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State-constrained optimal control of phase-field equations with obstacle

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Abstract

This paper is concerned with an optimal control problem for the phase-fieldtransition system with state constraint and obstacle. After showing the relationshipbetween the control problem and its approximation, we derive Pontryagin’smaximum principle for an optimal control of our original problem by using one of theapproximate problems.

MSC: 49K20, 49J20, 74N25.

1 Introduction

We consider the optimal control of solid-liquid phase transitions:

{ u t γ Δ u + δ I [ 1 , 1 ] ( u ) + φ ( u ) λ ( u ) v 0 in  Q = Ω × ( 0 , T ) , v t + ( λ ( u ) ) t k Δ v = B w in  Q = Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) in  Ω , u ν = v ν = 0 on  Σ = Ω × ( 0 , T ) ,
(1.1)

and

F(u)S,

where Ω is a bounded domain in R N (1N3)with a smooth boundary Ω, φ(u)= u 3 u,δ,γ,k>0denote given parameters, I [ 1 , 1 ] (u) is the subdifferential of the indicator function I [ 1 , 1 ] (u) on the closed interval [1,1], Bw is a givenforcing term on Q, ν is the outward normal derivative on Ω and u 0 , v 0 aregiven initial datums.

System (1.1) is a simplified model for a class of solid-liquid phase change problems. Inthe context of solid-liquid phase transitions, v and u represent theabsolute temperature and the order parameter which indicates the physical situation ofthe system, respectively. Therefore it is natural to assume that the range of uis bounded, say the closed interval [1,1] in this paper, and I [ 1 , 1 ] (u) denoting the range of the order parameteru is assumed to be a compact interval [1,1], u(t,x)1and u(t,x)1 mean,respectively, that the physical situation at (t,x) is of pure solid and pure liquid, while1<u(t,x)<1 means that thephysical situation at (t,x) is mushy.

A great deal of research has been done on the phase-field transition system, for whichwe refer to the book by Temam [1] and thereferences therein. Without the term I [ 1 , 1 ] (u), system (1.1) is the standard phase field modelwhich was studied in [2, 3].One of the most important characteristics of our model is the nonlinear term I [ 1 , 1 ] (u) (obstacle) which allows the coexistence of purephases in the dynamical phase transition process. The existence and uniqueness ofsolution for the phase field model with obstacle were discussed in [48]. In particular, the asymptotic behaviorof solutions to the non-isothermal phase-field transition system with obstacle wasconsidered in [6] and [7]. Recently, the Caginalp phase-field system with coupleddynamic boundary conditions, including the singular potentials, was presented in[9] and [10].

Throughout this paper, the Hilbert space H= L 2 (Ω) is equipped with the usualinner product (,) and the norm | | 2 .Define a closed subspace H 0 ofH by H 0 ={zH; Ω zdx=0}. We put V 0 =V H 0 with v V 0 = | v | L 2 ( Ω ) v 2 ,where V={v H 1 (Ω), v ν =0}. If we identify H and H 0 with their dual space, then we have VH V and V 0 H 0 V 0 ,where V 0 is the duality space of V 0 .

Throughout the paper, we suppose that the following assumptions hold.

Let U be a real Hilbert space, B:UH be a linear continuousoperator. Assuming that Z is a Banach space with the dual Z strictly convex, let SZ be aclosed convex subset with finite co-dimensionality.

(H1) F: L 2 (0,T;H)Z is in the class of C 1 .

(H2) g:[0,T]×H R + is measurable in t and for every σ>0, thereexists L σ >0independent of t such that g(0,u) L (0,T) and

| g ( t , y ) g ( t , z ) | L σ | y z | 2 for any t[0,T] with  | y | 2 + | z | 2 λ.

(H3) h:U R ¯ is lower semicontinuousand convex with the following growth property:

h(w) c 1 | w | U 2 + c 2 for any wU with  c 1 >0 and  c 2 R.

(H4) λ C 2 and there exists a constant κ>0 such that λ (s)κ forany sR.

We consider the following optimal control problem:

where

L(w)= 0 t [ g ( t , u ( t ) ) + h ( w ( t ) ) ] dt

and

(u,v) is the solution of (1.1) corresponding to w,F(u)S.

For any

(u,v,w)Y×Y× L 2 (0,T;U)

satisfying (1.1) is called a feasible pair, where Y= H 2 , 1 (Q)C(0,T;V).

The first question regarding problem (P) is if there is an admissible solution,i.e., if the set A a d is nonempty. Taking into account [11] similarly,we may assume in the sequel that for (u,v,w)Y×Y× L 2 (0,T;U), problem (P)admits at least one admissible solution.

