A relaxation approach to vector-valued Allen-Cahn MPEC problems

Universit¨ at Regensburg Mathematik

A relaxation approach to vector-valued Allen-Cahn MPEC problems

M. Hassan Farshbaf-Shaker

Preprint Nr. 27/2011

A relaxation approach to vector-valued Allen-Cahn MPEC problems

M.Hassan Farshbaf-Shaker

Abstract

In this paper we consider a vector-valued Allen-Cahn MPEC problem.

To derive optimality conditions we exploit a regularization-relaxation technique. The optimality system of the regularized-relaxed subprob- lems are investigated by applying the classical result of Zowe and Kur- cyusz. Finally we show that the stationary points of the regularized- relaxed subproblems converge to weak stationary points of the limit problem.

Key words. Vector-valued Allen-Cahn system, parabolic bi-obstacle prob- lems, MPECs, mathematical programs with complementarity constraints, optimality conditions.

AMS subject classification. 34G25, 35K86, 65K10, 49J20, 35R35

1 Introduction

The field of the mathematical and numerical analysis of systems of nonlin- ear PDE’s involving interfaces and free boundaries is a burgeoning area of research. Many such systems arise from mathematical models in material sci- ence and fluid dynamics such as phase separation in alloys, crystal growth, dynamics of multi-phase fluids and epitaxial growth. In applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimization prob- lems with PDE constraints including free boundaries, see [16]. Surveys and articles concerning the mathematical and numerical approaches to optimal control of free boundary problems may be found in [10, 5]. In this paper we consider an Allen-Cahn model as a phase-field model to describe the interface

∗Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

1

1 INTRODUCTION 2 evolution. Phase-field methods provide a natural method for dealing with the complex topological changes that occur, see [6]. The interface between the phases is replaced by a thin transitional layer of width O(ε) where ε is a small parameter. The underlying non-convex energy functional is based on the Ginzburg-Landau energy

E(y) :=

Z

Ω

ε

2|∇y|²+ 1 εΨ(y)

dx, ε >0, (1.1)

where Ω ⊂ R^d is an open and bounded domain, y : (0, T)×Ω → R^N is the phase field vector (in our setting the state variable) and Ψ is the bulk potential. Since each component ofy:= (y₁, . . . , y_N)^T stands for the fraction of one phase, the phase space for the order parameteryis the Gibbs simplex G:={v ∈R^N :v ≥0,v·1= 1}. (1.2) Here v ≥ 0 means v_i ≥ 0 for all i ∈ {1, . . . , N}, 1 = (1, . . . ,1)^T. For the bulk potential Ψ :R^N →R⁺0 ∪ {∞}we consider the multi obstacle potential

Ψ(v) := Ψ₀(v) +I_G =

(Ψ₀(v) := −¹₂kvk² for v∈G,

∞ otherwise,

whereI_G is the indicator function of the Gibbs simplex. We are interested in phase kinetics, so the next procedure is to minimize (1.1) under the constraint (1.2). For details, see [11, 12].

Notations. In the sequel we always denote by Ω ⊂ R^d a bounded domain (with spatial dimension d) with boundary Γ = ∂Ω. The outer unit normal on Γ is denoted by n. Vectors are defined by boldface letters. Moreover we define R^N+ :={v∈R^N |v ≥0} and the affine hyperplane

Σ:={v ∈R^N |v·1= 1},

which is indeed a convex subset of R^N. Its tangential space TΣ:={v ∈R^N |v·1= 0},

is a subspace of R^N. With these definitions we obtain for the Gibbs simplex G=R^N+∩Σ. We denote byL^p(Ω), W^k,p(Ω)for1≤p≤ ∞the Lebesgue- and Sobolev spaces of functions on Ωwith the usual normsk · k_L^p_(Ω),k · k_W^k,p_(Ω), and we write H^k(Ω) = W^k,2(Ω). For a Banach space X we denote its dual by X^∗, the dual pairing between f ∈ X^∗, g ∈ X will be denoted by hf, giX^∗,X. If X is a Banach space with the norm k · kX, we denote

1 INTRODUCTION 3 for T > 0 by L^p(0, T;X) (1 ≤ p ≤ ∞) the Banach space of all (equiva- lence classes of) Bochner measurable functions u : (0, T) −→ X such that ku(·)k_X ∈ L^p(0, T). We set Ω_T := (0, T)×Ω, Γ_T := (0, T)×Γ. ”Generic”

positive constants are denoted by C. Furthermore we define vector-valued function spaces by boldface letters, L²(Ω) := L²(Ω;R)^N. Moreover we de- fine L²₊(Ω) := {v ∈ L²(Ω) | v ∈ R^N+ a.e. inΩ} which is a convex cone in L²(Ω); L²_Σ(Ω) := {v ∈ L²(Ω) | v ∈ Σa.e. in Ω} which is a convex sub- set of L²(Ω) and L²_TΣ(Ω) := {v ∈ L²(Ω) | v ∈ TΣa.e. in Ω} which is a subspace of L²(Ω) and hence also a Hilbert space. Furthermore we have L²_G(Ω) := {v ∈ L²(Ω) | v ∈ G a.e. inΩ} and H_i¹(Ω) = H¹(Ω)∩L²_i(Ω) where i ∈ {+,Σ,TΣ,G}. Later we also use following special time depen- dent spaces L²(Ω_T) :=L²(0, T;L²(Ω)),

V :=L^∞(0, T;H¹(Ω))∩H¹(0, T;L²(Ω))∩L²(0, T;H²(Ω)) and

W(0, T) :=L²(0, T;H¹(Ω))∩H¹(0, T;H¹(Ω)^∗).

