Existence of Optimal Control for Dirichlet Boundary Optimization in a Phase Field Problem

Phase field modeling finds utility in various areas. In optimization theory in particular, the distributed control and Neumann boundary control of phase field models have been investigated thoroughly. Dirichlet boundary control in parabolic equations is commonly addressed using the very weak formulation or an approximation by Robin boundary conditions. In this paper, the Dirichlet boundary control for a phase field model with a non-singular potential is investigated using the Dirichlet lift technique. The corresponding weak formulation is analyzed. Energy estimates and problem-specific embedding results are provided, leading to the existence and uniqueness of the solution for the state equation. These results together show that the control to state mapping is well defined and bounded. Based on the preceding findings, the optimization problem is shown to have a solution.


Introduction
Phase field models (PFM) find utility in various areas, where interface tracking is of interest [7,9].Applications include, but are not limited to, the modeling of phase transitions in materials science [8,9,12,21,22,25,28,30,33], crack propagation [11] or tumor growth [14].The first of the three mentioned applications is addressed in this article.Namely, the controlled solidification of a pure substance by means of supercooling.To achieve this goal, the methods of PDE-constrained optimization are applied, where the PFM serves as the state equation.
This class of problems has been studied thoroughly since the 1990s.One of the first contributions addresses the distributed control (DC) of the phase field model [20].Since then, the DC of PFM has seen many theoretical developments and applications, including the simulation of isothermal alloy solidification [3], tumor growth modeling [26], inductive heating [35], and many others.Some of the theoretical advancements in the DC for the PFM include results on existence of the control and the derivation of optimality conditions for specific variants of the model that feature singular potentials [12,14].
In all of the contributions discussed so far, the problem formulations are typically supplemented by homogeneous Neumann (natural) or Dirichlet boundary conditions.As a consequence, the resulting state equation is of a variational type [23,29,32].Another type of control that may be considered for PFM is the Neumann or Robin Control (NoR) [13,34].This type of control naturally results in a variational formulation of the problem as well.To supplement the existing theoretical results, this paper focuses on the Dirichlet boundary control for the PFM.
A particular PFM that governs the solidification of a supercooled melt [4,27,28] is considered as the state equation of the system.It consists of the heat equation with a latent heat term and the phase field equation.In the optimization problem, a Dirichlet boundary condition for the heat equation that results in a particular solid shape (expressed by the phase field) at final time is found.The problem is described fully in Section 2. The state equation is shown to have a unique solution and the necessary auxiliary results, such as the boundedness of the solution operator with respect to the control and problem specific embedding statements are provided.These are then used to prove the existence of solution of the optimal control problem.