Optimal control problems of the phase transition system have been studied by severalauthors (for instance, see [1216]). In particular, let λ(u)= l 2 and δ=0 in(1.1), the optimal boundary controls for a phase field model and the state-constrainedoptimal control for the phase-field transition system were considered in [13] and [15],respectively. In [16], based on the energyestimates and the compact method, Ryu and Yagi considered the optimal control problemsof the adsorbate-induced phase transition model. It is noted that the optimal controlwithout state constraint or without obstacle of the phase field model was discussed in[14, 1720].

To the best of our knowledge, there are few papers concerned with the optimal controlproblems for the phase-field with obstacle although it is natural to have the obstaclein the solid-liquid phase transitions and related physics models, since the obstacle I [ 1 , 1 ] (u) brings the essential difficulty in gettingPontryagin’s maximum principle for corresponding models.

We state the maximum principle as follows.

Theorem 1.1 Suppose that (H1), (H2),(H3) and (H4) hold.Let( u , v , w )be optimal forproblem (P), then there exists a tetrad( μ 0 ,p,q, ζ 0 )R× L 2 (0,T;V) L (0,T;H)× L 2 (0,T;V) L (0,T;H)× Z with( μ 0 , ζ 0 )0and ameasureη L ( Q ) suchthat

{ p t γ Δ p + η + ( 3 ( u ) 2 1 ) p λ ( u ) p v λ ( u ) q [ F ( u ) ] ζ 0 μ 0 g ( t , u ) , q t + λ ( u ) p t k Δ q + λ ( u ) p u t = 0 , p ( T ) = 0 , q ( T ) = 0
(1.2)

and

{ B q ( t ) μ 0 h ( w ( t ) ) , ζ 0 , s F ( u ) Z , Z 0 s S .
(1.3)

Moreover, if F ( u )is injective,then( μ 0 ,p,q)0.

The rest of this paper is organized as follows. In Section 2, we provide existenceresults and a priori estimates in the form that is required to obtainPontryagin’s maximum principle for problem (P). Besides the existence ofan optimal control in problem ( P ε ),necessary optimality conditions for this problem and for problem (P) are provedin Section 3.

2 The approximation problem

This section is to show the existence of the optimal control of the approximationproblem corresponding to the phase transition system. To this end, we first show sometechnical lemmas, which are presented below for the sake of completeness.

In order to approximate I [ 1 , 1 ] (), we define a nondecreasing function β ε [21] on R by

β ε (r)=sign(r) 0 | r | min { 1 ε , [ s 1 ] + ε 2 } dsrR,

where [ ] + denotes the positive part of functions. Then β ε C 1 , ( β ε ) W 1 , (R) and

0 ( β ε ) (r) 1 ε , | ( β ε ) ( r ) | 1 ε ( [ r 1 ] + + [ 1 r ] + ) 1 2 for any rR.
(2.1)

We fix a primitive β ˆ ε of β ε such that

β ˆ ε (0)=0and β ˆ ε (r)0for any rR.
(2.2)

Without loss of generality, we may assume δ=1,therefore, the approximation of (1.1) is

{ u t γ Δ u + β ε ( u ) + φ ( u ) λ ( u ) v = 0 in  Q = Ω × ( 0 , T ) , v t + λ ( u ) u t k Δ v = B w in  Q = Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) in  Ω , u ν = v ν = 0 on  Σ = Ω × ( 0 , T ) .
(2.3)

Lemma 2.1 Suppose that β ε ()satisfies (2.1)-(2.2), w n L 2 (0,T;U), w n w ˜ weaklyin L 2 (0,T;U)and( u ˜ , v ˜ ), ( u n , v n )are the solutionsof (2.3) corresponding to w ˜ and w n ,respectively. Then there exists a subsequenceof( u n , v n ), still denoted byitself, such that

( u n , v n ) ( u ˜ , v ˜ ) weakly in Y 2 , β ε ( u n ) η ˜ weakly in L 2 ( 0 , T ; H ) as n .
(2.4)

Proof Replacing (u,v) and w by ( u n , v n ) and w n in (2.3),respectively, we obtain

{ u n , t γ Δ u n + β ε ( u n ) + φ ( u n ) λ ( u n ) v n = 0 in  Q = Ω × ( 0 , T ) , v n , t + λ ( u n ) u n , t k Δ v n = B w n in  Q = Ω × ( 0 , T ) , u n ( x , 0 ) = u 0 ( x ) , v n ( x , 0 ) = v 0 ( x ) in  Ω , u n ν = v n ν = 0 on  Σ = Ω × ( 0 , T ) .
(2.5)