Moreover we use L²_i(Ω_T) := L²(0, T;L²_i(Ω)), where i ∈ {+,Σ,TΣ}, V_Σ :=

V ∩L²_Σ(Ω_T) and W(0, T)_i := W(0, T)∩L²_i(Ω_T) where i ∈ {Σ,TΣ}. We also have V^hN_Σ :={u∈V_Σ |n· ∇u= 0 a.e. inΓ_T}. Here for vector-valued functions we define the L² inner product by

(ξ,y)_L² :=

N

X

i=1

(ξ_i, y_i)_L², (1.3) For the rest of the paper we make the following assumption

(H1) Assume Ω ⊂ R^d is a bounded domain and either convex or has a C^1,1−boundary and letT > 0be a positive time.

Hence, given an initial phase distribution y(0,·) = y₀ : Ω → G at time t = 0the interface motion can be modeled by the steepest descent of E with respect to theL²−norm which results, after suitable rescaling of time, in the following Allen-Cahn equation

ε∂ty=−grad_L2E(y) =ε∆y+ 1

ε(y−ζ^∗),

where ζ^∗ ∈∂I_G and ∂I_G denotes the subdifferential of I_G. As for the scalar case, see e.g [3, 8], this equation leads to the following Allen-Cahn variational inequality

1 INTRODUCTION 4 Let(H1)hold. For given initial datay₀ ∈H_G¹(Ω)findy∈L²(0, T;H_G¹(Ω))∩

H¹(0, T;L²(Ω)) such that y(0) =y₀ and

ε(∂_ty,χ−y)_L²_(Ω)+ε(∇y,∇(χ−y))_L²_(Ω)≥(1

εy,χ−y)_L²_(Ω), which has to hold for almost all t∈[0, T] and all χ∈H_G¹(Ω).

1.1 Allen-Cahn MPEC problem

Now we introduce our overall optimization problem. Our goal is to transform an initial phase distributiony₀ : Ω→Rwith minimal cost of control to some desired phase pattern y_T : Ω→Rat a given final timeT, where furthermore the distribution remains throughout the entire time interval close to a given distribution y_d.

Our upper level problem is







min J(y,u) := ^ν₂^dky−y_dk²_L2(ΩT)+^ν₂^Tky(T,·)−y_Tk²_L2(Ω)+ ^ν_2ε^ukuk²_L2(ΩT)

over (y,u)∈V^hN_G ×L²_TΣ(ΩT) s.t. (ACVI) holds .

Our lower level problem (ACVI) is:

Let (H1) hold. For given initial data y₀ ∈ H_G¹(Ω) and given control u ∈ L²_TΣ(ΩT) find y ∈ L²(0, T;H_G¹(Ω))∩H¹(0, T;L²(Ω)) such that y(0) = y0

and

ε(∂_ty,χ−y)_L²_(Ω)+ε(∇y,∇(χ−y))_L²_(Ω) ≥(1

εy+u,χ−y)_L²_(Ω), (1.4) which has to hold for almost all t∈[0, T] and all χ∈H_G¹(Ω).

Here, νd, νT, νu are positive constants. The resulting optimization prob- lem belongs to the problem class of the so-called MPECs (Mathematical Programs with Equilibrium Constraints) which are hard to handle for sev- eral reasons. Indeed, we note that due to the structure of the feasible set classical constraint qualifications such as the Mangasarian-Fromovitz con- straint qualifications do not hold true. As a result the existence of Lagrange multipliers of the upper level problem for characterizing first order optimality cannot be derived from standard KKT theory. These kinds of problems have

2 LOWER LEVEL PROBLEM: ALLEN-CAHN VARIATIONAL INEQUALITY5

been extensively studied by many authors, as for example V. Barbu [1], M.

Bergounioux [2] or more recently M. Hintermüller and I. Kopacka [13].

In this work our aim is to derive first order optimality conditions of C- stationarity-type (for different notions of stationarity for MPECs we refer to [15]). In contrast to [8] our approach in this paper consists of using first a relaxation technique to extend the feasible set of the resulting MPEC and secondly a Moreau-Yosida based regularization to avoid the lower reg- ularity of the Lagrange multiplier of the upper level problem corresponding to the state constraint in the relaxed problem. We derive first order opti- mality conditions of the regularized-relaxed subproblems using the classical result of Zowe and Kurcyusz [17] and we study the limit for vanishing re- laxation parameter and regularization parameter γ ↑ +∞. We derive the limit optimality system without considering global solutions (minimizers) of the regularized-relaxed subproblems. The approach reflects the typical sit- uation for nonlinear and non-convex minimization problems, where solution procedures guarantee stationarity points only rather than global minimizers.

The rest of the paper is organized as follows. In section 2 we analyze the vector-valued Allen-Cahn inequality as the lower level problem; the existence of a solution to the inequality is proven by a penalization technique, see for similar results in [4]. Furthermore the complementarity formulation for the Allen-Cahn inequality is given. In section 3 the MPCC (Mathematical pro- gramming with complementarity constraints) problem is formulated, which is a special case of an MPEC. To derive the optimality system for the MPCC we use a regularization relaxation technique in section 4. Furthermore we investigate the convergence behavior of minimizers with respect to the re- laxation and regularization parameters. We also derive first order optimality systems for the regularized-relaxed subproblems. In section 5 we investigate the convergence behavior of stationarity points to the original problem.

