Abstract
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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 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.
2 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 dx and dS 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 J, is to achieve a certain shape of the solid body at final time T. To formulate the problem, we let \({\varOmega }\subset \mathbb {R}^{n}\) for n = 1,2,3 be a bounded domain with a Lipschitz boundary and assume T > 0. Let \(u:\overline {{\varOmega }}\times \left [0,T\right ]\rightarrow \mathbb {R}\) represent the temperature field and let \(p:\overline {{\varOmega }}\times \left [0,T\right ]\rightarrow \mathbb {R}\) be the phase field which determines the shape of the solid subdomain by
Then, the formal statement of the problem reads
where β > 0 specifies the initial supercooling and α,Q are material-specific constants. The parameter ξ > 0 determines the phase interface thickness and b > 0 is a constant calculated so that the Gibbs-Thomson relation [18] is asymptotically recovered as ξ → 0. For this particular model, we get \(b=\frac {1}{6}\) (see, e.g. [28]). The initial conditions for the temperature and phase field are given by \(u_{\text {ini}}\in H^{1}\left ({\varOmega }\right )\), \(p_{\text {ini}}\in H^{1}\left ({\varOmega }\right )\). The solution of the minimization problem (1)–(8) is a function \(\theta :\partial {\varOmega }\times \left [0,T\right ]\rightarrow \mathbb {R}\) which controls the temperature at the boundary of Ω in such a way that p is as close as possible to the target phase field profile \(p_{\text {f}}\in L_{2}\left ({\varOmega }\right )\) at t = T. If necessary, the magnitude of the control 𝜃 can be reduced by increasing the regularization parameter γ ≥ 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
This high level of regularity is a consequence of the compact embedding statements used in the analysis that follows. Let 𝜃 in (3) be such that there exists η ∈ XC for which
Using the concept of a weak solution, we look for (see Section A.1 in the Appendix for Bochner space related notation)
such that (1)–(8) is satisfied in the distributional sense. To this end, it is purposeful to define
Using the principles of Dirichlet lift and (11), we can reformulate the optimization problem (1)–(8). Let Wad ⊂ XC be closed, convex and bounded, then the problem in question reads
where the equations (13) and (16) are in the sense of distributions. A pair of functions
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 XC and solution space \(\hat {X}_{S}\) be given by (9) and (20), respectively, and let Wad ⊂ XC 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 η ∈ XC. 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.
3 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.
Lemma 1
Let \({\varOmega }\subset \mathbb {R}^{n}\) be a bounded domain with a Lipschitz boundary for \(n\in \left \{ 1,2,3\right \} \) and T > 0. Let \(f\in H^{4}\left ({\varOmega }\times \left (0,T\right )\right )\). Then,
Proof
Since Ω has a Lipschitz boundary, so does \({\varOmega }\times \left (0,T\right )\). Using Theorem 3 and setting m = 4, p = 2 and k = 1 leads to
This means that there exists a Hölder continuous function \(g\in C^{1,\beta }\left (\overline {{\varOmega }}\times \left [0,T\right ]\right )\) such that f = g in \(H^{4}\left ({\varOmega }\times \left (0,T\right )\right )\) and specifically \(g\left (t\right )\in C^{1,\beta }\left (\overline {{\varOmega }}\right )\) for any \(t\in \left [0,T\right ]\). Since \(\left \Vert g\left (t\right )\right \Vert _{H^{1}\left ({\varOmega }\right )}=\left \Vert f\left (t\right )\right \Vert _{H^{1}\left ({\varOmega }\right )}\), the conclusion (21) holds. □
