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

Jiashan Zheng; Junjun Liu; Hao Liu

doi:10.1186/1687-2770-2013-234

Impact Factor 0.819

Research

Open Access

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

Jiashan Zheng¹,
Junjun Liu¹ and
Hao Liu²Email author

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 system optimal control Pontryagin’s maximum principle

1 Introduction

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

{\begin{matrix} u_{t} - γ Δ u + δ \partial I_{[- 1, 1]} (u) + φ (u) - λ^{'} (u) v ∋ 0 in Q = Ω \times (0, T), \\ v_{t} + {(λ (u))}_{t} - k Δ v = B w in Q = Ω \times (0, T), \\ u (x, 0) = u_{0} (x), v (x, 0) = v_{0} (x) in Ω, \\ \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = 0 on Σ = \partial Ω \times (0, T), \end{matrix}

(1.1)

and

F (u) \subset S,

where Ω is a bounded domain in $R^{N}$ ( $1 \leq N \leq 3$ )with a smooth boundary ∂ Ω, $φ (u) = u^{3} - u$ , $δ, γ, k > 0$ denote given parameters, $\partial I_{[- 1, 1]} (u)$ is the subdifferential of the indicator function $I_{[- 1, 1]} (u)$ on the closed interval $[- 1, 1]$ , Bw is a givenforcing term on Q, $\frac{\partial}{\partial ν}$ 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 $\partial I_{[- 1, 1]} (u)$ denoting the range of the order parameteru is assumed to be a compact interval $[- 1, 1]$ , $u (t, x) \equiv - 1$ and $u (t, x) \equiv 1$ 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 $\partial I_{[- 1, 1]} (u)$ , 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 $\partial I_{[- 1, 1]} (u)$ (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 [4–8]. 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 $(\cdot, \cdot)$ and the norm ${| \cdot |}_{2}$ .Define a closed subspace $H_{0}$ ofH by $H_{0} = {z \in H; \int_{Ω} z d x = 0}$ . We put $V_{0} = V \cap H_{0}$ with ${∥ v ∥}_{V_{0}} = {| \nabla v |}_{L^{2} (Ω)} \equiv {∥ v ∥}_{2}$ ,where $V = {v \in H^{1} (Ω), \frac{\partial v}{\partial ν} = 0}$ . If we identify $H^{*}$ and $H_{0}^{*}$ with their dual space, then we have $V \subset H \subset V^{*}$ and $V_{0} \subset H_{0} \subset V_{0}^{*}$ ,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, $B : U \to H$ be a linear continuousoperator. Assuming that Z is a Banach space with the dual $Z^{*}$ strictly convex, let $S \subset Z$ be aclosed convex subset with finite co-dimensionality.

(H₁) $F : L^{2} (0, T; H) \to Z$ is in the class of $C^{1}$ .

(H₂)

g : [0, T] \times H \to R^{+}

is measurable in t and for every

σ > 0

, thereexists

L_{σ} > 0

independent of t such that

g (0, u) \in L^{\infty} (0, T)

and

| g (t, y) - g (t, z) | \leq L_{σ} {| y - z |}_{2} for any t \in [0, T] with {| y |}_{2} + {| z |}_{2} \leq λ .

(H₃)

h : U \to \bar{R}

is lower semicontinuousand convex with the following growth property:

h (w) \geq c_{1} {| w |}_{U}^{2} + c_{2} for any w \in U with c_{1} > 0 and c_{2} \in R .

(H₄) $λ \in C^{2}$ and there exists a constant $κ > 0$ such that $λ^{″} (s) \leq κ$ forany $s \in R$ .

We consider the following optimal control problem:

where

L (w) = \int_{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) \subset S .

For any

(u, v, w) \in Y \times Y \times L^{2} (0, T; U)

satisfying (1.1) is called a feasible pair, where $Y = H^{2, 1} (Q) \cap C (0, T; V)$ .

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 $(u, v, w) \in Y \times Y \times L^{2} (0, T; U)$ , problem (P)admits at least one admissible solution.

Optimal control problems of the phase transition system have been studied by severalauthors (for instance, see [12–16]). In particular, let $λ (u) = \frac{l}{2}$ 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, 17–20].