2 Lower level problem: Allen-Cahn variational inequality

We begin with defining the operator A:V^hN_Σ →L²_TΣ(Ω_T)^∗ by (Ay,χ) := (−∆y,χ)_L²_(ΩT) for all χ∈L²_TΣ(ΩT).

Following [4] the problem (ACVI) can be reformulated with the help of the slack variable (Lagrange multiplier of the lower level problem) ξ corre- sponding to the inequality constraint y ≥ 0, which results in the following complementarity-problem (CCP):

2 LOWER LEVEL PROBLEM: ALLEN-CAHN VARIATIONAL INEQUALITY6

Let (H1) hold. For given initial data y₀ ∈ H_G¹(Ω) and u ∈ L²_TΣ(Ω_T) find y ∈V^hN_Σ such that y(0) =y₀ and

(ε∂_ty+εAy−1

ε(ξ+y)−u,χ)_L²_(ΩT) = 0, (2.1) which has to hold for all χ∈ L²_TΣ(Ω_T). Moreover we have the complemen- tarity conditions

(CC)







y ≥0 a.e. in Ω_T, ξ ≥0 a.e. in Ω_T, (ξ,y)_L²_(ΩT)= 0,

(2.2)

By Riesz representation theorem we indentify L²_TΣ(ΩT)^∗ with L²_TΣ(ΩT) and rewrite (2.1) as an operator equation

(LLP)







(y,u,ξ)∈V^hN_Σ ×L²_TΣ(Ω_T)×L²(Ω_T) ε∂_ty+εAy−¹_ε(y+ξ) =u in L²_TΣ(Ω_T) y(0) =y₀ a.e. inΩ.

Lemma 1. Let (H1) hold and (y₀,u) ∈ H_G¹(Ω) ×L²_TΣ(Ω_T) be given. A functiony ∈V^hN_Σ solves (ACVI) if there existsξ∈L²(Ω_T)such that (LLP) and (CC) are fulfilled.

Proof. Let y ∈ V^hN_Σ be the solution to (LLP) and (CC). For χ ∈H_G¹(Ω), the function (χ−y)∈ H_T¹_Σ(Ω) ⊂ L²_TΣ(Ω) is an admissible testfunction in (2.1). After partial integration we get

(ε∂_ty−1

ε(µ+y)−u,χ−y)_L²_(Ω)+ε(∇y,∇(χ−y))_L²_(Ω)= 0, for a.e. t ∈[0, T]. Using the property χ≥0 and (CC) gives

(ξ,χ−y)_L²_(ΩT) ≥0.

Hence we obtain for all χ∈H_G¹(Ω) and almost all t∈[0, T] (ε∂_ty− 1

εy−u,χ−y)_L²_(Ω)+ε(∇y,∇(χ−y))_L²_(Ω)≥0,

and hence y solves (ACVI). 2

Theorem 1. Let(H1)hold. Given(y₀,u)∈H_G¹(Ω)×L²_TΣ(Ω_T)there exists a unique solution (y,ξ)∈V^hN_Σ ×L²(Ω_T) to (CCP).

2 LOWER LEVEL PROBLEM: ALLEN-CAHN VARIATIONAL INEQUALITY7

Proof. We will give here a sketch of the main steps of the proof. For detailed calculations we refer to a similar proof in [4].

1. Step: Regularized problems We introduce the following regularization of the obstacle potential Ψ(y):

Ψ^δ(y) = Ψ₀(y) + 1 δ

Ψ(y),ˆ where

Ψ(y) =ˆ

N

X

i=1

min(y_i,0)².

Define the function Φ(r) = 2 min(r,ˆ 0) for all r∈R and note that Ψˆ⁰_y(y) :=

Φ(y) =ˆ {Φ(yˆ _i)}^N_i=1.

We now solve the following regularized Allen-Cahn equation (ACVI)_δ: Let (H1) hold. Given y₀ ∈H_G¹(Ω) and u ∈L²_TΣ(Ω_T) find y_δ ∈ V^hN_Σ such that y_δ(0) =y₀ and

ε(∂_ty_δ,χ)_L²_(Ω)+ε(∇y_δ,∇χ)_L²_(Ω)+ (1

εΨ^δ0y(y_δ)−u_δ,χ)_L²_(Ω) = 0, (2.3) which has to hold for almost all t∈[0, T] and all χ∈H_T¹_Σ(Ω).

For every δ ∈ (0,1] one can show the unique solvability of (2.3) by clas- sical theory of parabolic partial differential equations and then pass to the limit. Following [4] we reformulate (2.3) by usingΨ^δ0y(y_δ) = ¹_δΦ(yˆ _δ)−y_δ and defining ξ_δ :=−¹_δΦ(yˆ _δ). Hence, we have

ε(∂_ty_δ,χ)_L²_(Ω)+ε(∇y_δ,∇χ)_L²_(Ω)−(1

ε(y_δ+ξ_δ) +u_δ,χ)_L²_(Ω) = 0, (2.4) for all χ∈H_T¹_Σ(Ω).