Lemma 2
Let \({\varOmega }\subset \mathbb {R}^{n}\) be a bounded domain (with an arbitrary boundary) for \(n\in \left \{ 1,2,3\right \} \) and T > 0. Let \(f_{n}\rightharpoonup f\) in \(W\left (0,T;8,2;{H_{0}^{1}}\left ({\varOmega }\right ),L_{2}\left ({\varOmega }\right )\right )\). Then,
Proof
With the help of Theorem 4 (setting m = 1, k = 0, p = 2 and q = 5), the Gelfand triple
can be constructed. Applying Theorem 5 (Aubin-Lions) gives
It is sufficient to prove strong convergence of \({f_{n}^{3}}\rightarrow f^{3}\) in \(L_{2}\left (0,T;L_{\frac {5}{3}}\left ({\varOmega }\right )\right )\), since the reflexivity of \(L_{2}\left (0,T;L_{\frac {5}{3}}\left ({\varOmega }\right )\right )\) and the embedding
give rise to
For any two numbers \(a,b\in \mathbb {R}\), the estimate
may be derived. Using (26), the estimate
can be constructed. For any \(v,w\in L^{5}\left ({\varOmega }\right )\), Hölder’s inequality (Lemma 5) can be used with exponents \(\frac {3}{2}\) and 3 to arrive at
Combining (27) and (28) yields
Squaring, integrating over \(\left [0,T\right ]\) 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
Note that \(f_{n}\rightarrow f\) in \(L_{8}\left (0,T;L_{5}\left ({\varOmega }\right )\right )\), \(\left \Vert f_{n}\right \Vert _{L_{8}\left (0,T;L_{5}\left ({\varOmega }\right )\right )}^{4}\) is bounded and \(\left \Vert f\right \Vert _{L_{8}\left (0,T;L_{5}\left ({\varOmega }\right )\right )}^{4}\) is finite. Furthermore,
since \(L_{8}\left (0,T;L_{5}\left ({\varOmega }\right )\right )\hookrightarrow L_{4}\left (0,T;L_{5}\left ({\varOmega }\right )\right )\). Altogether,
which completes the proof of (22). To get the result (23), one just replaces the estimate (26) with
and uses the inequality
in the estimate analogous to (28). □
4 Analysis of the State Equation
We start by stating the weak formulation of the problem (13)–(19). To simplify the notation, we use u and p instead of \(\hat {u}\) and \(\hat {p}\), respectively. Using standard methods, we arrive at
where \(f_{0}:\mathbb {R}\rightarrow \mathbb {R}\) is defined as \(f_{0}\left (y\right )\equiv y\left (1-y\right )\left (y-\frac {1}{2}\right )\).
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 XC and solution space XS (the hat is omitted for the sake of readability) be given by (9) and (20), respectively, and let η ∈ XC. Then, (30)–(33) has a unique solution in XS.
Proof
The steps of the proof are as follows. First, the m 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 \(w_{i}:{\varOmega }\rightarrow \mathbb {R}\) such that
An example of functions that satisfy properties (34)–(35) are the eigenfunctions of the Laplacian on \({H_{0}^{1}}\left ({\varOmega }\right )\) [15]. For each \(m\in \mathbb {N}\), let there be two vector valued functions \(u_{m},p_{m}:\left [0,T\right ]\rightarrow {H_{0}^{1}}\left ({\varOmega }\right )\) such that
Additionally, let (36) be such that the problem
with the initial conditions
is satisfied for all \(m\in \mathbb {N}\). Standard ODE theory can be applied to show that (37)–(40) has a unique solution [5, 15]. Thus the functions (36) are well defined. We call the problem (37)–(40) the m th Galerkin approximation of (30)–(33).
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
for any \(f,g\in H^{1}\left ({\varOmega }\right )\). Additionally, the time derivative of the vector valued function \(f:\left [0,T\right ]\rightarrow H^{1}\left ({\varOmega }\right )\) is denoted by \(\dot {f}\) and the evaluation at a point (for any function) is also omitted, i.e. instead of \(f\left (t\right )=g\left (t\right )\text { a.e. in }\left [0,T\right ]\), we write \(f=g\text { a.e. in }\left [0,T\right ]\). In the following, the symbols \(c_{1},c_{2},\text {{\dots }}\) denote positive constants.