Dirichlet Control for the PFM Using the Dirichlet Lift
To the authors' best knowledge, there exist only a handful of publications addressing the Dirichlet boundary control of parabolic differential equations.The Dirichlet boundary condition optimization for a single parabolic PDE has been treated in the past by using a Robin boundary condition approximation [2].The same class of problems was addressed in [16,23], where the very weak formulation is utilized for the state equation.Dirichlet lifts have been used in the so called energy approach to optimization and first detailed for an elliptic problem in [10].More recently, this technique has been used for the optimization of a Dirichlet boundary condition for a parabolic PDE as well [17].
The presented method, although developed independently, bears some resemblances to [10,17].We use the technique of the Dirichlet lift to give a direct correspondence between the original state equation and the "lifted" reformulation.
For reader's convenience, a summary of notation used for Bochner spaces and fundamental statements that find utility in the proof are included in the Appendix.Also, note that the differential symbols d and d in volume and surface integrals are left out throughout the text to improve readability.
The state equation of the problem in question is a phase field system that can be used to simulate the solidification of a supercooled pure melt.It is composed of the heat equation and the phase field (Allen-Cahn) equation.The model is introduced in dimensionless form in accordance with the related works [6,27,28].The goal of the optimization, which is described by the cost functional , is to achieve a certain shape of the solid body at final time .(5) where 0 specifies the initial supercooling and are material-specific constants.The parameter 0 determines the phase interface thickness and 0 is a constant calculated so that the Gibbs-Thomson relation [18] is asymptotically recovered as 0. For this particular model, we get 1 6 (see, e.g.[28]).The initial conditions for the temperature and phase field are given by ini 1 , ini 1 .The solution of the minimization problem (1)-( 8) is a function [0 ] which controls the temperature at the boundary of in such a way that is as close as possible to the target phase field profile f 2 at .If necessary, the magnitude of the control can be reduced by increasing the regularization parameter 0 in the cost functional (1).Derivation of the dimensionless model and the meaning of the original dimensional quantities can be found in [28].
Next, we cast the problem into weak form, define the control and solution spaces and impose restrictions on the control set.To formulate the problem using the Dirichlet lift, consider the control space 4 0 .
This high level of regularity is a consequence of the compact embedding statements used in the analysis that follows.Let in (3) be such that there exists for which Tr .(10) Using the concept of a weak solution, we look for (see Section A. (16) where the equations ( 13) and ( 16) are in the sense of distributions.A pair of functions (20) that satisfies ( 13)-( 19) is called a solution of the state equation ( 13)- (19).It follows from the principles of the Dirichlet lift that a solution of ( 13)-( 19) is also a solution to (2)-( 8) when (10) holds.The statement that we aim to prove follows.

Theorem 1 Let the control space
and solution space be given by ( 9) and (20), respectively, and let ad be closed, convex and bounded.Then, there exists a (not necessarily unique) solution of the optimization problem (12)- (19).
The rest of the text is dedicated to the proof of Theorem 1.In Section 3, problem specific embedding results are provided.These are then used in Section 4 to show that there exists a unique solution of the state equation ( 13)-( 19) for any .The results of Sections 3 and 4 are then used to show the existence of optimal control for ( 12)- (19) in Section 5, which concludes the proof of Theorem 1.

Specific Embedding-Based Results
In this section, two specific results are derived, which find utility in the analysis of the solution operator of ( 13)- (19) and in the proof of the existence of optimal control. .Then,
Squaring, integrating over [0 ] and using Young's inequality (Lemma 6) then yields ( Applying the standard Hölder's inequality (Lemma 5) to the right-hand side of (29) gives in the estimate analogous to (28).

Analysis of the State Equation
We start by stating the weak formulation of the problem ( 13)- (19).To simplify the notation, we use and instead of and , respectively.Using standard methods, we arrive at a.e. in [0 ] (31) where 0 is defined as 0 1 1 2 .The following theorem states the existence and boundedness of the solution operator of the system ( 13)-( 19).This result is used later in Section 5 to prove Theorem 1.

Theorem 2 Let the control space
and solution space (the hat is omitted for the sake of readability) be given by ( 9) and (20), respectively, and let .Then, (30)-( 33) has a unique solution in .
Proof The steps of the proof are as follows.First, the th Galerkin approximation of the problem (30)-( 33) is defined (part A).Then, the key energy estimate is derived (part B).Lastly, the existence and uniqueness of the solution is proven (parts C and D).

Part A -Galerkin Approximation
Consider a countable set of functions such that form an orthonormal basis of 2 (34) form an orthogonal basis of 1 0 .

