. Moreover γ matches γd almost perfectly. 7.2. Example 2: Constrained Control. This example differs from Example 1 only due to the fact that now we impose u ∈ Uad . The metrics are shown in Table 7.2. We first remark that, as in the previous example, the control is locally unique and the control convergence rate is linear. λ J (¯ u) J 00 (¯ u) k¯ ukL2 (Γ) rate

∞ 0.07462 0 -

1 0.07317 0.8571 0.0516 1.4353

1e-1 0.06276 0.5143 0.4415 2.7840

1e-2 0.02773 0.8571 1.7092 1.2716

1e-3 0.00780 0.1429 2.9970 1.5117

1e-4 0.00375 8.49e-5 3 1.2134

1e-5 0.00334 9.74e-5 3 1.1942

Table 7.2 Constrained case: the values of the cost function J (¯ u), the smallest eigenvalue of J 00 (¯ u), the L2 -norm of u ¯ and the convergence rate of the optimal control as λ approaches 0. Notice that the constraint is active for λ = 1e-4 and 1e-5.

(¯ γ , y¯) u ¯ λ = 1, u ¯ ∈ (−.0780707, 0]

(¯ γ , y¯) u ¯ λ = 1e − 3, u ¯ ∈ (−5.37, 0.93)

λ = 1e − 1, u ¯ ∈ (−0.665452, 0]

λ = 1e − 4, u ¯ ∈ (−5.37, 0.93)

λ = 1e − 2, u ¯ ∈ (−2.65713, 0]

λ = 1e − 5, u ¯ ∈ (−5.37, 0.93)

Fig. 7.3. Example 2 (Constrained case): The optimal state solution (¯ γ , y¯), the applied control u ¯ in solid blue, and the previous control in dashed red for comparison. The pictures show the corresponding value of λ, from λ = 1 to λ = 1e-5, as well as the smallest and largest value of control. Notice that there is no visual difference between the optimal control for λ = 1e-3, 1e-4 and 1e-5. This is because the control constraints are active. 23

When λ = 1e-4 the control constraints become active and as a result the reduction in the cost function is severely impacted. This becomes clear after comparing the constrained and unconstrained cases for λ = 1e-5. Figure 7.3 shows the optimal state (¯ γ , y¯) and the two consecutive optimal controls (blue: current, red: previous). For λ = 1e-4 and 1e-5 the current and previous controls lie on top of each other because the constraints are active. We also remark that we can not get as close to the desired configuration γd as in the unconstrained case due to the constraint. Acknowledgments. We thank Abner Salgado for his constant support on deal.II. REFERENCES ´, Optimal control of a free boundary problem: Anal[1] H. Antil, R. H. Nochetto, and P. Sodre ysis with second order sufficient conditions, Submitted. arXiv preprint arXiv:1210.0031, (2012). ¨ ltzsch, Error estimates for the numerical approximation of [2] N. Arada, E. Casas, and F. Tro a semilinear elliptic control problem, Comput. Optim. Appl., 23 (2002), pp. 201–229. [3] W. Bangerth, R. Hartmann, and G. Kanschat, deal.II – a general purpose object oriented finite element library, ACM Trans. Math. Softw., 33 (2007), pp. 24/1–24/27. [4] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, Springer, New York, third ed., 2008. ¨ ltzsch, Error estimates for the numerical approximation of [5] E. Casas, M. Mateos, and F. Tro boundary semilinear elliptic control problems, Comput. Optim. Appl., 31 (2005), pp. 193– 219. [6] P. Deuflhard, Newton Methods for Nonlinear Problems - Affine Invariance and Adaptive Algorithms, vol. 35 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2004. [7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. [8] M. Hinze, A variational discretization concept in control constrained optimization: the linearquadratic case, Comput. Optim. Appl., 30 (2005), pp. 45–61. [9] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE constraints, vol. 23 of Mathematical Modelling: Theory and Applications, Springer, New York, 2009. ¨ sch, Error estimates for linear-quadratic control problems with control constraints, Op[10] A. Ro tim. Methods Softw., 21 (2006), pp. 121–134. [11] P. Saavedra and L. R. Scott, Variational formulation of a model free-boundary problem, Math. Comp., 57 (1991), pp. 451–475. [12] V. A. Solonnikov, On the stokes equations in domains with non-smooth boundaries and on viscous incompressible flow with a free surface, in Nonlinear partial differential equations and their applications: Coll` ege de France seminar, vol. 3, Pitman Publishing (UK), 1982, pp. 340–423.

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