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 $\partial I_{[- 1, 1]} (u)$ 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}) \in R \times L^{2} (0, T; V) \cap L^{\infty} (0, T; H) \times L^{2} (0, T; V) \cap L^{\infty} (0, T; H) \times Z^{*}

with

(μ_{0}, ζ_{0}) \neq 0

and ameasure

η \in L^{\infty} {(Q)}^{*}

suchthat

{\begin{matrix} - p_{t} - γ Δ p + η + (3 {(u^{*})}^{2} - 1) p - λ^{″} (u^{*}) p v^{*} - λ^{'} (u^{*}) q \\ \in - {[\partial F (u^{*})]}^{*} ζ_{0} - μ_{0} \partial g (t, u^{*}), \\ - q_{t} + λ^{'} (u^{*}) p_{t} - k Δ q + λ^{″} (u^{*}) p u_{t}^{*} = 0, \\ p (T) = 0, q (T) = 0 \end{matrix}

(1.2)

and

{\begin{matrix} B^{*} q (t) \in μ_{0} \partial h (w^{*} (t)), \\ {〈 ζ_{0}, s - \partial F (u^{*}) 〉}_{Z^{*}, Z} \leq 0 \forall s \in S . \end{matrix}

(1.3)

Moreover, if $F^{'} (u^{*})$ is injective,then $(μ_{0}, p, q) \neq 0$ .

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

\partial I_{[- 1, 1]} (\cdot)

, we define a nondecreasing function

β^{ε}

[21] on R by

β^{ε} (r) = sign (r) \int_{0}^{| r |} min {\frac{1}{ε}, \frac{{[s - 1]}^{+}}{ε^{2}}} d s \forall r \in R,

where

{[\cdot]}^{+}

denotes the positive part of functions. Then

β^{ε} \in C^{1}

,

{(β^{ε})}^{'} \in W^{1, \infty} (R)

and

0 \leq {(β^{ε})}^{'} (r) \leq \frac{1}{ε}, | (β^{ε}) (r) | \geq \frac{1}{ε} ({[r - 1]}^{+} + {[- 1 - r]}^{+}) - \frac{1}{2} for any r \in R .

(2.1)

We fix a primitive

{\hat{β}}^{ε}

of

β^{ε}

such that

{\hat{β}}^{ε} (0) = 0 and {\hat{β}}^{ε} (r) \geq 0 for any r \in R .

(2.2)

Without loss of generality, we may assume

δ = 1

,therefore, the approximation of (1.1) is

{\begin{matrix} u_{t} - γ Δ u + β^{ε} (u) + φ (u) - λ^{'} (u) v = 0 in Q = Ω \times (0, T), \\ v_{t} + λ^{'} (u) u_{t} - k Δ v = B w in Q = Ω \times (0, T), \\ u (x, 0) = u_{0} (x), v (x, 0) = v_{0} (x) in Ω, \\ \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = 0 on Σ = \partial Ω \times (0, T) . \end{matrix}

(2.3)

Lemma 2.1 Suppose that

β^{ε} (\cdot)

satisfies (2.1)-(2.2),

w_{n} \in L^{2} (0, T; U)

,

w_{n} \to \tilde{w}

weaklyin

L^{2} (0, T; U)

and

(\tilde{u}, \tilde{v})

,

(u_{n}, v_{n})

are the solutionsof (2.3) corresponding to

\tilde{w}

and

w_{n}

,respectively. Then there exists a subsequenceof

(u_{n}, v_{n})

, still denoted byitself, such that

\begin{aligned} (u_{n}, v_{n}) \to (\tilde{u}, \tilde{v}) weakly in Y^{2}, \\ β^{ε} (u_{n}) \to \tilde{η} weakly in L^{2} (0, T; H) as n \to \infty . \end{aligned}

(2.4)

Proof Replacing

(u, v)

and w by

(u_{n}, v_{n})

and

w_{n}

in (2.3),respectively, we obtain

{\begin{matrix} u_{n, t} - γ Δ u_{n} + β^{ε} (u_{n}) + φ (u_{n}) - λ^{'} (u_{n}) v_{n} = 0 in Q = Ω \times (0, T), \\ v_{n, t} + λ^{'} (u_{n}) u_{n, t} - k Δ v_{n} = B w_{n} in Q = Ω \times (0, T), \\ u_{n} (x, 0) = u_{0} (x), v_{n} (x, 0) = v_{0} (x) in Ω, \\ \frac{\partial u_{n}}{\partial ν} = \frac{\partial v_{n}}{\partial ν} = 0 on Σ = \partial Ω \times (0, T) . \end{matrix}

(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

\begin{matrix} \frac{d}{d t} (\frac{γ}{2} {| \nabla u_{n} |}_{2}^{2} + {| v_{n} |}_{2}^{2}) + {| u_{n, t} |}_{2}^{2} + k {| \nabla v_{n} |}_{2}^{2} + \int_{Ω} (β^{ε} (u_{n}) + φ (u_{n})) u_{n, t} d x \\ \leq - \int_{Ω} B w_{n} v_{n} d x . \end{matrix}