Consider a fixed but arbitrary \(m\in \mathbb {N}\). Then, multiply (37) by \(\dot {\alpha }_{m}^{k}\)and (38) by \(\dot {\beta }_{m}^{k}\) for each k = 1,…,m. Adding up the equations multiplied by \(\dot {\alpha }_{m}^{k}\) and \(\dot {\beta }_{m}^{k}\) separately gives
Some of the terms in (45) can be rewritten as
where \(\omega _{0}\left (y\right )=\frac {1}{4}\left (\left (y-\frac {1}{2}\right )^{2}-\frac {1}{4}\right )^{2}\) for any \(y\in \mathbb {R}\). Using (46) and (47), the system (44)–(45) may be reformulated as
The bilinearity of \(B\left [\cdot ,\cdot \right ]\) gives rise to the identity
Using (50), we can rewrite (48) as
Applying the Young’s (Lemma 6) and Schwarz inequalities on the right-hand side of (51), we get the estimates
where δ1,δ2 > 0. Setting \(\delta _{1}=\delta _{2}=\frac {1}{2}\) leads to the estimate
Estimating the right-hand side of (49) using analogous methods with δ4 > 0 yields
Setting \(\delta _{4}=\frac {b\beta \xi }{\alpha \xi ^{2}}\) gives rise to the estimate (see (49))
Multiplying (52) by \(\frac {\alpha \xi ^{2}}{4Q^{2}}\) and adding the result to (53) gives
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 \(\text {Tr}:H^{1}\left ({\varOmega }\right )\rightarrow L_{2}\left (\partial {\varOmega }\right )\) and \(u_{m}\in {H_{0}^{1}}\left ({\varOmega }\right )\) to give the estimate
Note that \(\left \Vert \eta \right \Vert _{H^{1}\left ({\varOmega }\right )}^{2}\) and \(\left \Vert \dot {\eta }\right \Vert _{H^{1}\left ({\varOmega }\right )}^{2}\) must be finite almost everywhere in \(\left (0,T\right )\) since \(\eta \in H^{4}\left ({\varOmega }\times \left (0,T\right )\right )\). Dropping the positive terms \(\frac {\alpha \xi ^{2}}{8Q^{2}}\left \Vert \dot {u}_{m}\right \Vert _{L_{2}\left ({\varOmega }\right )}^{2},\frac {\alpha \xi ^{2}}{4}\left \Vert \dot {p}_{m}\right \Vert _{L_{2}\left ({\varOmega }\right )}^{2}\) from the left hand side of (54) and applying (55) leads to
Adding the non-negative terms \(\left \Vert \nabla p_{m}\right \Vert _{L_{2}\left ({\varOmega }\right )}^{2}\) and \(\underset {{\varOmega }}{\int \limits }\omega _{0}\left (p_{m}\right )\) to the right-hand side of (56) and adjusting the constants yields
Using Grönwall’s lemma (Lemma 3) with the setting
leads to
where \(t\in \left [0,T\right ]\). Taking t = T on the right-hand side of (59) (this overestimates the right-hand side for any \(t\in \left [0,T\right ]\)) gives the estimate
The initial condition for the Galerkin approximation (39) gives
Taking \(\eta \in H^{4}\left ({\varOmega }\times \left (0,T\right )\right )\) into account, (61) may be refined further. From Lemma 1, it follows that \(\left \Vert \eta \left (0\right )\right \Vert _{H^{1}\left ({\varOmega }\right )}^{2}<\infty \). To obtain an estimate in the \(H^{4}\left ({\varOmega }\times \left (0,T\right )\right )\)-norm, one can write
where the continuous embedding from Lemma 1 was used once again. The term \(\left \Vert \nabla p_{m}\left (0\right )\right \Vert _{L_{2}\left ({\varOmega }\right )}^{2}\) on the right-hand side of (59) can be estimated using (40) as
Lastly, the term \(\underset {{\varOmega }}{\int \limits }\omega _{0}\left (p_{m}\left (0\right )\right )\) in (59) is estimated as follows. Viewing \(\omega _{0}\left (y\right )\) as a function from \(\mathbb {R}\) to \(\mathbb {R}\) and using Young’s inequality (Lemma 6), constants c12,c13,c14,c15 can be found such that
This estimate gives
The continuous embedding given by Theorem 4 for n = 1,2,3 yields
Using this embedding along with (63) leads to
Applying (60), (61), (62), (64), and (65) gives rise to the estimate
Since
and ω0 is positive (see (47)) taking the essential supremum of (66) shows that
Using Poincaré’s (Lemma 4) and Young’s inequality (Lemma 6) gives the estimate
Similarly,
The estimates (67), (68) along with (66) imply that
Furthermore, the bound on the time derivatives is provided. Letting \({\varrho }\left (t\right )\) be defined as in (58) and using a similar estimate to (60), the inequality (59) can be viewed as
Integrating (71) with respect to t over \(\left [0,T\right ]\) gives rise to
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
Adding the non-negative terms \(\frac {8Q^{2}c_{7}}{\alpha }\left \Vert \nabla p_{m}\right \Vert _{L_{2}\left ({\varOmega }\right )}^{2}\) and \(\frac {8Q^{2}c_{7}}{\alpha \xi ^{2}}\underset {{\varOmega }}{\int \limits }\omega _{0}\left (p_{m}\right )\) to the right-hand side of (73), adjusting the constants and integrating with respect to t over \(\left [0,T\right ]\) yields
where (72) was used. Since \(\left (e^{Tc_{8}}-1\right )\) is positive, the right hand side of (74) can be further estimated by
Since c7 can be chosen such that \(\frac {1}{c_{8}}\frac {8Q^{2}c_{7}}{\alpha \xi ^{2}}>1\) (see (57)), adding \(\rho \left (0\right )\) to both sides of (74) and dropping the positive term \(\rho \left (T\right )\) leads to
The inequality (75) can be rewritten as
where the estimates (62), (63), (64), and (65) were used once again to estimate \(\rho \left (0\right )\).
