Constrained optimization problems governed by PDE models of grain boundary motions

Abstract: In this article, we consider a class of optimal control problems governed by state equations of Kobayashi-Warren-Carter-type. The control is given by physical temperature. The focus is on problems in dimensions less than or equal to 4. The results are divided into four Main Theorems, concerned with: solvability and parameter dependence of state equations and optimal control problems; the first-order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness.


Introduction
, as the base spaces for this work. Moreover, we set: In this article, we consider a class of optimal control problems, denoted by ( ) OP ε K , which are labeled by , the optimal control problem ( ) OP ε K is described as follows: 2 , called the optimal control, such that is a cost functional on H [ ] 2 , defined as follows: (4) The state system ( ) S ε is based on a phase field model of planar grain boundary motion, known as Kobayashi-Warren-Carter system (cf. [1,2] is a perturbation for the orientation order η, and > ν 0 is a fixed constant to relax the diffusion of the orientation angle θ. The first part (3) of the state system ( ) S ε is the initial-boundary value problem of an Allen-Cahn-type equation, so that the forcing term u can be regarded as a temperature control of the grain boundary formation. Also, the second problem (4) is the initial-boundary value problem to reproduce crystalline micro-structure of polycrystal, and the case of = ε 0 is the closest to the original setting adopted by Kobayashi et al. [1,2]. Indeed, when = ε 0, the quasi-linear diffusion as in (4) is described in a singular , and it is known that this type of singularity is effective to reproduce the facet, i.e., the locally uniform (constant) phase in each oriented grain (cf. [1]). Hence, the systems ( ) S ε , for positive ε, can be regarded as regularized approximating systems, that are to approach to the physically realistic situation ( ) S 0 , in the limit ↓ ε 0. Meanwhile, in the optimal control problem ( ) Moreover, are fixed constants. The objective of this article is to significantly extend the results of our previous work [15], which dealt with: ♯1) Key properties of the state systems ( ) S ε with one-dimensional domain ( ) = Ω 0,1 ; ♯2) Mathematical analysis of the optimal control problem ( ) OP ε K , for ≥ ε 0, but with one-dimensional domain ( ) ⊂ Ω 0,1 without any control constraints, i.e., In light of this, the novelty of this work is in: ♯3) The development of a mathematical analysis to obtain optimal controls of grain boundaries under the higher dimensional setting { } ∈ N 2, 3, 4 of the spatial domain, and the temperature constraint In addition, the presence of constraints makes the mathematical analysis further challenging. Notice that such constraints are meaningful from a practical point of view. We further emphasize that in the main part of this work, the ∞ L -boundedness of η will be essential, and the main results will be valid under the following assumption on the data: , s u c h t h a t , 0 ,1.
Hence, in general cases of constraints K ∈ K (including no constraint case), we will be forced to adopt some limiting (approximating) approach on the basis of the results under the restricted situation (r.s.0). Now, in view of ♯1)-♯3), we set the goal of this article to prove four Main theorems, summarized as follows. Main Theorem 1: Mathematical results concerning the following items.
(I-A) (Solvability of state systems): Existence and uniqueness for the state system ( ) This article is organized as follows. The Main Theorems are stated in Section 3, after the preliminaries in Section 1, and the auxiliary lemmas in the Appendix. The part after Section 3 will be divided into Sections 4-7, and these four sections will be devoted to the proofs of the respective four Main Theorems 1-4.

Preliminaries
We begin by prescribing the notations used throughout this article. Basic notations. For  For any dimension ∈ d , we denote by d the d-dimensional Lebesgue measure. The measure theoretical phrases, such as "a.e.," " t d ," " x d ," and so on, are all with respect to the Lebesgue measure in each corresponding dimension. Abstract notations. For an abstract Banach space X, we denote by | | ⋅ X the norm of X, and denote by ⟨⋅ ⋅⟩ , X the duality pairing between X and its dual * X . In particular, when X is a Hilbert space, we denote by ( ) ⋅ ⋅ , X the inner product of X. Moreover, when there is no possibility of confusion, we uniformly denote by | | ⋅ the norm of Euclidean spaces, and for any dimension ∈ d , we write the inner product (scalar product) of d , as follows: In particular, we denote by ∂ t , ∇, and div the distributional time derivative, the distributional gradient, and distributional divergence, respectively.
On this basis, we define among the Hilbert spaces H , V , V 0 , H, V, and V 0 , and the respective dual spaces * H , * V , * V 0 , H * , V * , and V * 0 . Additionally, in this article, we define the topology of the Hilbert space V 0 by using the following inner product: for any ,ˆ,ˆˆˆ,ˆˆˆˆˆ, for anyˆ; Finally, we define: . , Let X be an abstract Hilbert space X. Then, any closed and convex set ⊂ K X defines a single-valued operator ⟶ X K proj : The operator proj K is called the orthogonal projection (or projection in short) onto K .