Multiplying (2.5)1 and (2.5)2 by u n , t and v n ,respectively, integrating over Ω and adding the resulting equations, we end up with

d d t ( γ 2 | u n | 2 2 + | v n | 2 2 ) + | u n , t | 2 2 + k | v n | 2 2 + Ω ( β ε ( u n ) + φ ( u n ) ) u n , t d x Ω B w n v n d x .
(2.6)

Therefore, we conclude with the help of Young’s inequality and the properties of β ε that

d d t ( γ 2 | u n | 2 2 + | v n | 2 2 + 1 4 | u n | L 4 4 + | β ˆ ε | L 1 ) + k | v n | 2 2 + | u n , t | 2 2 1 4 | B w n | 2 2 + | v n | 2 2 + 1 2 | u n , t | 2 2 + 1 2 | u n | 2 2 1 4 | B w n | 2 2 + | v n | 2 2 + 1 2 | u n , t | 2 2 + | u n | L 4 4 + C .
(2.7)

Here and throughout the proof of Lemma 2.1, we shall denote by C severalpositive constants independent of n. Applying Gronwall’s inequality to(2.7), we derive

| u n | L ( 0 , T ; V L 4 ) + | v n | L ( 0 , T ; H ) + | v n | L 2 ( 0 , T ; V ) + | u n | H 1 ( 0 , T ; H ) + | β ˆ ε | L ( 0 , T ; L 1 ) C.
(2.8)

Now, testing (2.5)1 by u n ,we derive

1 2 d d t | u n | 2 2 + γ | u n | 2 2 + Ω ( β ε ) | u n | 2 + 3 u n 2 | u n | 2 d x = Ω λ ( u n ) v n u n d x γ 2 | u n | 2 2 + C ( | v n | L 6 6 + Ω | λ ( u n ) | 3 d x ) ,
(2.9)

which together with λ (t)κ,(2.2), (2.8), Nirenberg’s inequality and Gronwall’s inequality implies that

| u n | L ( 0 , T ; V ) + | u n | L 2 ( 0 , T ; H 2 ) C.
(2.10)

Next, multiplying (2.5)1 by β ε ,integrating over [0,T] and invoking Young’s inequality, wederive

d d t | β ˆ ε | L 1 + | β ε | 2 2 = Ω ( γ Δ u n φ ( u n ) + λ ( u n ) v n ) β ε ( u n ) d x 1 2 | β ε | 2 2 + C ( | u n | L 6 6 + | u n | 2 2 + γ 2 | Δ u n | 2 2 + Ω | λ ( u n ) | 3 d x + | v n | L 6 6 ) .
(2.11)

Thanks to (2.8), λ (t)κ andGronwall’s inequality, we derive

| β ˆ ε | L ( 0 , T ; L 1 ) + | β ε | L 2 ( 0 , T ; H ) C.
(2.12)

Inserting (2.10) and (2.12) into (2.5)1, we have

| u n , t | L 2 ( 0 , T ; H ) C.
(2.13)

Now, differentiating (2.5)1 with respect to t and multiplying theresult by u n , t ,then multiplying (2.5)2 by v t , adding theresulting equations and integrating over Ω leads to

d d t ( 1 2 | u n , t ( t ) | 2 2 + k 2 | v n ( t ) | 2 2 ) + γ | u n , t | 2 2 + Ω ( β ε ) u n , t 2 ( x , t ) d x + | v n , t | 2 2 | u n , t ( t ) | 2 2 + Ω κ v n ( t ) u n , t 2 ( t ) d x + Ω B w n v n , t d x | u n , t ( t ) | 2 2 + Ω | κ v n ( t ) u n , t 2 ( t ) | d x + 1 2 | v n , t ( t ) | 2 2 + 1 2 | B w n | 2 2 .
(2.14)

On the other hand, with the help of (2.8), (2.10), Hölder’s inequality andNirenberg’s inequality, we get

| 0 t Ω κ v n ( t ) u n , t 2 d x | | κ | 0 t | v n ( s ) | L 6 | u n , t ( s ) | L 12 5 2 d s | κ | ( 0 t | v n ( s ) | L 6 2 d s ) 1 2 ( 0 t | u n , t ( s ) | L 12 5 4 d s ) 1 2 C 1 ( 0 t | v n ( s ) | H 1 2 d s ) 1 2 × ( 0 t ( | u n , t ( s ) | | u n , t ( s ) | 3 + | u n , t ( s ) | 4 ) d s ) 1 2 C 2 sup 0 s t | u n , t ( s ) | × ( 0 t ( | u n , t ( s ) | | u n , t ( s ) | + | u n , t ( s ) | 2 ) d s ) 1 2 C 3 sup 0 s t | u n , t ( s ) | [ 1 + ( 0 t | u n , t ( s ) | 2 d s ) 1 4 ] μ sup 0 s t | u n , t ( s ) | 2 2 + μ 0 t | u n , t ( s ) | 2 d s + C 4 ,
(2.15)

where μ is a small positive constant and C i (i=1,2,3,4)are independent of n. Inserting (2.15) into (2.14), we derive