Estimate (76) implies that
Combining (69), (70) with (76) yields that
where \(q\in \mathbb {N}\) is arbitrary.
Part C - Existence of a Solution
Since the space \(W\left (0,T;q,2;{H_{0}^{1}}\left ({\varOmega }\right ),L_{2}\left ({\varOmega }\right )\right )\) is reflexive for any \(q\in \mathbb {N}\), from (78), it follows that there exist sub-sequences such that
Picking sub-sequences \(\left (u_{h_{m}}\right ),\left (p_{h_{m}}\right )\) that converge weakly in \(W\left (0,T;8,2;{H_{0}^{1}}\left ({\varOmega }\right ),\right .\) \(\left .L_{2}\left ({\varOmega }\right )\right )\) to \(u^{\left (8\right )},p^{\left (8\right )}\), respectively, and then choosing sub-sequences of \(\left (u_{h_{m}}\right ),\left (p_{h_{m}}\right )\) that converge weakly in \(W\left (0,T;2,2;{H_{0}^{1}}\left ({\varOmega }\right ),L_{2}\left ({\varOmega }\right )\right )\) to \(u^{\left (2\right )},p^{\left (2\right )}\), respectively, one arrives at
where the sub-sequence notation has been omitted for simplicity. Applying Theorem 5 with the Gelfand triple \({H_{0}^{1}}\left ({\varOmega }\right )\hookrightarrow \hookrightarrow L_{2}\left ({\varOmega }\right )\hookrightarrow {H_{0}^{1}}\left ({\varOmega }\right )^{*}\), we find that (83), (84) lead to \(u^{\left (2\right )}=u^{\left (8\right )}\equiv u,p^{\left (2\right )}=p^{\left (8\right )}\equiv p\).
To show that the pair \(\left (u,p\right )\) is a solution to (30)–(33), consider two functions of the form
where \(a^{i},b^{i}:\left [0,T\right ]\rightarrow \mathbb {R}\) are smooth coefficients. Functions of the form (85) are dense in \(L_{2}\left (0,T;{H_{0}^{1}}\left ({\varOmega }\right )\right )\) (see [15]). Multiplying (37) by \(a^{i}\left (t\right )\) and (38) by \(b^{i}\left (t\right )\) for all \(i\in \left \{ 1,\ldots ,m\right \} \) and summing each of them separately leads to
Integrating (86)–(87) with respect to t over \(\left [0,T\right ]\) gives rise to
To complete the proof of existence of the solution, the convergence of all the terms in (88)–(89) needs to be shown. The terms
converge since \(\varphi ,\psi \in L_{2}\left (0,T;{H_{0}^{1}}\left ({\varOmega }\right )^{*}\right )\) (using the canonical pairing). Moreover, (83) and (84) ensure that
The convergence
follows from \(\psi \in L_{2}\left (0,T;{H_{0}^{1}}\left ({\varOmega }\right )^{*}\right )\) (in the sense of the canonical embedding) and (84). The remaining linear terms \(\underset {0}{\overset {T}{\int \limits }}\underset {{\varOmega }}{\int \limits }\nabla u_{m}\cdot \nabla \varphi ,\underset {0}{\overset {T}{\int \limits }}\underset {{\varOmega }}{\int \limits }\nabla p_{m}\cdot \nabla \psi \) are handled similarly. It can be shown that
which results in
Furthermore, Lemma 2 can be applied since
which gives the convergence of the term \(\underset {0}{\overset {T}{\int \limits }}\underset {{\varOmega }}{\int \limits }f_{0}\left (p_{m}\right )\psi .\) More specifically, the cubic and quadratic terms converge:
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 \(\eta \in H^{4}\left ({\varOmega }\times \left (0,T\right )\right )\).