2. Step: A priori estimates Let(H1)hold and y₀ ∈H_G¹(Ω). For a sequence {u_δ}δ∈(0,1] uniformly bounded in L²_TΣ(Ω_T)it is shown in [4] that

y_δ bounded in V^hN_Σ uniformly in δ ∈(0,1],

ξ_δ bounded in L²(Ω_T) uniformly in δ ∈(0,1]. (2.5) 3. Step: Passing to the limit From Step 2 we get the convergence results as δ &0

y_δ −→ y weakly in L²(0, T;H²(Ω)), yδ −→ y weakly in H¹(0, T;L²(Ω)), y_δ −→ y weak-star in L^∞(0, T;H¹(Ω)), ξ_δ −→ ξ weakly in L²(Ω_T),

(2.6)

2 LOWER LEVEL PROBLEM: ALLEN-CAHN VARIATIONAL INEQUALITY8

The set {ξ_δ ∈ L²(Ω_T) : ξ_δ ≥ 0 a.e. in Ω_T} is convex and closed and hence weakly closed and we obtain ξ≥0 a.e. in Ω_T. Furthermore, the convex and closed subsetV^hN_Σ is weakly closed and we obtain thaty∈V^hN_Σ . For proving y ≥0 we refer the reader to [4, 9]. We get moreover as δ &0

(ξ,y)_L²_(Ω) ←−(ξ_δ,y_δ)_L²_(Ω) =−1

δ( ˆΦ(y_δ),y_δ)_L²_(Ω) ≤0,

and hence (ξ,y)_L²_(Ω) ≤ 0. However, since ξ ≥ 0 and y ≥ 0 we have that (ξ,y)_L²_(Ω) = 0 a.e. in (0, T). Hence, (y,ξ) ∈V^hN_Σ ×L²(Ω_T) is the solution to (CCP). For uniqueness we refer the reader to [4]. 2 The following proposition will be useful for establishing the next results.

Proposition 1. Let (u_k)_k≥1 be an uniformly bounded sequence in L²_TΣ(Ω_T) and(y_k,ξ_k)k≥1 the corresponding solutions of (CCP). Then there exits(y,ξ)∈ V^hN_Σ ×L²(Ω_T)and a subsequence still denoted by (y_k,u_k,ξ_k)k≥0 such that as k ↑ ∞

yk −→ y weakly in L²(0, T;H²(Ω)), y_k −→ y weakly in H¹(0, T;L²(Ω)), y_k −→ y weak-star in L^∞(0, T;H¹(Ω)), ξk −→ ξ weakly in L²(ΩT),

(2.7)

and (y,ξ)∈V^hN_Σ ×L²(Ω_T) fulfil (CCP).

Proof. For every u_k the corresponding solutions to (2.4) are given by (yδ,k,ξδ,k)k≥1. By (2.5) we have

(y_δ,k,ξ_δ,k) bounded in V^hN_Σ ×L²(Ω_T) uniformly in δ and k.

By virtue of the lower semi-continuity of the norm we get

(yk,ξk) bounded in V^hN_Σ ×L²(ΩT) uniformly in k.

Continuing as in the proof of Theorem 1 we get (2.1). We get furthermore as δ&0

(ξ,y)_L²_(Ω) ←−(ξ_k,y_k)_L²_(Ω) = 0,

because of the strong and weak convergence ofy_k and ξ_k inL²(Ω). The rest of the proof is similar to the proof of Theorem 1. 2

3 UPPER LEVEL PROBLEM: OPTIMAL CONTROL PROBLEM 9

3 Upper level problem: Optimal control prob- lem

We consider the time dependent vectorial Allen-Cahn-MPCC problem:

(P₀) min{J(y,u)|(y,u,ξ)∈ D₀} where D₀ is the feasible set given by

(D₀)











(y,u,ξ)∈V^hN_Σ ×L²_TΣ(Ω_T)×L²(Ω_T) ε∂_ty+εAy− ¹_ε(y+ξ) = u inL²_TΣ(Ω_T) y(0) =y₀ a.e. in Ω

y≥0a.e. in Ω_T, ξ ≥0 a.e. inΩ_T, (ξ,y)_L²_(ΩT)= 0.

Theorem 2. The problem (P₀) has at least one solution.

Proof. Let (y_k,u_k,ξ_k)k≥0 be a minimizing sequence for (P₀) such that inf(P₀)≤J(y_k,u_k)≤inf(P₀) + 1

k.

Then (uk)k≥0 is bounded in L²_TΣ(ΩT) and by Proposition 1 there exists (y,u,ξ) ∈ V^hN_Σ ×L²_TΣ(Ω_T)×L²(Ω_T) and a subsequence still denoted by (y_k,u_k,ξ_k)k≥0 such that (2.7) holds. Moreover we easily can check by the same proof-techniques as in the proof of Theorem 1 that(y,u,ξ)∈ D0 which implies that (y,u,ξ) is a feasible point for (P₀). On the other hand (2.7) and the weak lower semi-continuity of norms yield

J(y,u)≤lim inf

k↑∞ J(y_k,u_k)≤inf(P₀).

Consequently (y,u,ξ) is an optimal solution of(P₀). 2 Following [2], we add from now on an explicit constraint to (P₀) involving the multiplier ξ in L²(Ω_T). The new time dependent vectorial Allen-Cahn- MPEC problem reads

(P) min{J(y,u)|(y,u,ξ)∈ D}, where

(D)



















(y,u,ξ)∈V^hN_Σ ×L²_TΣ(Ω_T)×L²(Ω_T) ε∂_ty+εAy− ¹_ε(y+ξ) = u inL²_TΣ(Ω_T) y(0) = y₀ a.e. in Ω

y≥0 a.e. in Ω_T, ξ ≥0a.e. in Ω_T, (ξ,y)_L²_(ΩT) = 0,

1

2kξk²_L2(ΩT) ≤R,

4 REGULARIZED-RELAXED UPPER LEVEL PROBLEMS 10 where R is a sufficiently large positive constant. For instance, R may be the largest positive number that may be computed by the machine, see [2].