(2.6)

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

β^{ε}

that

\begin{matrix} \frac{d}{d t} (\frac{γ}{2} {| \nabla u_{n} |}_{2}^{2} + {| v_{n} |}_{2}^{2} + \frac{1}{4} {| u_{n} |}_{L^{4}}^{4} + {| {\hat{β}}^{ε} |}_{L^{1}}) + k {| \nabla v_{n} |}_{2}^{2} + {| u_{n, t} |}_{2}^{2} \\ \leq \frac{1}{4} {| B w_{n} |}_{2}^{2} + {| v_{n} |}_{2}^{2} + \frac{1}{2} {| u_{n, t} |}_{2}^{2} + \frac{1}{2} {| u_{n} |}_{2}^{2} \\ \leq \frac{1}{4} {| B w_{n} |}_{2}^{2} + {| v_{n} |}_{2}^{2} + \frac{1}{2} {| u_{n, t} |}_{2}^{2} + {| u_{n} |}_{L^{4}}^{4} + C . \end{matrix}

(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^{\infty} (0, T; V \cap L^{4})} + {| v_{n} |}_{L^{\infty} (0, T; H)} + {| v_{n} |}_{L^{2} (0, T; V)} + {| u_{n} |}_{H^{1} (0, T; H)} + {| {\hat{β}}^{ε} |}_{L^{\infty} (0, T; L^{1})} \leq C .

(2.8)

Now, testing (2.5)₁ by

- △ u_{n}

,we derive

\begin{matrix} \frac{1}{2} \frac{d}{d t} {| \nabla u_{n} |}_{2}^{2} + γ {| △ u_{n} |}_{2}^{2} + \int_{Ω} {(β^{ε})}^{'} {| \nabla u_{n} |}^{2} + 3 u_{n}^{2} {| \nabla u_{n} |}^{2} d x \\ = - \int_{Ω} λ^{'} (u_{n}) v_{n} △ u_{n} d x \\ \leq \frac{γ}{2} {| △ u_{n} |}_{2}^{2} + C ({| v_{n} |}_{L^{6}}^{6} + \int_{Ω} {| λ^{'} (u_{n}) |}^{3} d x), \end{matrix}

(2.9)

which together with

λ^{″} (t) \leq κ

,(2.2), (2.8), Nirenberg’s inequality and Gronwall’s inequality implies that

{| u_{n} |}_{L^{\infty} (0, T; V)} + {| u_{n} |}_{L^{2} (0, T; H^{2})} \leq C .

(2.10)

Next, multiplying (2.5)₁ by

β^{ε}

,integrating over

[0, T]

and invoking Young’s inequality, wederive

\begin{matrix} \frac{d}{d t} {| {\hat{β}}^{ε} |}_{L^{1}} + {| β^{ε} |}_{2}^{2} \\ = \int_{Ω} (γ Δ u_{n} - φ (u_{n}) + λ^{'} (u_{n}) v_{n}) β^{ε} (u_{n}) d x \\ \leq \frac{1}{2} {| β^{ε} |}_{2}^{2} + C ({| u_{n} |}_{L^{6}}^{6} + {| u_{n} |}_{2}^{2} + γ^{2} {| Δ u_{n} |}_{2}^{2} + \int_{Ω} {| λ^{'} (u_{n}) |}^{3} d x + {| v_{n} |}_{L^{6}}^{6}) . \end{matrix}

(2.11)

Thanks to (2.8),

λ^{″} (t) \leq κ

andGronwall’s inequality, we derive

{| {\hat{β}}^{ε} |}_{L^{\infty} (0, T; L^{1})} + {| β^{ε} |}_{L^{2} (0, T; H)} \leq C .