Remark 2.
(Key properties of the projection) Let K be a closed and convex set in a Hilbert space X. Then, the following facts hold.
(Fact 1) The projection ⟶ X K proj : K is a nonexpansive operator from X into itself, i.e., proj , for all , 1, 2.
a.e. , , for any . , on a Hilbert space X, we denote by ( ) D Ψ the effective domain of Ψ. Also, we denote by ∂Ψ the subdifferential of Ψ. The subdifferential ∂Ψ corresponds to a weak differential of convex function Ψ, and it is known as a maximal monotone graph in the product space × X X. The set is called the domain of ∂Ψ. But, it should be noted that the converse inclusion of (6) is not true, in general.
be a continuous and convex function, defined as: When = ε 0, the convex function f 0 of this case coincides with the d-dimensional Euclidean norm | | ⋅ , and hence, the subdifferential ∂f 0 coincides with the set valued signal function ⟶ Sgn : 2 In the meantime, when > ε 0, the convex function f ε belongs to ∞ C -class, and the subdifferential ∂f ε is identified with the (single-valued) usual gradient: it will be estimated that f y f y ε y ε y ε ε y y ε ε y ỹ ,˜,˜˜˜, for all ,˜0 and ,˜, , in the sense of Mosco, as → ∞ n .
be the sequence of nonexpansive convex functions, as in (8) and (9). Then, the uniform estimate (9a) immediately leads to: , in the sense of Mosco, as , for any 0.
be a C 1 -function, which is Lipschitz continuous on . Also, g has a nonnegative primitive ( ) ≤ ∈ G C 0 2 , i.e., the derivative ′ = G G η d d coincides with g on . Moreover, g satisfies that: and limsup .  (1) and (5), respectively, and for any constraint , , which is defined as Moreover, the following extra assumption will be adopted to verify the dependence of optimal controls with respect to the constraint satisfies that: , for 0, 1, is a sequence of constraints such that: , , as , for a.e. , , and 0, 1,  The assumption (A4) leads to the boundedness of the second derivative ″ α of α. In fact, from the Lipschitz continuity of α and ′ αα , one can see that The assumption (A6) prescribes general settings of the constraint and the approaching sequence of constraints , as in Main Theorems 2 and 3, and as a special case, it contains the invariant (constant) setting In particular, we note that the result of Main Theorem 2 (II-B) will be enhanced in Main theorem 3 (III-B), under an additional assumption labeled by (23). , and a forcing pair Then, the following hold. 2 , in the sense that: and n n n 1 2 be given sequences such that: In particular, if: then Constrained optimization  1257 Remark 7. As a consequence of (16) and Remark 5, we further find a subsequence { } , , , , , in the pointwise sense a.e. in , , and in the pointwise sense a.e. in , i n ,a n d in , for a.e. 0, , as . , and any constraint K . Then, the following two items hold.
Let us assume the extra assumption (A6), for the sequence of constraints as in (14). In addition, for any n be the optimal control of ( ) OP ε K n n in the case when the initial pair of corre- Main Theorem 3. In addition to the assumptions (A1)-(A5), let us suppose the restricted situation (r.s.0) as in the Introduction, i.e., (III-A) (Necessary condition for ( ) ε is a unique solution to the following variational system: subject to the terminal condition: be sequences as in (14). Also, let be a sequence of constraints, fulfilling (A6). In addition, let us assume Remark 8. Note that the conclusion (24) of Main Theorem 3(III-B) is an enhanced version of that of Main Theorem 2(II-B).
, as in Proposition 5, in the case when: On this basis, let us define Then, since the embedding we can obtain the unique solution 2 to the variational system (20)-(22) as follows: Main Theorem 4. Let us assume (A1)-(A5), and let us assume that the situation is not under (r.s.0), i.e., it is under: Also, let us define a Hilbert space W 0 as follows: Then, there exists an optimal control In particular, if > ε 0, i.e., the situation is under: , a.e. in .