However, as (P) lacks constraint regularity, for deriving stationarity condi- tions for (P₀) in the next section we relax the constraints of (P) such that the relaxed version of (P) satisfies well-known constraint qualifications of mathematical programming in Banach spaces [17]. In this context, it turns out that the well posedness of the relaxed version of (P)depends on the new constraint for ξ, see [2].

4 Regularized-relaxed upper level problems

In this section we introduce and study a regularized-relaxed version of the optimal control problem (P). Following the approaches in [13], [14], our objective is to characterize some type of C-stationarity of critical points of (P). This is achieved by passing to the limit with respect to the regularization and relaxation parameters. The regularized-relaxed problems are defined as follows:

(Pγ) min{Jγ(y,u)|(y,u,ξ)∈ Dγ}, where J_γ(y,u) :=J(y,u) + _2γ¹

N

P

i=1

kmax(0, λ−γy_i)k²_L2(ΩT) and

(Dγ)























(y,u,ξ)∈V^hN_Σ ×L²_TΣ(Ω_T)×L²(Ω_T) ε∂_ty+εAy−¹_ε(y+ξ) =u in L²_TΣ(Ω_T) y(0) =y₀ a.e. inΩ

ξ ≥0a.e. in Ω_T, α_γ ≥(ξ,y)_L²_(ΩT), R≥ ¹₂kξk²_L2(ΩT),

where λ ∈ L²(Ω_T), which mimics a regular version of the multiplier associ- ated to y ≥ 0, is arbitrary fixed with λ ≥ 0 a.e. inΩT. Note that we add a regularization term _2γ¹

N

P

i=1

kmax(0, λ+γyi)k²_L2(ΩT) toJ(y,u)with γ denot- ing the associated regularization parameter. This step relaxes the pointwise state constraint y ≥ 0a.e. in Ω_T. The derivative of the regularization-term serves as a regular (i.e, L²(Ω_T)−) approximation of the multiplier associ- ated with y ≥ 0a.e. in ΩT. Further we relax (ξ,y)_L²_(ΩT) = 0 by allowing (ξ,y)_L²_(ΩT) ≤ α_γ for some α_γ > 0. These modifications motivate the de- scription of (P_γ) as the regularized-relaxed version of (P). Subsequently we

4 REGULARIZED-RELAXED UPPER LEVEL PROBLEMS 11 are interested in γ ↑ ∞ and α_γ ↓ 0 as γ ↑ ∞. Let D_γ and D denote the feasible sets of (P_γ) and (P), respectively. Observe that we have

D_γ ⊇ D 6=∅. (4.1)

4.1 Minimizers of the upper level problems

Theorem 3. For every γ > 0, the regularized-relaxed problem (P_γ) ad- mits at least one minimizer (globally optimal solution) which is denoted by (y_γ,u_γ,ξ_γ).

Proof. For the proof let γ > 0 be arbitrary but fixed. Since J_γ ≥ 0 and because of (4.1) D_γ 6=∅ the infimum d :=J_γ(y_γ,u_γ) in D_γ exists and hence we find a minimizing sequence (y^k_γ,u^k_γ,ξ_γ^k)k≥1 ⊂ D_γ with

k↑∞limJ_γ(y_γ^k,u^k_γ) =d.

As {J_γ(y_γ^k,u^k_γ)} is bounded, {u^k_γ} is bounded in L²_TΣ(Ω_T). Then by virtue of Proposition 1 there exits(y_γ,ξ_γ)∈V^hN_Σ ×L²(Ω_T)and a subsequence still denoted by (y_γ^k,u^k_γ,ξ^k_γ)k≥1 such that

y_γ^k −→ y_γ weakly in L²(0, T;H²(Ω)), y_γ^k −→ y_γ weakly in H¹(0, T;L²(Ω)), y_γ^k −→ y_γ weak-star in L^∞(0, T;H¹(Ω)), ξ^k_γ −→ ξ_γ weakly in L²(Ω_T).

(4.2)

We next show that the limit point (y_γ,ξ_γ)∈ D_γ. It is clear that α_γ ≥(y_k,ξ_k)_L²_(ΩT) →(y,ξ)_L²_(ΩT).

The rest of the proof is similar to the proof of Theorem 1. The weak con- vergence of (y_γ^k,u^k_γ,ξ^k_γ)ask ↑ ∞, the feasibility of(y_γ,u_γ,ξ_γ) and the lower semi-continuity of J_γ give

d= lim inf

k↑∞ J_γ(y_γ^k,u^k_γ)≥J_γ(y_γ,u_γ,ξ_γ)≥d.

Therefore (y_γ,u_γ,ξ_γ)∈ D_γ is an optimal solution of (P_γ)for every γ >0.2 Next we are interested in the convergence behavior of optimal solutions with respect to the regularization and relaxation parameters. For each γ > 0, let α_γ satisfy α_γ ↓ 0 as γ ↑ ∞. We now show that the minimizers of the relaxed-regularized problems (P_γ) converge to a minimizer of (P).

4 REGULARIZED-RELAXED UPPER LEVEL PROBLEMS 12 Theorem 4. For every γ > 0, let (y_γ,u_γ,ξ_γ) be a solution of (P_γ). Then there exist

(y^∗,u^∗,ξ^∗)∈V^hN_Σ ×L²_TΣ(Ω_T)×L²(Ω_T)

and a subsequence still denoted by (yγ,uγ,ξγ)γ>0 such that as γ ↑ ∞ y_γ −→ y^∗ weakly in L²(0, T;H²(Ω)),

y_γ −→ y^∗ weakly in H¹(0, T;L²(Ω)), y_γ −→ y^∗ weak-star in L^∞(0, T;H¹(Ω)), u_γ −→ u^∗ strongly in L²_TΣ(Ω_T),

ξ_γ −→ ξ^∗ weakly in L²(Ω_T).