(2.12)

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

{| u_{n, t} |}_{L^{2} (0, T; H)} \leq 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

\begin{matrix} \frac{d}{d t} (\frac{1}{2} {| u_{n, t} (t) |}_{2}^{2} + \frac{k}{2} {| \nabla v_{n} (t) |}_{2}^{2}) + γ {| \nabla u_{n, t} |}_{2}^{2} + \int_{Ω} {(β^{ε})}^{'} u_{n, t}^{2} (x, t) d x + {| v_{n, t} |}_{2}^{2} \\ \leq {| u_{n, t} (t) |}_{2}^{2} + \int_{Ω} κ v_{n} (t) u_{n, t}^{2} (t) d x + \int_{Ω} B w_{n} v_{n, t} d x \\ \leq {| u_{n, t} (t) |}_{2}^{2} + \int_{Ω} | κ v_{n} (t) u_{n, t}^{2} (t) | d x + \frac{1}{2} {| v_{n, t} (t) |}_{2}^{2} + \frac{1}{2} {| B w_{n} |}_{2}^{2} . \end{matrix}

(2.14)

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

\begin{matrix} | \int_{0}^{t} \int_{Ω} κ v_{n} (t) u_{n, t}^{2} d x | \\ \leq | κ | \int_{0}^{t} {| v_{n} (s) |}_{L^{6}} {| u_{n, t} (s) |}_{L^{\frac{12}{5}}}^{2} d s \\ \leq | κ | {(\int_{0}^{t} {| v_{n} (s) |}_{L^{6}}^{2} d s)}^{\frac{1}{2}} {(\int_{0}^{t} {| u_{n, t} (s) |}_{L^{\frac{12}{5}}}^{4} d s)}^{\frac{1}{2}} \\ \leq C_{1} {(\int_{0}^{t} {| v_{n} (s) |}_{H^{1}}^{2} d s)}^{\frac{1}{2}} \times {(\int_{0}^{t} (| \nabla u_{n, t} (s) | {| u_{n, t} (s) |}^{3} + {| u_{n, t} (s) |}^{4}) d s)}^{\frac{1}{2}} \\ \leq C_{2} sup_{0 \leq s \leq t} | u_{n, t} (s) | \times {(\int_{0}^{t} (| \nabla u_{n, t} (s) | | u_{n, t} (s) | + {| u_{n, t} (s) |}^{2}) d s)}^{\frac{1}{2}} \\ \leq C_{3} sup_{0 \leq s \leq t} | u_{n, t} (s) | [1 + {(\int_{0}^{t} {| \nabla u_{n, t} (s) |}^{2} d s)}^{\frac{1}{4}}] \\ \leq μ sup_{0 \leq s \leq t} {| u_{n, t} (s) |}_{2}^{2} + μ \int_{0}^{t} {| \nabla u_{n, t} (s) |}^{2} d s + C_{4}, \end{matrix}

(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

\begin{matrix} \frac{1}{2} {| u_{n, t} (t) |}_{2}^{2} + \frac{k}{2} {| \nabla v_{n} (t) |}_{2}^{2} \\ + (γ - μ) {| \nabla u_{n, t} (t) |}_{2}^{2} + \int_{0}^{t} \int_{Ω} {(β^{ε})}^{'} (u_{n} (t)) u_{n, t}^{2} (t) d x d s \\ \leq \int_{0}^{t} {| u_{n, s} (s) |}_{2}^{2} d s + μ sup_{0 \leq s \leq t} {| u_{n, t} (s) |}_{2}^{2} + \frac{1}{2} \int_{0}^{t} {| B w_{n} (s) |}_{2}^{2} d s + C . \end{matrix}

(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 \leq s \leq t} {| u_{n, t} (t) |}_{2} + sup_{0 \leq s \leq t} {| \nabla v_{n} (t) |}_{2} + {| \nabla u_{n, t} (t) |}_{2}^{2} \leq C_{T},

(2.17)

which combined with (2.13) implies that

λ^{'} (u_{n}) u_{n, t} \in 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^{\infty} (0, T; V)} + {| v_{n} |}_{L^{2} (0, T; H^{2})} \leq 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} (\cdot)

satisfies (2.1)-(2.2),let

w_{ε} \in L^{2} (0, T; U)

with

w_{ε} \to w^{*}

weaklyin

L^{2} (0, T; U)

as

ε \to 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) \in Y \times Y \times L^{2} (0, T; H)

such that

η \in \partial I_{[- 1, 1]} (u) a.e. L^{2} (0, T; H),

(2.19)

while

(u_{ε_{n}}, v_{ε_{n}}) \to (u, v) weakly in {(L^{\infty} (0, T; V) \cap L^{2} (0, T; H^{2} (Ω)))}^{2},

(2.20)

(u_{ε_{n}}, v_{ε_{n}}) \to (u, v) strongly in {(L^{2} (0, T; V) \cap C (0, T; H))}^{2},

(2.21)

(u_{ε_{n}}^{'}, v_{ε_{n}}^{'}) \to (u^{'}, v^{'}) weakly in {(L^{2} (0, T; H))}^{2},

(2.22)

β^{ε} (u_{ε_{n}}) \to η weakly in L^{2} (0, T; H)

(2.23)

as

ε_{n} \to 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} \leq C,

(2.24)

where $C > 0$ isindependent of ε, n.