1 2 | u n , t ( t ) | 2 2 + k 2 | v n ( t ) | 2 2 + ( γ μ ) | u n , t ( t ) | 2 2 + 0 t Ω ( β ε ) ( u n ( t ) ) u n , t 2 ( t ) d x d s 0 t | u n , s ( s ) | 2 2 d s + μ sup 0 s t | u n , t ( s ) | 2 2 + 1 2 0 t | B w n ( s ) | 2 2 d s + C .
(2.16)

Taking the supremum with respect to t in (2.16), choosingμ>0sufficiently small and applying Gronwall’s inequality, we end up with

sup 0 s t | u n , t ( t ) | 2 + sup 0 s t | v n ( t ) | 2 + | u n , t ( t ) | 2 2 C T ,
(2.17)

which combined with (2.13) implies that λ ( u n ) u n , t L 2 (0,T;H). Therefore, employing thestandard parabolic theory to (2.5)2 leads to

| v n , t | L 2 ( 0 , T ; H ) + | v n | L ( 0 , T ; V ) + | v n | L 2 ( 0 , T ; H 2 ) C.
(2.18)

Now we may combine the estimates (2.8), (2.10), (2.12), (2.13) and (2.18) to concludethe results. This completes the proof. □

Lemma 2.2 Suppose that β ε ()satisfies (2.1)-(2.2),let w ε L 2 (0,T;U)with w ε w weaklyin L 2 (0,T;U)asε0,( u ε , v ε )be the solution of(2.3) corresponding to w ε .Then, on some subsequence( u ε n , v ε n )of( u ε , v ε ), there exists aquad(u,v,η)Y×Y× L 2 (0,T;H)such that

η I [ 1 , 1 ] (u)a.e. L 2 (0,T;H),
(2.19)

while

( u ε n , v ε n )(u,v)weakly in ( L ( 0 , T ; V ) L 2 ( 0 , T ; H 2 ( Ω ) ) ) 2 ,
(2.20)
( u ε n , v ε n )(u,v)strongly in ( L 2 ( 0 , T ; V ) C ( 0 , T ; H ) ) 2 ,
(2.21)
( u ε n , v ε n ) ( u , v ) weakly in ( L 2 ( 0 , T ; H ) ) 2 ,
(2.22)
β ε ( u ε n )ηweakly in L 2 (0,T;H)
(2.23)

as ε n 0and(u,v,η)is a solution of(1.1) satisfying the following estimates:

| u | Y 2 + | v | Y 2 + | η | L 2 ( Q T ) 2 C,
(2.24)

whereC>0isindependent of ε, n.

Proof Rewrite (2.3) as follows:

{ u ε , t γ Δ u ε + β ε ( u ε ) + φ ( u ε ) λ ( u ε ) v ε = 0 in  Q = Ω × ( 0 , T ) , v ε , t + λ ( u ε ) u ε , t k Δ v ε = B w ε in  Q = Ω × ( 0 , T ) , u ε ( x , 0 ) = u 0 ( x ) , v ε ( x , 0 ) = v 0 ( x ) in  Ω , u ε ν = v ε ν = 0 on  Σ = Ω × ( 0 , T ) .
(2.25)

Employing almost exactly the same arguments as in the proof of Lemma 2.1, weconclude the results (2.20)-(2.22). Furthermore, by a standard argument in[22], we get η I [ 1 , 1 ] (u) a.e. in L 2 (0,T;H). This completes theproof. □

Now, we assume that ( u , v , w ) is optimal for problem(P). For each ε>0, let( u ε , v ε , w ε ) be the solution to

{ u t γ Δ u + β ε ( u ) + φ ( u ) λ ( u ) v = 0 in  Q = Ω × ( 0 , T ) , v t + λ ( u ) u t k Δ v = B w in  Q = Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) in  Ω , u ν = v ν = 0 on  Σ = Ω × ( 0 , T ) .
(2.26)

It follows from Lemma 2.2 that

δ(ε) | u ε u | L 2 ( 0 , T ; H ) 0.
(2.27)

Now, the approximating optimal control problems ( P ε ) are asfollows:

Minimize  L ε (w)over w L 2 (0,T;U),

where L ε : L 2 (0,T;U)R, by

L ε (w)= 0 T [ g ε ( t , u ε ) + h ( w ) ] dt+ 1 2 | w w | L 2 ( 0 , T ; U ) 2 + 1 2 δ ( ε ) [ d S ( F ( u ε ) ) + δ ( ε ) ] 2
(2.28)

and (u,v) is the solution of (2.3). Here, d S (F(u)) denotes the distance of F(u) to S,

g ε (t,y)= R n g(t, P n yε Λ n τ) ρ n (τ)dτ
(2.29)

is the approximations of g[23], wheren=[ 1 ε ], ρ n is amollifier in R n , P n :H X n is the projection of H on X n , which isthe finite dimensional space generated by { e i } i = 1 n , { e i } i = 1 isan orthonormal basis in H, Λ n : R n X n is the operator defined by Λ n (τ)= i = 1 n τ i e i ,τ=( τ 1 , τ 2 ,, τ n ).

First of all, we show the existence of optimal solutions for ( P ε ).

Lemma 2.3 ( P ε )has at least one optimal solution.

Proof Let ε>0 befixed. It is clear that inf L ε (w)>. Let d ε =inf{ L ε (w):w L 2 (0,T;U)} and w n be aminimizing sequence such that

d ε L ε ( w n ) d ε + 1 n ,
(2.30)

which together with (H2), (H3) and (2.28) implies that w n isbounded in L 2 (0,T;U). Without loss ofgenerality, we may assume that w n w ˜ in L 2 (0,T;U). Let( u n , v n ) and ( u ˜ , v ˜ ) be the solutions of (2.3)corresponding to w n and w ˜ , respectively. It follows fromLemma 2.1 that on some subsequence of ( u n , v n ), still denoted by itself,

( u n , v n ) ( u ˜ , v ˜ ) weakly in  Y × Y and strongly in  ( C ( 0 , T ; H 1 ( Ω ) ) L 2 ( 0 , T ; H ) ) 2 .
(2.31)

With the help of (H2), (2.29) and (2.31), we also obtain

0 T | g ε ( t , u n ) g ε ( t , u ˜ ) | L 2 dtC 0 T | u n u ˜ | L 2 dt0as n.
(2.32)

On the other hand, due to (2.31) and (H1), we have that

lim n F( u n )=F( u ˜ )
(2.33)

and therefore

lim n 1 2 δ ( ε ) [ d S ( F ( u n ) ) + δ ( ε ) ] 2 = 1 2 δ ( ε ) [ d S ( F ( u ˜ ) ) + δ ( ε ) ] 2 .
(2.34)

Finally, (2.28) and (2.32)-(2.34) imply that ( u ˜ , v ˜ , w ˜ ) is the optimal pair forproblem ( P ε ).This concludes the proof of Lemma 2.3. □

Lemma 2.4 Let w ε beoptimal for problem ( P ε )and( u ε , v ε )be the solution of(2.3) corresponding to w ε .Then, on some subsequence ε n ,

( u ε n , v ε n ) ( u , v ) strongly in ( L 2 ( 0 , T ; V ) C ( 0 , T ; H ) ) 2 ,
(2.35)
w ε n w strongly in L 2 (0,T;U).
(2.36)

Proof Since w ε is asolution to ( P ε ),we have

L ε ( w ε ) 0 T [ g ε ( t , u ε ) + h ( w ) ] dt+ 1 2 δ ( ε ) [ d S ( F ( u ε ) ) + δ ( ε ) ] 2 ,
(2.37)

which together with (2.27) implies that

1 2 δ ( ε ) [ d S ( F ( u ε ) ) + δ ( ε ) ] 2 1 2 δ ( ε ) [ | F ( u ε ) F ( u ) | Z + δ ( ε ) ] 2 1 2 δ ( ε ) [ C | u ε u | L 2 ( 0 , T ; H ) + δ ( ε ) ] 2 ( 1 + C ) 2 2 δ ( ε ) 0 as  ε 0 ,
(2.38)

which combined with (2.37) implies that

lim sup ε 0 L( w ε ) 0 T [ g ( t , u ) + h ( w ) ] dt,
(2.39)

which implies that (2.39), that w ε isbounded in L 2 (0,T;U). Without loss generality,we may assume that w ε w ˜ weakly in L 2 (0,T;U), which together withLemma 2.2 implies that there exists a sequence of ε n suchthat

( u ε n , v ε n )( u ˜ , v ˜ )strongly in  ( L 2 ( 0 , T ; V ) C ( 0 , T ; H ) ) 2 .
(2.40)

On the other hand, (2.28) and (2.39) imply that

lim ε n 0 d S ( F ( u ε n ) ) =0,
(2.41)

and therefore

lim ε n 0 d S ( F ( u ˜ ) ) =0.
(2.42)

Thus, we conclude from (2.28), (2.40) and (2.42) that

lim inf ε n 0 L ε n ( w ε n ) 0 T [ g ( t , u ˜ ) + h ( w ˜ ) ] dt.
(2.43)

Finally, it follows from (2.39), (2.43) and Lemma 2.2 that

( u ε n , v ε n , w ε n ) ( u , v , w ) strongly in  ( L 2 ( 0 , T ; V ) C ( 0 , T ; H ) ) 2 × L 2 ( 0 , T ; U ) .
(2.44)

This completes the proof. □

3 The optimality condition for ( P ε ) and(P)

In the following we derive the optimality condition for problem (P) by showingthe relation between approximation problem ( P ε ) andproblem (P). We start this section with the necessary conditions for w ε to be optimal for ( P ε ).