Furthermore _2γ¹

N

P

i=1

kmax(0, λ+γy_i)k²_L2(Ω_T) →0 as γ ↑ ∞ and (y^∗,u^∗,ξ^∗) is a solution of (P).

Proof. We consider the point (y_γ,u_γ,ξ_γ) that is a solution to the problem (P_γ). Then (y_γ,u_γ,ξ_γ) ∈ D_γ for all γ > 0. Hence for each γ ≥ 1 we can estimate

J_γ(y_γ,u_γ)≤J_γ(0,0)

≤ 1

2ky_Tk²_L2(Ω)+1

2ky_dk²_L2(ΩT)+1 2

N

X

i=1

kmax(0, λ)k²_L2(ΩT). Hence

u_γ is bounded in L²_TΣ(Ω_T) uniformly in γ ∈(0,∞) (4.3) and for all 1≤i≤N

√1

2γmax(0, λ−γyⁱ_γ) is bounded in L²(Ω_T) uniformly in γ ∈(0,∞).

(4.4) By virtue of (4.3) and Proposition 1 there exist(y^∗,ξ^∗)∈V^hN_Σ ×L²(Ω_T)and a subsequence still denoted by (y_γ,u_γ,ξ_γ)γ≥0 such that as k↑ ∞

yγ −→ y^∗ weakly in L²(0, T;H²(Ω)), y_γ −→ y^∗ weakly in H¹(0, T;L²(Ω)), y_γ −→ y^∗ weak-star in L^∞(0, T;H¹(Ω)), ξ_γ −→ ξ^∗ weakly in L²(Ω_T),

(4.5)

4 REGULARIZED-RELAXED UPPER LEVEL PROBLEMS 13 and(y^∗,ξ^∗)∈V^hN_Σ ×L²(Ω_T)∈ D. The set{ξ_γ ∈L²(Ω_T) :ξ_γ ≥0 a.e. inΩ_T} is weakly closed and we obtain

ξ^∗ ≥0 a.e. in Ω_T. (4.6)

Furthermore, we have

(y^∗,ξ^∗)_L²_(ΩT) = lim

γ↑∞(ξ_γ,y_γ)_L²_(ΩT) ≤ lim

γ↑∞α_γ = 0. (4.7) and

R≥ 1

2kξ^∗k²_L2(ΩT). Moreover from (4.4) we obtain

kmax(0,λ

γ −γy_γⁱ)k_L²_(ΩT)→0, as γ ↑ ∞ ∀1≤i≤N.

Since y_γ converges strongly in L²(Ω_T), without loss of generality we may assume that y_γ converges to y^∗ a.e. in Ω_T. Taking the limit and applying Fatou’s lemma we conclude that

kmax(0,−(yⁱ)^∗)k²_L2(ΩT) =klim inf

γ↑∞ max(0,λ

γ −γyⁱ_γ)k²_L2(ΩT)

≤lim inf

γ↑∞ kmax(0,λ

γ −γy_γⁱ)k²_L2(ΩT)≤ lim

γ↑∞

2c γ = 0.

Consequently

kmax(0,−(yⁱ)^∗)k²_L2(ΩT) = 0 ∀1≤i≤N.

and

y^∗ ≥0 a.e. inΩT. (4.8)

This with (4.6) and (4.7) implies

(y^∗,ξ^∗)_L²_(ΩT) = 0.

Now let( ˜y,u,˜ ξ)˜ be an optimal control of(P). Note that by (4.1) ( ˜y,u,˜ ξ)˜ ∈ D_γ and (y^∗,u^∗,ξ^∗)∈ D. We therefore conclude

J( ˜y,u)˜ ≤J(y^∗,u^∗),

J_γ(y_γ,u_γ)≤J_γ( ˜y,u)˜ ∀γ >0.

4 REGULARIZED-RELAXED UPPER LEVEL PROBLEMS 14 Using the lower semi-continuity of J, the definition of J_γ and the non- negativity of y˜ it follows that

J(y^∗,u^∗)≤lim inf

γ↑∞ J(y_γ,u_γ)

≤lim inf

γ↑∞ Jγ(yγ,uγ)≤lim sup

γ↑∞

Jγ(yγ,uγ)≤lim sup

γ↑∞

Jγ( ˜y,u) =˜ J( ˜y,u)˜

≤J(y^∗,u^∗).

Therefore

γ↑∞limJ_γ(y_γ,u_γ) =J(y^∗,u^∗) = J( ˜y,u).˜

and (y^∗,u^∗,ξ^∗) ∈ D is optimal for (P). The convergence of the objective function values yields as γ ↑ ∞

1 2γ

N

X

i=1

kmax(0, λ−γy_γⁱ)k²_L2(Ω_T) →0 ∧ kuγk²_L2(Ω_T) → ku^∗k²_L2(Ω_T). As weak convergence together with norm-convergence inL²(Ω_T)imply strong convergence, this yields the strong convergence of {uγ} inL²(ΩT). 2

4.2 First order optimality conditions

In the previous section, our analysis required minimizers (global solutions) of the regularized-relaxed problems. However, finding globally optimal solu- tions (in particular by means of numerical algorithms) is difficult in practice.