Proof Rewrite (2.3) as follows:

{\begin{matrix} u_{ε, t} - γ Δ u_{ε} + β^{ε} (u_{ε}) + φ (u_{ε}) - λ^{'} (u_{ε}) v_{ε} = 0 in Q = Ω \times (0, T), \\ v_{ε, t} + λ^{'} (u_{ε}) u_{ε, t} - k Δ v_{ε} = B w_{ε} in Q = Ω \times (0, T), \\ u_{ε} (x, 0) = u_{0} (x), v_{ε} (x, 0) = v_{0} (x) in Ω, \\ \frac{\partial u_{ε}}{\partial ν} = \frac{\partial v_{ε}}{\partial ν} = 0 on Σ = \partial Ω \times (0, T) . \end{matrix}

(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 $η \in \partial I_{[- 1, 1]} (u)$ 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

{\begin{matrix} u_{t} - γ Δ u + β^{ε} (u) + φ (u) - λ^{'} (u) v = 0 in Q = Ω \times (0, T), \\ v_{t} + λ^{'} (u) u_{t} - k Δ v = B w^{*} in Q = Ω \times (0, T), \\ u (x, 0) = u_{0}^{*} (x), v (x, 0) = v_{0}^{*} (x) in Ω, \\ \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = 0 on Σ = \partial Ω \times (0, T) . \end{matrix}

(2.26)

It follows from Lemma 2.2 that

δ (ε) \equiv {| u_{ε}^{*} - u^{*} |}_{L^{2} (0, T; H)} \to 0 .

(2.27)

Now, the approximating optimal control problems (

P^{ε}

) are asfollows:

Minimize L_{ε} (w) over w \in L^{2} (0, T; U),

where

L_{ε} : L^{2} (0, T; U) \to R

, by

L_{ε} (w) = \int_{0}^{T} [g_{ε} (t, u_{ε}) + h (w)] d t + \frac{1}{2} {| w - w^{*} |}_{L^{2} (0, T; U)}^{2} + \frac{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) = \int_{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 $n = [\frac{1}{ε}]$ , $ρ_{n}$ is amollifier in $R^{n}$ , $P_{n} : H \to X_{n}$ is the projection of H on $X_{n}$ , which isthe finite dimensional space generated by ${e_{i}}_{i = 1}^{n}$ , ${e_{i}}_{i = 1}^{\infty}$ isan orthonormal basis in H, $Λ_{n} : R^{n} \to X_{n}$ is the operator defined by $Λ_{n} (τ) = \sum_{i = 1}^{n} τ_{i} e_{i}$ , $τ = (τ_{1}, τ_{2}, \dots, τ_{n})$ .

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) > - \infty

. Let

d_{ε} = inf {L_{ε} (w) : w \in L^{2} (0, T; U)}

and

w_{n}

be aminimizing sequence such that

d_{ε} \leq L_{ε} (w_{n}) \leq d_{ε} + \frac{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} \to \tilde{w}

in

L^{2} (0, T; U)

. Let

(u_{n}, v_{n})

and

(\tilde{u}, \tilde{v})

be the solutions of (2.3)corresponding to

w_{n}

and

\tilde{w}

, respectively. It follows fromLemma 2.1 that on some subsequence of

(u_{n}, v_{n})

, still denoted by itself,

\begin{array}{rcl} (u_{n}, v_{n}) \to (\tilde{u}, \tilde{v}) & weakly in Y \times Y and \\ strongly in {(C (0, T; H^{1} (Ω)) \cap L^{2} (0, T; H))}^{2} . \end{array}

(2.31)

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

\int_{0}^{T} {| g_{ε} (t, u_{n}) - g_{ε} (t, \tilde{u}) |}_{L^{2}} d t \leq C \int_{0}^{T} {| u_{n} - \tilde{u} |}_{L^{2}} d t \to 0 as n \to \infty .

(2.32)

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

lim_{n \to \infty} F (u_{n}) = F (\tilde{u})

(2.33)

and therefore

lim_{n \to \infty} \frac{1}{2 δ (ε)} {[d_{S} (F (u_{n})) + δ (ε)]}^{2} = \frac{1}{2 δ (ε)} {[d_{S} (F (\tilde{u})) + δ (ε)]}^{2} .