Part D - Uniqueness of the Solution
To show that the solution found in part C is unique, let \(\left (u_{1},p_{1}\right )\) and \(\left (u_{2},p_{2}\right )\) be two solutions of (30)–(33). Define u12 ≡ u1 − u2 and p12 ≡ p1 − p2. Subtracting the two systems leads to
Setting \(v=u_{12}\left (t\right )-Qp_{12}\left (t\right )\) and \(w=p_{12}\left (t\right )\) in (91) and (92), respectively, and using the notation (41)-(43) leads to
The regularity properties of the solution guarantee that there exists a constant \(C_{f_{0}}\) such that
Using the Young’s (Lemma 6) and Schwarz inequalities, the following estimates can be made:
In (97), Cemb is the constant of the continuous embedding \({H_{0}^{1}}\left ({\varOmega }\right )\hookrightarrow L_{6}\left ({\varOmega }\right )\). Combining (94) with (96) results in
Similarly, the estimate
arises by blending (98), (97) with (95), where ε in (97) is set so that
Adding the \(\frac {\xi ^{2}}{2Q^{2}}\) multiple of (99) to (100) along with using the estimate
gives rise to
Dropping the positive terms on the left hand side of (101) and increasing the constants on the right-hand side so that
for some constant c34 > 0 sets the stage for using Grönwall’s lemma (Lemma 3). We get
Integrating (102) over \(\left [0,T\right ]\) and using the initial conditions of the problem (91)–(92) gives
□
5 Existence of Optimal Solution
Utilizing the notation (9), (20) and letting Wad ⊂ XC 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):
-
(c1)
The solution operator of the system \(S:W_{\text {ad}}\rightarrow \hat {X}_{S}\) is a bounded operator.
-
(c2)
The state equation operator
$$ e:\hat{X}_{S}\times W_{\text{ad}}\rightarrow L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)^{*}\right)^{2}\times L_{2}\left( {\varOmega}\right)^{2} $$is weakly sequentially continuous.
-
(c3)
J (defined by (12)) is convex and continuous.
Note that condition (c3) can be weakened to J being sequentially lower semi-continuous [19].
As for condition (c1), the existence of the solution operator S is ensured by Theorem 2. To prove its boundedness, the boundedness of Wad is used. Taking the supremum of (66) while using (67) and (68) yields
where M is the constant bounding Wad in XC. This means that there exists a constant M1 dependent only on the Sobolev norms of the initial conditions uini,pini, final time T and the constant M such that
Similarly, using (76) leads to
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
Take
and \(\eta _{1},\eta _{2}\in H^{4}\left ({\varOmega }\times \left (0,T\right )\right )=X_{C}\) and make the estimate
Considering \(\left (p_{1},u_{1},\eta _{1}\right )\in \hat {X}_{S}\times X_{C}\) such that \(\left \Vert \left (p_{1},u_{1},\eta _{1}\right )-\left (p_{2},u_{2},\eta _{2}\right )\right \Vert _{\hat {X}_{S}\times X_{C}}<1\) makes
bounded thanks to the embedding statements (106) and the continuity of the trace operator. Using the same embedding statements, it is possible to make
arbitrarily small by choosing an appropriate 0 < δ < 1 such that
This proves the continuity of J. Lastly, J is convex due to the linearity of the integral and convexity of the square function. □
6 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 (\(H^{4}\left ({\varOmega }\times \left (0,T\right )\right )\)). This is a consequence of the compact embedding statements used in the analysis.
