Open Access

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

Boundary Value Problems20132013:234

https://doi.org/10.1186/1687-2770-2013-234

Received: 23 June 2013

Accepted: 25 September 2013

Published: 8 November 2013

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.

Keywords

phase-field transition systemoptimal controlPontryagin’s maximum principle

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 ( 1 N 3 )with a smooth boundary Ω, φ ( u ) = u 3 u , δ , γ , k > 0 denote 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 ) 1 and u ( t , x ) 1 mean,respectively, that the physical situation at ( t , x ) is of pure solid and pure liquid, while 1 < 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 = { z H ; Ω z d x = 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 V H 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 : U H be a linear continuousoperator. Assuming that Z is a Banach space with the dual Z strictly convex, let S Z 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 > 0 independent 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  w U  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 s R .

We consider the following optimal control problem:
where
L ( w ) = 0 t [ g ( t , u ( t ) ) + h ( w ( t ) ) ] d t
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 ) 0 and ameasure η 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 } d s r R ,
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  r R .
(2.1)
We fix a primitive β ˆ ε of β ε such that
β ˆ ε ( 0 ) = 0 and β ˆ ε ( r ) 0 for any  r R .
(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), choosing μ > 0 sufficiently 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 ) as ε 0 , ( 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 ) η weakly in L 2 ( 0 , T ; H )
(2.23)
as ε n 0 and ( 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)

where C > 0 isindependent 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 ) ] d t + 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 y ε Λ n τ ) ρ n ( τ ) d τ
(2.29)

is the approximations of g[23], where n = [ 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 d t C 0 T | u n u ˜ | L 2 d t 0 as  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 ) ] d t + 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 ) ] d t ,
(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 ˜ ) ] d t .
(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 + χ w and ( 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  χ 0 .
(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 d t + ( 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 ) d t = 0 T g ε ( t , u ε ) , y d t ,
(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 and h ( 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 d t ,
(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 η weakly star in  ( L ( Q T ) ) .
(3.16)
It follows from (3.10) and (3.11) that
1 μ ε + | ζ ε | Z 2 for any  ε > 0 .
(3.17)
Therefore, there exist generalized subsequences of μ ε and ζ ε such that
μ ε μ 0 as  ε 0
(3.18)
and
ζ ε ζ 0 weakly star in  Z  as  ε 0 .
(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  ε 0 ,
(3.20)
where ρ ( t ) g ( t , u ) for all most t ( 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  ε 0 .
(3.21)
Now we claim that
( 3 u ε 2 1 ) p ε ( 3 ( u ) 2 1 ) p weakly star in  L 2 ( 0 , T ; V )  as  ε 0 .
(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 > 0 and δ > 0 suchthat
δ | ζ ε | Z 2 for any  ε < ε 1
(3.25)
and
ζ ε , w F ( u ) Z , Z ζ ε , F ( u ) F ( u ε ) Z , Z 0 uniformly in  w S .
(3.26)

Since S X isa closed convex subset with finite co-dimensionality, so is S F ( 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 = 0 and ( 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. □

Declarations

Authors’ Affiliations

(1)
School of Mathematics, Beijing Institute of Technology
(2)
School of Science, University of Science and Technology Liaoning

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