(2.34)

Finally, (2.28) and (2.32)-(2.34) imply that $(\tilde{u}, \tilde{v}, \tilde{w})$ 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}}) \to (u^{*}, v^{*}) strongly in {(L^{2} (0, T; V) \cap C (0, T; H))}^{2},

(2.35)

w_{ε_{n}} \to 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}) \leq \int_{0}^{T} [g_{ε} (t, u_{ε}^{*}) + h (w^{*})] d t + \frac{1}{2 δ (ε)} {[d_{S} (F (u_{ε}^{*})) + δ (ε)]}^{2},

(2.37)

which together with (2.27) implies that

\begin{array}{rcl} \frac{1}{2 δ (ε)} {[d_{S} (F (u_{ε}^{*})) + δ (ε)]}^{2} & \leq & \frac{1}{2 δ (ε)} {[{| F (u_{ε}^{*}) - F (u^{*}) |}_{Z} + δ (ε)]}^{2} \\ \leq & \frac{1}{2 δ (ε)} {[C {| u_{ε}^{*} - u^{*} |}_{L^{2} (0, T; H)} + δ (ε)]}^{2} \\ \leq & \frac{{(1 + C)}^{2}}{2} δ (ε) \to 0 as ε \to 0, \end{array}

(2.38)

which combined with (2.37) implies that

\underset{ε \to 0}{lim sup} L (w_{ε}) \leq \int_{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} \to \tilde{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}}) \to (\tilde{u}, \tilde{v}) strongly in {(L^{2} (0, T; V) \cap C (0, T; H))}^{2} .

(2.40)

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

lim_{ε_{n} \to 0} d_{S} (F (u_{ε_{n}})) = 0,

(2.41)

and therefore

lim_{ε_{n} \to 0} d_{S} (F (\tilde{u})) = 0 .

(2.42)

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

\underset{ε_{n} \to 0}{lim inf} L_{ε_{n}} (w_{ε_{n}}) \geq \int_{0}^{T} [g (t, \tilde{u}) + h (\tilde{w})] d t .

(2.43)

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

\begin{matrix} (u_{ε_{n}}, v_{ε_{n}}, w_{ε_{n}}) \to (u^{*}, v^{*}, w^{*}) \\ strongly in {(L^{2} (0, T; V) \cap C (0, T; H))}^{2} \times L^{2} (0, T; U) . \end{matrix}

(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}) \in R \times L^{2} (0, T; V) \cap C (0, T; H) \times L^{2} (0, T; V) \cap C (0, T; H) \times Z^{*}

suchthat

{\begin{matrix} - 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} \\ = - {[\partial F (u^{*})]}^{*} ζ_{0} - μ_{0} \partial 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 \end{matrix}

(3.1)

B^{*} q_{ε} = μ_{ε} [\nabla h (w_{ε}) + w_{ε} - w^{*}] a.e. t \in [0, T]

(3.2)

and

ζ_{ε} \in \partial d_{S} (F (u_{ε})),

(3.3)

where $\partial d_{S}$ isthe sub-differential of $d_{S}$ .

Proof Let

w \in 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}) \to (u_{ε}, v_{ε}) strongly in C (0, T; H) \cap L^{2} (0, T; V) as χ \to 0 .

(3.4)

Now, owing to the fact that

w_{ε}

isoptimal for problem (

P_{ε}

), wehave

\frac{L_{ε} (w_{ε}^{χ}) - L_{ε} (w_{ε})}{χ} \geq 0

(for all

χ > 0

),hence

0 \leq μ_{ε} \int_{0}^{T} 〈 \nabla g_{ε} (t, u_{ε}), y_{ε} 〉 + {〈 \nabla h (w_{ε}) + w_{ε} - w^{*}, w 〉}_{U} d t + 〈 {(F^{'} (u_{ε}))}^{*} ζ_{ε}, y_{ε} 〉,

(3.5)

where

(y_{ε}, {\bar{y}}_{ε})

is the solution to

{\begin{matrix} 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}) {\bar{y}}_{ε} = 0, \\ {\bar{y}}_{ε, t} + λ^{'} (u_{ε}) y_{ε, t} - k Δ {\bar{y}}_{ε} + λ^{″} (u_{ε}) y_{ε} u_{ε, t} = 0, \\ y_{ε} (0) = 0, {\bar{y}}_{ε} (0) = 0 . \end{matrix}

(3.6)

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

lim_{χ \to 0} \frac{1}{χ} \int_{0}^{T} g_{ε} (t, u_{ε}^{χ}) - g_{ε} (t, u_{ε}) d t = \int_{0}^{T} 〈 \nabla g_{ε} (t, u_{ε}), y 〉 d t,