Often, one rather has to rely on stationary points, i.e. points satisfying first order optimality conditions, or on local solutions. In this subsection we de- rive the first order optimality system for the regularized-relaxed problems (P_γ)_γ>0 using the mathematical programming approach in Banach spaces due to Zowe and Kurcyusz [17]. Let X and Z be real Banach spaces. For

F : X −→ R Frechét-differentiable functional, g : X −→ Z continuously Frechét-differentiable, we consider the following mathematical program:

min{F(x)| g(x)∈M, x∈C}, (4.9) whereC is a convex closed subset ofX andM a closed cone inZ with vertex at 0. We define the notion of local optimality

4 REGULARIZED-RELAXED UPPER LEVEL PROBLEMS 15 Definition 1. We call xˆ a local solution of (4.9) if there is some σ >0 such that

F(ˆx)≤F(x) for all x∈C with g(x)∈M and kˆx−xkX ≤σ.

Now we suppose that the problem (4.9) has an local optimal solutionx, andˆ we introduce the conical hulls of C− {ˆx} and M − {z}, respectively, by

C(ˆx) = {x∈ X | ∃β ≥0, ∃c∈C, x=β(c−x)},ˆ M(z) = {ζ ∈ Z | ∃λ ≥0, ∃k ∈M, ζ =k−λz}.

The main result in [17] on the existence of a Lagrange multiplier for (4.9) is stated next.

Theorem 5. Letxˆ be an optimal solution of the problem (4.9) satisfying the following constraints qualification

g⁰(ˆx)·C(ˆx)−M(g(ˆx)) =Z. (4.10) Then there exists a Lagrange multiplier z^∗ ∈ Z^∗ such that

hz^∗, ζi_Z∗,Z ≥0 ∀ζ ∈M, (4.11)

hz^∗, g(ˆx)i_Z∗,Z = 0, (4.12)

F⁰(ˆx)−z^∗◦g⁰(ˆx)∈C(ˆx)₊, (4.13) where A₊ ={x^∗ ∈ X^∗ :hx^∗, ai_X∗,X ≥0∀a∈A}, Z^∗ and X^∗ are the topolog- ical dual spaces of Z and X, respectively, and (z^∗◦g⁰(ˆx))d=hz^∗, g⁰(ˆx)di_Z∗,Z

∀d∈ X.

We apply Theorem 5 to (P_γ). For this purpose we set X =V^hN ×L²_TΣ(Ω_T)×L²(Ω_T), C =V^hN_Σ ×L²_TΣ(ΩT)×L²(ΩT),

Z =L²_TΣ(Ω_T)×L²(Ω)×L²(Ω_T)×R×R, M ={0} × {0} ×L²₊(Ω_T)×R+×R+, ˆ

x= (y_γ,u_γ,ξ_γ), F(ˆx) =J_γ(y_γ,u_γ),

g(x) =



















ε∂_ty_γ+εAy_γ−¹_εy_γ−u_γ− ¹_εξ_γ, y_γ(0)−y₀,

ξ_γ,

α_γ−(ξ_γ,y_γ)_L²_(ΩT), R−¹₂kξ_γk²_L2(ΩT).

4 REGULARIZED-RELAXED UPPER LEVEL PROBLEMS 16 Then we have for the convex hull ofV^hN_Σ ×L²_TΣ(Ω_T)×L²(Ω_T)−{(y_γ,u_γ,ξ_γ)^T}

C



 y_γ u_γ ξ_γ



=









 c d e



∈ X | ∃β ≥0, ∃





˜ c d˜

˜ e



∈C,



 c d e



=β





˜ c−y_γ d˜−u_γ

˜ e−ξ_γ









 .

The constraint qualification (4.10) in our setting requires the existence of c := (c₁,· · · , c_N)^T ∈ V^hN_TΣ,

d := (d₁,· · · , d_N)^T ∈ L²_TΣ(Ω_T), e := (e1,· · · , eN)^T ∈ L²(ΩT), k := (k₁,· · · , k_N)^T ∈ L²₊(Ω_T), and (k_N+1, k_N+2, λ)^T ∈R³+ such that for arbitrary given

z₁ := (z₁,· · · , z_N)^T ∈ L²_TΣ(Ω_T), z₂ := (z_N+1,· · · , z_2N)^T ∈ L²_TΣ(Ω), z₃ := (z_2N+1,· · · , z_3N)^T ∈ L²(Ω_T), and (z3N+1, z3N+2)∈R² the following system holds

z₁ =ε∂_tc+εAc− 1

εc−d−1

εe in L²_TΣ(Ω_T), (4.14)

z2 =c(0) inL²_TΣ(Ω), (4.15)

z₃ =e−(k−λξ_γ) in L²(Ω_T), (4.16)

z_3N+1 =−(c,ξ_γ)_L²_(ΩT)−(e,y_γ)_L²_(ΩT)−(k_N+1−λ(α_γ−(ξ_γ,y_γ)_L²_(ΩT))), (4.17) z_3N+2 =−(ξ_γ,e)_L²_(ΩT)−(k_N+2−λ(R− 1

2kξ_γk²_L2(ΩT))). (4.18) By virtue of (H1)and by the classical theory of parabolic partial differential equations (see [7], for example), the system

ε∂_tc+εAc− 1

εc=z₁− 1

εe inL²_TΣ(Ω_T), (4.19)

z₂ =c(0) inL²_TΣ(Ω), (4.20)

admits a unique solution c∈V^hN_TΣ for everyz₁ ∈L²_TΣ(Ω_T) and e∈L²(Ω_T).