References
Adams R, Fournier J. Sobolev spaces. Amsterdam: Elsevier; 2003.
Belgacem FB, Bernardi C, Fekih HE. Dirichlet boundary control for a parabolic equation with a final observation I: a space-time mixed formulation and penalization. Asymptot Anal 2011;71:101–121.
Belmiloudi A. Robust and optimal control problems to a phase-field model for the solidification of a binary alloy with a constant temperature. J Dyn Control Syst 2004;10 :453–499. https://doi.org/10.1023/B:JODS.0000045361.82698.7f.
Beneš M. Anisotropic phase-field model with focused latent-heat release. FREE BOUNDARY PROBLEMS: Theory and applications II, GAKUTO International Series in Mathematical Sciences and Applications; 2000. p. 18–30.
Beneš M. Mathematical analysis of phase-field equations with numerically efficient coupling terms. Interface Free Bound 2001;3:201–221.
Beneš M. Mathematical and computational aspects of solidification of pure substances. Acta Math Univ Comenianae 2001;70(1):123–151.
Beneš M., Hilhorst D, Weidenfeld R. Interface dynamics for an anisotropic Allen-Cahn equation. Nonlocal Elliptic and Parabolic Problems 2004;66: 39–45.
Boettinger WJ, Coriell S, Greer AL, Karma A, Kurz W, Rappaz M, Trivedi R. Solidification microstructures: recent developments, future directions. Acta Mater 2000;48:43–70.
Bragard J, Karma A, Lee YH. Linking phase-field and atomistic simulations to model dendritic solidification in highly undercooled melts. Interface Sci 2002;10:121–136.
Chowdhury S, Gudi T, Nandakumaran A. Error bounds for a dirichlet boundary control problem based on energy spaces. Math Comput 2015;86: 1103–1126. https://doi.org/10.1090/mcom/3125.
Christian Miehe Lisa-Marie Schanzel HU. 2014. Phase field modeling of fracture in multi-physics problems. part i. balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids Computer Methods in Applied Mechanics and Engineering.
Colli P, Gilardi G, Marinoschi G, Rocca E. Optimal control for a conserved phase field system with a possibly singular potential. Evol Equ 2018;7:95–116.
Colli P, Gilardi G, Sprekels J. Analysis and optimal boundary control of a nonstandard system of phase field equations. Milan J Math 2012;80:119–149.
Colli P, Signori A, Sprekels J. Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials. Appl Math Optim 2019;83:2017–2049.
Evans LC. 1998. Partial differential equations, Graduate Studies in Mathematics, vol 18, American Mathematical Society.
Gong W, Hinze M, Zhou Z. Finite element method and a priori error estimates for dirichlet boundary control problems governed by parabolic pdes. J Sci Comput 2016;66:941–967. https://doi.org/10.1137/100795632.
Gudi T, Mallik G, Sau R. 2021. Finite element analysis of the dirichlet boundary control problem governed by linear parabolic equation. Arxiv. https://doi.org/10.48550, arXiv:2111.02039.
Gurtin ME. On the two-phase Stefan problem with interfacial energy and entropy. Arch Ration Mech An 1986;96(3):199–241. https://doi.org/10.1007/BF00251907.
Hinze M, Pinnau R, Ulbrich M, Ulbrich S. 2009. Optimization with PDE constraints. Springer.
Hoffman KH, Jiang L. Otimal control of a phase field model for solidification. Numer Funct Anal Optim 1992;13:11–27.
Karma A, Rappel WJ. Numerical simulation of three-dimensional dendritic growth. Phys Rev Lett 1996;77(19):4050–4053.
Karma A, Rappel WJ. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys Rev E 1998;57(4):4.
Kunisch K, Vexler B. Constrained dirichlet boundary control in l2 for a class of evolution equations. SIAM J Control Optim 2007;46:1726–1753. https://doi.org/10.1137/060670110.
Leugering G, Engell S, Griewank A, Hinze M, Rannacher R, Schulz V, Ulbrich M, Ulbrich S. 2012. Constrained optimization and optimal control for partial differential equations. https://doi.org/10.1007/978-3-0348-0133-1.