(3.7)

\begin{matrix} lim_{χ \to 0} \frac{1}{χ} \int_{0}^{T} [(h (w_{ε}^{χ}) - h (w_{ε})) + \frac{1}{2} ({| w_{ε}^{χ} - w^{*} |}_{U}^{2} - {| w_{ε} - w^{*} |}_{U}^{2})] d t \\ = \int_{0}^{T} 〈 \nabla h (w_{ε}) + w_{ε} - w^{*}, w 〉 d t \end{matrix}

(3.8)

and

lim_{χ \to 0} ({[d_{S} F (u_{ε}^{χ}) + ε]}^{2} - {[d_{S} F (u_{ε}) + ε]}^{2}) = \frac{F (u_{ε}) + ε}{ε} {〈 ζ_{ε}, F^{'} (u_{ε}) y 〉}_{Z^{*}, Z},

(3.9)

where

\nabla g_{ε} (t, u_{ε})

denotes the gradient of

g_{ε}

to the second variable at

u_{ε}

and

\nabla h (w_{ε})

denotes the gradient ofh at

w_{ε}

.Here,

ζ_{ε} \in \partial d_{S} (F (u_{ε}))

and

\partial 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^{*}} = {\begin{matrix} 1 & if F (u_{ε}) \notin S, \\ 0 & if F (u_{ε}) \in S . \end{matrix}

(3.10)

Let

μ_{ε} = \frac{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 \leq \int_{0}^{T} - 〈 B^{*} q_{ε}, w 〉 + μ_{ε} {〈 \nabla 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}) \to (u, v) weakly in {(L^{\infty} (0, T; V) \cap L^{2} (0, T; H^{2}))}^{2},

(3.13)

(u_{ε}, v_{ε}) \to (u, v) strongly in {(C (0, T; H) \cap L^{2} (0, T; V))}^{2},

(3.14)

(u_{ε}^{'}, v_{ε}^{'}) \to (u^{'}, v^{'}) weakly in {(L^{2} (0, T; H))}^{2},

(3.15)

β^{'} (u_{ε}) y_{ε} \to η weakly star in {(L^{\infty} (Q_{T}))}^{*} .

(3.16)

It follows from (3.10) and (3.11) that

1 \leq μ_{ε} + {| ζ_{ε} |}_{Z^{*}} \leq 2 for any ε > 0 .

(3.17)

Therefore, there exist generalized subsequences of

μ_{ε}

and

ζ_{ε}

such that

μ_{ε} \to μ_{0} as ε \to 0

(3.18)

and

ζ_{ε} \to ζ_{0} weakly star in Z^{*} as ε \to 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

\nabla g_{ε} (t, u_{ε}) \to ρ (t) weakly in L^{2} (0, T; H) as ε \to 0,

(3.20)

where

ρ (t) \in \partial g (t, u^{*})

for all most

t \in (0, T)

. Thanks to (H₁) andLemma 2.2, we also infer

{[F^{'} (u_{ε})]}^{*} ζ_{ε} \to {[F^{'} (u^{*})]}^{*} ζ_{0} weakly in L^{2} (0, T; H) as ε \to 0 .

(3.21)

Now we claim that

(3 u_{ε}^{2} - 1) p_{ε} \to (3 {(u^{*})}^{2} - 1) p weakly star in L^{2} (0, T; V^{*}) as ε \to 0 .

(3.22)

Indeed, let

w \in 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

\begin{matrix} \int_{0}^{T} | (3 u_{ε}^{2} - 1) p_{ε} - (3 {(u^{*})}^{2} - 1) p, w 〉 | d t \\ \leq \int_{0}^{T} | 〈 3 (u_{ε}^{2} - {(u^{*})}^{2}) p + [3 {(u_{ε})}^{2} - 1] (p_{ε} - p), w 〉 | d t \\ \leq \int_{0}^{T} | 〈 3 (u_{ε} + u^{*}) (u_{ε} - u^{*}) p_{ε} + [3 {(u_{ε})}^{2} - 1] (p_{ε} - p), w 〉 | d t \\ \leq 2 (ν^{6} + 1) [max_{0 \leq s \leq T} {| u_{ε} - u^{*} |}_{2}^{2} + \int_{0}^{T} {| p_{ε} - p |}_{2}^{2} d t] \to 0 as ε \to 0 . \end{matrix}

(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) \in {(L^{2} (0, T; V) \cap C (0, T; H))}^{2}

andsatisfies (1.2). On the other hand, observing that

ζ_{ε} \in \partial d_{S} (F (u_{ε}))

, we derive

{〈 ζ_{ε}, w - F (u^{*}) 〉}_{Z^{*}, Z} \leq {〈 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}) \neq 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 \leq {| ζ_{ε} |}_{Z^{*}} \leq 2 for any ε < ε_{1}

(3.25)

and

{〈 ζ_{ε}, w - F (u^{*}) 〉}_{Z^{*}, Z} \leq {〈 ζ_{ε}, F (u^{*}) - F (u_{ε}) 〉}_{Z^{*}, Z} \to 0 uniformly in w \in S .

