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

Jiashan Zheng; Junjun Liu; Hao Liu

doi:10.1186/1687-2770-2013-234

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, Beijing, China

(2)

School of Science, University of Science and Technology Liaoning, Anshan, 114051, China

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