Ramirez JC, Beckermann C, Karma A, Diepers HJ. Phase-field modeling of binary alloy solidification with coupled heat and solute diffusion. Phys Rev E 2004;69(051):607.
Sprekels J, Tröltzsch F. 2020. Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth. Arxiv. https://doi.org/10.48550, arXiv:2005.02784.
Strachota P, Wodecki A, Beneš M. Efficiency of a hybrid parallel algorithm for phase-field simulation of polycrystalline solidification in 3D. ALGORITMY 2020, 21th Conference on Scientific Computing, Vysoké Tatry - Podbanské, Slovakia, September 10 - 15, 2020. Proceedings of contributed papers and posters, pp 131–140. SPEKTRUM STU. In: Frolkovič P, Mikula K, and Ševčovič D, editors; 2020.
Strachota P, Wodecki A, Beneš M. Focusing the latent heat release in 3D phase field simulations of dendritic crystal growth. Modelling Simul Mater Sci Eng 2021;29(065):009.
Strang G, Fix G. 2008. An analysis of the finite element method, 2ed. Prentice Hall series in automatic computation. Wellesley-Cambridge Press.
Suwa Y, Saito Y. Computer simulation of grain growth by the phase field model. effect of interfacial energy on kinetics of grain growth. Mater Trans 2003;44:2245–2251. https://doi.org/10.2320/matertrans.44.2245.
Temam R. 1977. Navier-stokes equations : theory and numerical analysis. Elsevier.
Wait R, Mitchell A. 1985. Finite element analysis and applications. A Wiley-Interscience publication.
Wheeler AA, Murray BT, Schaefer RJ. Computation of dendrites using a phase field model. Physica D 1993;66:243–262.
Yang W, Sun J, Zhang S. Analysis of optimal boundary control for a three-dimensional reaction-diffusion system. Control Optim 2017;7:325–344.
Zonghong X, Wei W, Ying Z, Yue W, Yumei L. Optimal control for a phase field model of melting arising from inductive heating. AIMS Math 2022;7:121–142.
Acknowledgments
This work has been supported by the projects:
– Centre of Advanced Applied Sciences (Reg. No. CZ.02.1.01/0.0/0.0/16_019/0000778),
– Research Center for Informatics (Reg. No. CZ.02.1.01/0.0/0.0/16_019/0000765),
co-financed by the European Union. Partial support of:
– grant No. SGS20/184/OHK4/3T/14 of the Grant Agency of the Czech Technical University in Prague,
– the project No. NV19-08-00071 of the Ministry of Health of the Czech Republic.
Funding
Open access publishing supported by the National Technical Library in Prague.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
In this section, the notation used for Bochner spaces and fundamental statements that find utility in the above proof are listed.
1.1 A.1 Bochner Spaces \(L_{p}\left (0,T;Y\right )\), \(W\left (0,T;p,\widetilde {p},X_{0},X,X_{1}\right )\) and \(W\left (0,T;p,\widetilde {p},X_{0},X_{1}\right )\)
Let Y be a Banach space. Suppose that \(f:\left [0,T\right ]\rightarrow Y\) is a vector valued function and define the norms
For \(1\leq p\leq \infty \), the Bochner space is defined as
Let X0,X,X1 be Banach Spaces, X0,X1 be reflexive and
Additionally, consider \(p,\widetilde {p}\in \left (1,\infty \right )\) and T > 0. Then, define the set
where the distributional derivative \(\dot {v}\) is taken in the space \(L_{1}\left (0,T;X_{1}\right )\). This is possible since for any \(f\in L_{p}\left (0,T;X_{0}\right )\), we get \(f\in L_{1}\left (0,T;X_{0}\right )\) and \(f\in L_{1}\left (0,T;X_{1}\right )\) by (108). If \(\dot {f}\in L_{1}\left (0,T;X_{1}\right )\) exists and the additional condition \(\dot {f}\in L_{\widetilde {p}}\left (0,T;X_{1}\right )\) is fulfilled, then f ∈ W. In this setting, (108) is called a Gelfand (or Hilbert) triple. For any \(f:\left [0,T\right ]\rightarrow X_{0}\) define the norm
The set W along with (109) is a Banach space denoted \(\left (W,\left \Vert \cdot \right \Vert _{W}\right )\).