(3.26)

Since $S \subset X$ isa closed convex subset with finite co-dimensionality, so is $S - F (u^{*})$ , which together with (3.25)and (3.26) implies that $(μ_{0}, ζ_{0}) \neq 0$ [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 $(μ_{0}, ζ_{0}) \neq 0$ . Thus, if $F^{'} (u^{*})$ is injective, then $(μ_{0}, p) \neq 0$ . We completethe proof. □

Declarations

Authors’ Affiliations

(1)

School of Mathematics, Beijing Institute of Technology

(2)

School of Science, University of Science and Technology Liaoning

References

Temam R Applied Mathematical Sciences. In Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd edition. Springer, New York; 1997.View ArticleGoogle Scholar
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.Google Scholar
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-HMathSciNetView ArticleMATHGoogle Scholar
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.Google Scholar
Damlamian AL, Kenmochi N: Evolution equations associated with non-isothermal phase separation:subdifferential approach. Ann. Mat. Pura Appl. 1999, 4: 167-190.MathSciNetView ArticleGoogle Scholar
Ito A, Kenmochi N: Asymptotic behaviour of solutions to phase field models with constraints. Funkc. Ekvacioj 1996, 39: 123-142.MathSciNetMATHGoogle Scholar
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/05020491MathSciNetView ArticleMATHGoogle Scholar
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.View ArticleGoogle Scholar
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.MathSciNetView ArticleMATHGoogle Scholar
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.MathSciNetView ArticleMATHGoogle Scholar
Barbu V: Local controllability of the phase field system. Nonlinear Anal., Theory Methods Appl. 2002, 50: 363-372. 10.1016/S0362-546X(01)00767-2MathSciNetView ArticleMATHGoogle Scholar
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/s00161-011-0215-8MathSciNetView ArticleMATHGoogle Scholar
Chen Z: Optimal boundary controls for a phase field model. IMA J. Math. Control Inf. 1993, 10: 157-176. 10.1093/imamci/10.2.157View ArticleMATHGoogle Scholar
Lefter C, Sprekels J: Optimal boundary control of a phase field system modeling non-isothermal phasetransitions. Adv. Math. Sci. Appl. 2007, 17: 181-194.MathSciNetMATHGoogle Scholar
Moroşanu C, Wang G: State-constrained optimal control for the phase-field transition system. Numer. Funct. Anal. Optim. 2007, 28: 379-403. 10.1080/01630560701250119MathSciNetView ArticleMATHGoogle Scholar
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.2005.01.044MathSciNetView ArticleMATHGoogle Scholar
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/s10957-010-9743-9MathSciNetView ArticleMATHGoogle Scholar
Horn, W, Sokolowskiy, J, Sprekels, J: Control problems with state constraints forthe Penrose-Fife phase-field model. Weierstrass-Inst. für Angewandte Analysisund Stochastik (1995)Google Scholar
Wang Q, Nakagiri S: Research Institute for Mathematical Sciences Kokyuüroku. 2000, 1128: 172-180.MathSciNetMATHGoogle Scholar
Zheng S: Global existence for a thermodynamically consistent model of phase field type. Differ. Integral Equ. 1992, 5: 241-253.MATHGoogle Scholar
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.MathSciNetView ArticleMATHGoogle Scholar
Barbu V: Analysis and control of nonlinear infinite dimensional systems. In Mathematics in Science and Engineering Edited by: Ames WF. 1993. Georgia Institute of TechnologyGoogle Scholar
Barbu V: Optimal Control of Variational Inequalities. Pitman Research Notes in Mathematics, Boston; 1984.MATHGoogle Scholar
Lions JL: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin; 1971.View ArticleMATHGoogle Scholar

Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permitsunrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

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

Abstract

Keywords

1 Introduction

2 The approximation problem

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

Declarations

Authors’ Affiliations

References

Copyright

3 The optimality condition for ( $P^{ε}$ ) and(P)