Lemma 3.1 Suppose that (H1), (H2), (H3)and (H4) hold. Let w ε beoptimal for problem ( P ε )and( u ε , v ε )be the solution of(2.3) corresponding to w ε .Then there exists a tetrad( μ ε , p ε , q ε , ζ ε )R× L 2 (0,T;V)C(0,T;H)× L 2 (0,T;V)C(0,T;H)× Z suchthat

{ p ε , t γ Δ p ε + β ( u ε ) p ε + ( 3 u ε 2 1 ) p ε λ ( u ε ) p ε v ε λ ( u ε ) q ε = [ F ( u ) ] ζ 0 μ 0 g ( t , u ) , q ε , t + λ ( u ε ) p ε , t k Δ q ε + λ ( u ε ) p ε u ε , t = 0 , p ( T ) = 0 , q ( T ) = 0
(3.1)
B q ε = μ ε [ h ( w ε ) + w ε w ] a.e.t[0,T]
(3.2)

and

ζ ε d S ( F ( u ε ) ) ,
(3.3)

where d S isthe sub-differential of d S .

Proof Let w L 2 (0,T;U), w ε χ = w ε +χwand ( u ε χ , v ε χ ) be the solution of (2.3)corresponding to w ε χ .Then it is clear that

( u ε χ , v ε χ ) ( u ε , v ε )strongly in C(0,T;H) L 2 (0,T;V) as χ0.
(3.4)

Now, owing to the fact that w ε isoptimal for problem ( P ε ), wehave L ε ( w ε χ ) L ε ( w ε ) χ 0(for all χ>0),hence

0 μ ε 0 T g ε ( t , u ε ) , y ε + h ( w ε ) + w ε w , w U dt+ ( F ( u ε ) ) ζ ε , y ε ,
(3.5)

where ( y ε , y ¯ ε ) is the solution to

{ y ε , t γ Δ y ε + β ( u ε ) y ε + ( 3 u ε 2 1 ) y ε λ ( u ε ) y ε v ε λ ( u ε ) y ¯ ε = 0 , y ¯ ε , t + λ ( u ε ) y ε , t k Δ y ¯ ε + λ ( u ε ) y ε u ε , t = 0 , y ε ( 0 ) = 0 , y ¯ ε ( 0 ) = 0 .
(3.6)

Next, employing the same arguments as in the proof of [23], we conclude that

lim χ 0 1 χ 0 T g ε ( t , u ε χ ) g ε (t, u ε )dt= 0 T g ε ( t , u ε ) , y dt,
(3.7)
lim χ 0 1 χ 0 T [ ( h ( w ε χ ) h ( w ε ) ) + 1 2 ( | w ε χ w | U 2 | w ε w | U 2 ) ] d t = 0 T h ( w ε ) + w ε w , w d t
(3.8)

and

lim χ 0 ( [ d S F ( u ε χ ) + ε ] 2 [ d S F ( u ε ) + ε ] 2 ) = F ( u ε ) + ε ε ζ ε , F ( u ε ) y Z , Z ,
(3.9)

where g ε (t, u ε ) denotes the gradient of g ε to the second variable at u ε andh( w ε ) denotes the gradient ofh at w ε .Here, ζ ε d S (F( u ε )) and d S is the sub-differential of d S , whichimplies (3.3). Thanks to S being convex, closed and Z beingstrictly convex, we may also infer that

| ζ ε | Z ={ 1 if  F ( u ε ) S , 0 if  F ( u ε ) S .
(3.10)

Let

μ ε = δ ( ε ) δ ( ε ) + d S ( F ( u ε ) )
(3.11)

and ( p ε , q ε ) be the solution of (3.1).It follows from (3.1), (3.5) and (3.6) that

0 0 T B q ε , w + μ ε h ( w ε ) + w ε w , w U dt,
(3.12)

which implies (3.2). This completes the proof. □

The proof of Theorem 1.1 By using the properties of α ε and β ε and Lemma 2.2, we have that, on a sequence of ε still denoted byε,