4 REGULARIZED-RELAXED UPPER LEVEL PROBLEMS 17 Therefore, a solution of (4.14)-(4.18) is obtained by choosing

d=0,

λ=ρ³, e=ρf −ρ²ξ_γ c solution of(4.19)−(4.20), k= (ρ³−ρ²)ξ_γ+ρf −z₃,

kN+1 =ρ³(αγ−(ξγ,yγ)_L²_(ΩT)) +ρ²(ξγ,yγ)_L²_(ΩT)−ρ(yγ,f)_L²_(ΩT)

−(c,ξγ)_L²_(ΩT)−z3N+1, k_N+2 =ρ³(R− 1

2kξ_γk²_L2(ΩT)) +ρ²kξ_γk²_L2(ΩT)−ρ(ξ_γ,f)_L²_(ΩT)−z_3N+2 for some f ∈ L²(Ω_T) with f > 0 a.e. in Ω_T, and ρ > 0 large enough such

that k, k_N+1 and k_N+2 are nonnegative. 2

Consequently problem(P_γ)satisfies the constraint qualification (4.10). Hence, according to Theorem 5, the set of Lagrange multipliers is nonempty and bounded, i.e. introducing

λⁱ_γ := max(0, λ−γy_γⁱ), λ_γ := (λ¹_γ, . . . , λ^N_γ)^T, we have the following

Proposition 2. Let (y_γ,u_γ,ξ_γ) be a solution for the problem (P_γ). Then there exists a Lagrange multiplier vector (p_γ,µ_γ, r_γ, κ_γ) in W(0, T)_TΣ × L²(Ω_T)×R×R such that the following first order optimality system holds

−ε∂_tp_γ+εA^∗p_γ−1

εp_γ−r_γξ_γ+

+λ_γ =ν_d(y_γ−y_d) in L²(0, T;H¹(Ω)^∗), (4.21) p_γ(T,·) = ν_T(y_γ(T,·)−y_T), a.e. in Ω, (4.22) p_γ+ν_u

ε u_γ =0 a.e. in Ω_T, (4.23)

κ_γξ_γ+ 1

εp_γ−µ_γ+r_γy_γ =0 a.e. in Ω_T, (4.24) ξ_γ ≥0 a.e. in Ω_T, µ_γ ≥0 a.e. in Ω_T, (ξ_γ,µ_γ)_L²_(ΩT)= 0, (4.25) κ_γ ≥0, 1

2kξ_γk²_L2(ΩT)≤R, κ_γ

2 kξ_γk²_L2(ΩT) =κ_γR, (4.26) r_γ ≥0, (y_γ,ξ_γ)_L²_(ΩT) ≤α_γ, r_γ(y_γ,ξ_γ)_L²_(ΩT)=r_γα_γ, (4.27) ε∂_ty_γ+εAy_γ− 1

εy_γ−u_γ− 1

εξ_γ =0 in L²_TΣ(Ω_T), (4.28)

y_γ(0) = y₀ a.e. in Ω, (4.29)

5 OPTIMALITY FOR THE LIMIT PROBLEM(P) 18 Proof. For every fixed 0 < γ we know by virtue of Theorem 5 that p_γ ∈ L²_TΣ(Ω_T). Furthermore we obtain p_γ ∈ W(0, T)_TΣ by the classical theory of parabolic partial differential equations, see for example [7]. 2

5 Optimality for the limit problem (P )

In this section we investigate the convergence of a sequence

(yγ,uγ,ξγ,pγ,µγ, rγ, κγ)γ>0satisfying the optimality conditions (4.21)-(4.29).

For this purpose we make the following assumptions:

• (O1) Let{u_γ}be bounded in L²_TΣ(Ω_T) uniformly inγ ∈(0,∞),

• (O2) we choose αγ such that _α¹

γ√

γ ≤C,

• (O3) we assume that κγγ ≤C.

Here and in what follows, C denotes a generic positive constant that may take different values at different occurrences but not depending on γ. We also introduce the notations

ϑⁱ_γ =r_γξ_γⁱ −λⁱ_γ, ϑ_γ := (ϑⁱ_γ)^N_i=1, N_γⁱ ={(t, x)∈Ω_T :y_γⁱ <0}, P_γⁱ = Ω_T \N_γⁱ,

Πⁱ_γ ={(t, x)∈Ω_T :λ−γy_γⁱ ≥0}.

Lemma 2. Let γ >0, (O1)-(O3) hold and let (yγ,uγ,ξγ,pγ,µγ, rγ, κγ) be a solution of the optimality system (4.21)-(4.29). Then we have

1.) y_γ is bounded in V^hN_Σ uniformly in γ ∈(0,∞), 2.) y_γ(T) is bounded in L²_Σ(Ω) uniformly in γ ∈(0,∞), 3.) p_γ is bounded in L²(0, T;H_T¹_Σ(Ω)) uniformly in γ ∈(0,∞), 4.) p_γ(0) is bounded in L²_TΣ(Ω) uniformly in γ ∈(0,∞), 5.) u_γ is bounded in L²(0, T;H¹(Ω)) uniformly in γ ∈(0,∞), 6.) ^√1_γλ_γ is bounded in L²(Ω_T) uniformly in γ ∈(1,∞), 7.) ∂_tp_γ is bounded in W(0, T)^∗ uniformly in γ ∈(0,∞), 8.) ϑ_γ is bounded in W(0, T)^∗ uniformly in γ ∈(0,∞), Proof. By virtue of (O1) Proposition 1 gives the estimates 1) and 2).