Part B -Energy Estimates
Many of the techniques featured in this section have been inspired by previous works on PDE analysis [5,15].
To simplify the notation, consider  Moreover, the generalized Poincaré's inequality (Lemma 4) is applied to the right-hand side of (54) along with the continuity of the trace operator Tr Adding the non-negative terms where the continuous embedding from Lemma 1 was used once again.The term 0 2 2 on the right-hand side of (59) can be estimated using (40) as Lastly, the term 0 0 in (59) is estimated as follows.Viewing 0 as a function from to and using Young's inequality (Lemma 6), constants 12 13 14 15 can be found such that (64) The continuous embedding given by Theorem 4 for 1 2 3 yields The estimates (67), (68) along with (66) imply that is bounded in 0 Furthermore, the bound on the time derivatives is provided.Letting be defined as in (58) and using a similar estimate to (60), the inequality (59) can be viewed as Returning to the inequality (54), applying (55) and retaining all the terms on the left-hand side (as opposed to the estimate 66), one arrives at The inequality (75) can be rewritten as , we find that (83), (84) lead to (see [15]).Multiplying (37) by and (38) by for all The relations (90) imply that which concludes the proof that the limit functions satisfy ( 30)- (31).Applying integration by parts to the time derivatives in ( 30)- (31), equations ( 88)-(89) and taking the limit shows that the initial conditions are satisfied by the limit functions.This concludes the proof of existence of a solution for ( 30)-( 33) for an arbitrary 4 0 .

Part D -Uniqueness of the Solution
To show that the solution found in part C is unique, let 1 1 and 2 2 two solutions of ( 30)- (33) Utilizing the notation ( 9), ( 20) and letting ad be closed, convex and bounded, we provide the proof of Theorem 1.
Proof (of Theorem 1) In order to ensure the existence of optimal control, we prove the following sufficient conditions (Theorem 7): is weakly sequentially continuous.(c3) (defined by (12)) is convex and continuous.
Note that condition (c3) can be weakened to being sequentially lower semi-continuous [19].
As for condition (c1), the existence of the solution operator is ensured by Theorem 2. To prove its boundedness, the boundedness of ad is used.Taking the supremum of (66) while using (67) and (68) yields Taking the limit inferior of the estimates (104), (105), adding them and adjusting the constants yields which is the desired boundedness property of the solution operator.
To prove (c2), only the convergence of the nonlinear terms is checked.Lemma 2 provides the necessary convergence property.
To prove (c3), consider the embedding given by Theorem 6 and observe that This proves the continuity of .Lastly, is convex due to the linearity of the integral and convexity of the square function.

Conclusion
In this paper, a weak formulation of the Dirichlet Boundary condition optimization for a phase field problem is proposed.Energy estimates along with some specific compact embedding results are provided.These results together with the proof of uniqueness of the solution for a given control give rise to a bounded solution operator.Using the aforementioned results, the existence of optimal control is proven.
It is worthwhile to note that quite strong regularity assumptions are imposed on the control ( 4 0 ).This is a consequence of the compact embedding statements used in the analysis.comes into play when dealing with non-linear PDEs (see Section 3).Lastly, an example configuration for the Gelfand triple and space is presented.
Lastly, other spaces that are associated with vector valued functions are discussed.

A.3 Embedding Theorems
Selected statements about embeddings of functional spaces [1,15,31] are listed in this section.The reader should note that even though these embeddings are only discussed for bounded subdomains of , some of these results hold for unbounded domains as well (possibly under further technical assumptions) [1,15].[0 ] 2 .

A.3 Existence of Optimal Control
The following theorem can be used to prove the existence of optimal control [19,24].
on the right-hand side of (59) (this overestimates the right-hand side for any [ positive (see (47)) taking the essential supremum of (66) side of (73), adjusting the constants and integrating with respect to over [0 where the estimates (62), (63), (64), and (65) were used once again to estimate 0 .Estimate (76) implies that are bounded in 2 are smooth coefficients.Functions of the form (85) are dense in 2 0 1 0 complete the proof of existence of the solution, the convergence of all the terms in (88)-(89) needs to be shown.The terms sense of the canonical embedding) and (84).)-(84) was used.Furthermore, Lemma 2 can be applied since in

1
Let be a bounded domain.We say that has a Lipschitz boundary if every point has an open neighborhood such that is the graph of a Lipschitz continuous function.
If has a feasible point, the state equation (111) has a bounded solution operator, is weakly sequentially continuous, and is convex and continuous, the problem (110)-(111) has a solution.