( u ε , v ε )(u,v)weakly in  ( L ( 0 , T ; V ) L 2 ( 0 , T ; H 2 ) ) 2 ,
(3.13)
( u ε , v ε )(u,v)strongly in  ( C ( 0 , T ; H ) L 2 ( 0 , T ; V ) ) 2 ,
(3.14)
( u ε , v ε ) ( u , v ) weakly in  ( L 2 ( 0 , T ; H ) ) 2 ,
(3.15)
β ( u ε ) y ε ηweakly star in  ( L ( Q T ) ) .
(3.16)

It follows from (3.10) and (3.11) that

1 μ ε + | ζ ε | Z 2for any ε>0.
(3.17)

Therefore, there exist generalized subsequences of μ ε and ζ ε such that

μ ε μ 0 as ε0
(3.18)

and

ζ ε ζ 0 weakly star in  Z  as ε0.
(3.19)

Using Lemma 2.2, we may pass to the limit in (3.2) and derive(1.3)1.

On the other hand, due to Lemma 2.2 and the same argument as in [23], we can conclude that

g ε (t, u ε )ρ(t)weakly in  L 2 (0,T;H) as ε0,
(3.20)

where ρ(t)g(t, u ) for all mostt(0,T). Thanks to (H1) andLemma 2.2, we also infer

[ F ( u ε ) ] ζ ε [ F ( u ) ] ζ 0 weakly in  L 2 (0,T;H) as ε0.
(3.21)

Now we claim that

( 3 u ε 2 1 ) p ε ( 3 ( u ) 2 1 ) pweakly star in  L 2 ( 0 , T ; V )  as ε0.
(3.22)

Indeed, let w L 2 (0,T;V) and ν=max{ | p | L 2 ( 0 , T ; V ) +1, | u | L 2 ( 0 , T ; V ) +1, | w | L 2 ( 0 , T ; V ) }, then we derive

0 T | ( 3 u ε 2 1 ) p ε ( 3 ( u ) 2 1 ) p , w | d t 0 T | 3 ( u ε 2 ( u ) 2 ) p + [ 3 ( u ε ) 2 1 ] ( p ε p ) , w | d t 0 T | 3 ( u ε + u ) ( u ε u ) p ε + [ 3 ( u ε ) 2 1 ] ( p ε p ) , w | d t 2 ( ν 6 + 1 ) [ max 0 s T | u ε u | 2 2 + 0 T | p ε p | 2 2 d t ] 0 as  ε 0 .
(3.23)

With the help of (3.13), (3.16), (3.18)-(3.23), we can pass to the limit in (3.1) toderive that (p,q) ( L 2 ( 0 , T ; V ) C ( 0 , T ; H ) ) 2 andsatisfies (1.2). On the other hand, observing that ζ ε d S (F( u ε )), we derive

ζ ε , w F ( u ) Z , Z ζ ε , F ( u ) F ( u ε ) Z , Z ,
(3.24)

which together with (3.19) and Lemma 2.2 implies (1.3)2 (the secondinequality of (1.3)).

Finally, we are in a position to prove that ( μ 0 , ζ 0 )0. To this end,we suppose that μ 0 =0.It follows from (3.17), (3.19) and (3.24) that there exist ε 1 >0and δ>0 suchthat

δ | ζ ε | Z 2for any ε< ε 1
(3.25)

and

ζ ε , w F ( u ) Z , Z ζ ε , F ( u ) F ( u ε ) Z , Z 0uniformly in wS.
(3.26)

Since SX isa closed convex subset with finite co-dimensionality, so is SF( u ), which together with (3.25)and (3.26) implies that ( μ 0 , ζ 0 )0 [24].

Assuming that F ( u ) is injective and( μ 0 ,p,q)=0, thanks to (1.2), wederive ( F ( u ) ) ζ 0 =0,which yields ζ 0 =0and ( μ 0 , ζ 0 )=0. This is a contradictionwith ( μ 0 , ζ 0 )0. Thus, if F ( u ) is injective, then( μ 0 ,p)0. We completethe proof. □

References

  1. 1.

    Temam R Applied Mathematical Sciences. In Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd edition. Springer, New York; 1997.

  2. 2.

    Elliott C, Zheng S: Global existence and stability of solutions to the phase field equations. Intern. Ser. Numer. Math. 95. In Free Boundary Problems. Birkhäuser, Basel; 1990:48-58.

  3. 3.

    Penrose O, Fife PC: Thermodynamically consistent models of phase-field type for the kinetics of phasetransitions. Physica D 1990, 43: 44-62. 10.1016/0167-2789(90)90015-H

  4. 4.