N
In the meantime, when > ε 0, (29)-(32) imply that the pair of functions [ ] ∘ ∘ p z , solves the following system: in the sense of distribution on Q. Note that the above system corresponds to the distributional form of the variational system (20)- (22), as in Main Theorem 3(III-A).
Remark 11. In the light of (19a), (28a), and Remark 3 (Fact 4), we will observe that In this section, we give the proof of the first Main Theorem 1. Before the proof, we refer to the reformulation method as in [22], and consider to reduce the state system ( ) Note that the assumptions (A2) and (A4) guarantee the lower semi-continuity and convexity of Φ ε R on [ ] H 2 .
Remark 12. As consequences of standard variational methods, we easily check the following facts.
R is a single-valued operator such that: respectively. Then, based on (Fact 7) and (Fact 8), it is verified that the state system ( ) S ε is equivalent to the following Cauchy problem.
In the context, "′" is the time derivative, and -, is the forcing term of the Cauchy problem .
Proof. We set: and prove that R 0 is the required constant.
In the light of (6), it is immediately verified that Then, by using (9a), (Fact 7), (Fact 8), (A4), and Young's inequality, we compute that: Due to (37), the inequalities in (38) lead to and for any Also, as a straightforward consequence of (Fact 8), it is seen that In the meantime, invoking (33), [ Now, let us take the constant > R 0 0 obtained in Key-Lemma 1. Then, owing to (40)-(42), and Key-Lemma 1, we can compute that In the light of (6), the above (43) is sufficient to conclude this Corollary. □ Lemma 1. Let us assume (A1)-(A4), and fix functions . Then, the initialboundary value problem: , and in particular, if: then it holds that ( ) ∈ ∞ η L Q .
. Then, referring to the general theories of nonlinear evolution equations (e.g., [19,23,24]), we immediately find a solution ( , in the variational sense: Next, we assume ( ) ∈ ∩ ∞ η V L Ω 0 and ( ) ∈ ∞ u L Q and verify the ∞ L -regularity of the solution η as in (45). To this end, we invoke the assumption (A3) and take a large constant ,a n d .
On this basis, we set our remaining task to show that a.e. in .
Due to (47) and (A4), the constants L 0 and −L 0 fulfill that respectively, together with the initial values L 0 and −L 0 , and the zero-Neumann boundary conditions. Now, let us take the difference between partial differential equations in (44) and (49a) (resp. (49b) and (44)), and multiply both the sides by ). Then, from (A2)-(A4), it is inferred that: Applying Gronwall's lemma, and invoking (47), we obtain which implies the validity of (48). □ Remark 13. Let ≥ ε 0 be arbitrary constant. Then, as a consequence of (Fact 7), (Fact 8), Key-Lemma 1, Corollary 1, and Lemma 1, we can say that the state system ( ) S ε is equivalent to the following Cauchy problem of evolution equation, denoted by ( ) for any ≥ r 0.

Proof of Main Theorem 2
In this section, we prove the second Main Theorem 2. Before the proof, we prepare the following lemma.
Lemma 2. Let us assume (A5) and (A6), and let us fix the function ∈ ⋂ = ∞ κ K n n 1 as in (A6). Besides, let us take any function ∈ u K , and define a sequence , by setting: n K n n n 0 1 n Then, it holds that: Proof. As is easily seen, a.e. , , 1, 2, 3, ,  ), a.e. in , i.e.¯¯¯¯, a.e. in , n n which leads to: The convergence (56) will be deduced as a straightforward consequence of (58), (59 ] ∈ u v,¯K ad , and let us invoke the definition of the cost function ε , defined in (2), to estimate that: i n f¯,¯, for all 0. Proof of Main Theorem 2 (II-A). Let us fix any ≥ ε 0. Then, from the estimate (60), we immediately find a sequence of forcing pairs , a s , , w e a k l y i n , a s , On account of (61a), (62), and (63), it is computed that: and this leads to Thus, we conclude the item (II-A). □

Proof of Main Theorem 2 (II-B). Let us take
as in (14). Besides, for the pair of functions [ ] ∈ u v,¯K ad as in (60), let us define ,¯0 ,¯0 in , as .
The convergences (64) and (65) enable us to estimate u v , n n of the optimal control of ( ) OP ε K n n . Then, in the light of (60) and (66), it is observed that Additionally, for every ℓ = 0, 1, the convex functionals on ( | | ( )) ⧹ ∞ ℓ − L Q κ 1 1 , defined as: are weakly lower semi-continuous. Therefore, we can observe from (67) and (A6) that: Since the limit * * u , when = M 0 u , can be taken arbitrary, the estimates as in (68) enable us to suppose that , . , one can see that , for any , a.e. 0, , subject to 0 0 in , Incidentally, the above linearization formulas can be verified as consequences of the assumptions (A1)-(A4) and the mean-value theorem (cf. [  : X ⟶ is Gâteaux differentiable over X. Moreover, for any 2 , such that: is a bounded linear operator, which is given as a restriction , as in Proposition 5, in the case when: Therefore, as a consequence of Main Theorem 1 (I-B), it is observed that Here, let us set: Then, in the light of (74) and Remark 14, one can say that: and for a.e. 0, , any , , and any 0, 1 , with the use of the constant  h k  p z  δ   , , , , ,  , , , , ,  , 0, ,¯,¯,¯,   ,  ,  0 ,0, ,  ,¯, ,  0,0 ,   , ,¯,¯, , 0,0¯0, 0 , for 0, 1 . Then, we estimate that for a.e. 0, , , such that 1 , a n d 0 , a s ,    Remark 17. In the previous work [16], one of the essential requirements is to use the continuous embedding ( ) ⊂ V L Ω 4 , as in Remark 1, which is satisfied under the restriction ≤ N 4 of the spatial dimension ∈ N . Therefore, under the assumption { } ∈ N 2, 3, 4 of this article, Propositions 3-6 will be applicable, although the previous results as in [16] were obtained under strict assumption { } ∈ N 1, 2, 3 .
Finally, we recall an auxiliary result, which was indirectly obtained in the proof of [