Commonly, the spaces \(\left (W,\left \Vert f\right \Vert _{W}\right )\) are used as the solution space for time dependent partial differential equations (PDEs). Moreover, \(\left (W,\left \Vert f\right \Vert _{W}\right )\hookrightarrow \hookrightarrow L_{p}\left (0,T;X\right )\), which comes into play when dealing with non-linear PDEs (see Section 3). Lastly, an example configuration for the Gelfand triple and space W is presented.
Lastly, other spaces that are associated with vector valued functions are discussed. Let X be a Banach space and T > 0. Define the set
The space \(C\left (\left [0,T\right ];X\right )\) together with the norm
is a Banach space denoted \(\left (C\left (\left [0,T\right ];X\right ),\left \Vert \cdot \right \Vert _{C\left (0,T;X\right )}\right )\) (since X is complete).
1.2 A.2 Inequalities
Lemma 3
(Grönwall’s lemma) Let \({\varrho }\in C\left (\left [0,T\right ],\mathbb {R}\right )\), \(\phi ,\psi \in L_{1}\left (\left [0,T\right ],\mathbb {R}\right )\) be non-negative and let
Then,
Lemma 4
(Poincaré’s inequality) Let Ω be a bounded Lipschitz domain and \(f\in H^{1}\left ({\varOmega }\right )\). Then, there exists a constant cp such that
Lemma 5
(Hölder’s inequality) Let Ω be a measure space, \(f,g:{\varOmega }\rightarrow \mathbb {R}\) be measurable functions and p,q be Hölder conjugates (i.e. \(\frac {1}{q}+\frac {1}{p}=1\)). Then,
Lemma 6
(Young’s inequality) Let a,b ≥ 0 and ε > 0. Then,
1.3 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 \(\mathbb {R}^{n}\), some of these results hold for unbounded domains as well (possibly under further technical assumptions) [1, 15].
Definition 1
Let \({\varOmega }\subset \mathbb {R}^{n}\) be a bounded domain. We say that Ω has a Lipschitz boundary if every point x ∈ ∂Ω has an open neighborhood Ux such that Ux ∩ ∂Ω is the graph of a Lipschitz continuous function.
Theorem 3
(Hölder-Sobolev embedding) Let \({\varOmega }\subset \mathbb {R}^{n}\) be a bounded domain with a Lipschitz boundary. Let \(m\in \mathbb {N}\) and \(p\in \left [1,\infty \right )\). Then, for any \(k\in \mathbb {N}_{0}\), \(\beta \in \left (0,1\right )\) such that
the embedding
holds.
Theorem 4
(Sobolev compact embedding) Let \({\varOmega }\subset \mathbb {R}^{n}\) be a bounded domain. Let \(m\in \mathbb {N}\), \(p\in \left [1,\infty \right )\), q ≥ 1, \(k\in \mathbb {N}_{0}\), m > k and
hold. Then,
Theorem 5
(embedding of Bochner spaces I.) Let X0,X,X1 form a Gelfand triple as defined by (108). Then, for any \(p,\widetilde {p}\in \left (1,\infty \right )\) and T > 0,
Theorem 6
(embedding of Bochner spaces II.) Let T > 0. Then,
1.4 A.3 Existence of Optimal Control
The following theorem can be used to prove the existence of optimal control [19, 24].
Theorem 7
(existence of optimal control) Let Z be a Banach space and Xc and Xs be reflexive Banach spaces. Suppose \(J:X_{s}\times X_{c}\rightarrow \mathbb {R},e:X_{s}\times X_{c}\rightarrow Z\) are continuous maps and assume W ⊂ Xc is convex bounded and closed and V ⊂ Xs is a convex and closed. Consider the problem
If V has a feasible point, the state equation (111) has a bounded solution operator \(\eta \in W\rightarrow u\left (\eta \right )\in X_{s}\), e is weakly sequentially continuous, and J is convex and continuous, the problem (110)-(111) has a solution.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wodecki, A., Balázsová, M., Strachota, P. et al. Existence of Optimal Control for Dirichlet Boundary Optimization in a Phase Field Problem. J Dyn Control Syst 29, 1425–1447 (2023). https://doi.org/10.1007/s10883-023-09642-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-023-09642-4