    Damlamian A, Kenmochi N, Sato N: Phase field equations with constraints. Gakuto. Inter. Ser. Math. Sci. Appl. In Nonlinear Mathematical Problems in Industry. Gakkotosho, Tokyo; 1993:391-404.

  5. 5.

    Damlamian AL, Kenmochi N: Evolution equations associated with non-isothermal phase separation:subdifferential approach. Ann. Mat. Pura Appl. 1999, 4: 167-190.

  6. 6.

    Ito A, Kenmochi N: Asymptotic behaviour of solutions to phase field models with constraints. Funkc. Ekvacioj 1996, 39: 123-142.

  7. 7.

    Ito A, Kenmochi N: Asymptotic behaviour of solutions to non-isothermal phase separation model withconstraint in one-dimensional space. J. Math. Soc. Jpn. 1998, 50: 491-519. 10.2969/jmsj/05020491

  8. 8.

    Kenmochi N: Systems of nonlinear PDEs arising from dynamical phase transitions. Lecture Notes in Math. 1584. In Phase Transitions and Hysteresis. Edited by: Visintin A. Springer, Berlin; 1994:39-86.

  9. 9.

    Conti M, Gatti S, Miranville A: Asymptotic behavior of the Caginalp phase-field system with coupled dynamicboundary conditions. Discrete Contin. Dyn. Syst., Ser. S 2012, 5: 485-505.

  10. 10.

    Grasselli M, Miranville A, Schimperna G: The Caginalp phase-field system with coupled dynamic boundary conditions andsingular potentials. Discrete Contin. Dyn. Syst. 2010, 28: 67-98.

  11. 11.

    Barbu V: Local controllability of the phase field system. Nonlinear Anal., Theory Methods Appl. 2002, 50: 363-372. 10.1016/S0362-546X(01)00767-2

  12. 12.

    Colli P, Gilardi G, Podio-Guidugli P, Sprekels J: Distributed optimal control of a nonstandard system of phase field equations. Contin. Mech. Thermodyn. 2012, 24: 437-459. 10.1007/s00161-011-0215-8

  13. 13.

    Chen Z: Optimal boundary controls for a phase field model. IMA J. Math. Control Inf. 1993, 10: 157-176. 10.1093/imamci/10.2.157

  14. 14.

    Lefter C, Sprekels J: Optimal boundary control of a phase field system modeling non-isothermal phasetransitions. Adv. Math. Sci. Appl. 2007, 17: 181-194.

  15. 15.

    Moroşanu C, Wang G: State-constrained optimal control for the phase-field transition system. Numer. Funct. Anal. Optim. 2007, 28: 379-403. 10.1080/01630560701250119

  16. 16.

    Ryu S-U, Yagi A: Optimal control for an adsorbate–induced phase transition model. Appl. Math. Comput. 2005, 171: 420-432. 10.1016/j.amc.2005.01.044

  17. 17.

    Benincasa T, Favini A, Moroşanu C: A product formula approach to a nonhomogeneous boundary optimal control problemgoverned by nonlinear phase-field transition system part II: Lie-Trotter productformula. J. Optim. Theory Appl. 2011, 148: 31-45. 10.1007/s10957-010-9743-9

  18. 18.

    Horn, W, Sokolowskiy, J, Sprekels, J: Control problems with state constraints forthe Penrose-Fife phase-field model. Weierstrass-Inst. für Angewandte Analysisund Stochastik (1995)

  19. 19.

    Wang Q, Nakagiri S: Research Institute for Mathematical Sciences Kokyuüroku. 2000, 1128: 172-180.

  20. 20.

    Zheng S: Global existence for a thermodynamically consistent model of phase field type. Differ. Integral Equ. 1992, 5: 241-253.

  21. 21.

    Ohtsuka T, Shirakawa K, Yamazaki N: Optimal control problem for Allen-Cahn type equation associated with totalvariation energy. Discrete Contin. Dyn. Syst. 2012, 5: 159-181.

  22. 22.

    Barbu V: Analysis and control of nonlinear infinite dimensional systems. In Mathematics in Science and Engineering Edited by: Ames WF. 1993. Georgia Institute of Technology

  23. 23.

    Barbu V: Optimal Control of Variational Inequalities. Pitman Research Notes in Mathematics, Boston; 1984.

  24. 24.

    Lions JL: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin; 1971.

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Correspondence to Hao Liu.

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The authors declare that they have no competing interests.

Authors’ contributions

JZ carried out the optimal control problem for the phase-field transition system withstate constraint and obstacle and drafted the manuscript. HL and JL participated in thedesign of the study and examined the results carefully. All authors read and approvedthe final manuscript.

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Keywords

  • phase-field transition system
  • optimal control
  • Pontryagin’s maximum principle