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

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.

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

and

where Ω is a bounded domain in R^N(1\le N\le 3)with a smooth boundary ∂ Ω, \phi (u)=u^3-u,\delta ,\gamma ,k>0denote given parameters, \partial I_{[-1,1]}(u) is the subdifferential of the indicator functionI_{[-1,1]}(u) on the closed interval [-1,1], Bw is a givenforcing term on Q, \frac{\partial }{\partial \nu }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 -1and u(t,x)\equiv 1 mean,respectively, that the physical situation at (t,x) is of pure solid and pure liquid, while 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(\mathrm{\Omega }) 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 }_{\mathrm{\Omega }}zdx=0\}. We putV_0=V\cap H_0with {\parallel v\parallel }_{V_0}={|\mathrm{\nabla }v|}_{L^2(\mathrm{\Omega })}\equiv {\parallel v\parallel }_2,where V=\{v\in H^1(\mathrm{\Omega }),\frac{\partial v}{\partial \nu }=0\}. If we identifyH^{\ast }and H_0^{\ast }with their dual space, then we have V\subset H\subset V^{\ast }and V_0\subset H_0\subset V_0^{\ast },where V_0^{\ast }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 dualZ^{\ast }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 ofC^1.

(H₂) g:[0,T]\times H\to R^{+}is measurable in t and for every \sigma >0, thereexists L_{\sigma }>0independent of t such that g(0,u)\in L^{\mathrm{\infty }}(0,T) and

(H₃) h:U\to \overline{R} is lower semicontinuousand convex with the following growth property:

(H₄) \lambda \in C^2and there exists a constant \kappa >0 such that{\lambda }^{″}(s)\le \kappa forany s\in R.

We consider the following optimal control problem:

where

and

For any

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_{ad}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 \lambda (u)=\frac{l}{2}and \delta =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^{\ast },v^{\ast },w^{\ast })be optimal forproblem (P), then there exists a tetrad({\mu }_0,p,q,{\zeta }_0)\in R\times L^2(0,T;V)\cap L^{\mathrm{\infty }}(0,T;H)\times L^2(0,T;V)\cap L^{\mathrm{\infty }}(0,T;H)\times Z^{\ast }with({\mu }_0,{\zeta }_0)\ne 0and ameasure\eta \in L^{\mathrm{\infty }}{(Q)}^{\ast }suchthat

and

Moreover, ifF^{\prime }(u^{\ast })is injective,then({\mu }_0,p,q)\ne 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^{\epsilon }),necessary optimality conditions for this problem and for problem (P) are provedin Section 3.

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{\beta }^{\epsilon }[21] on R by

where {[\cdot ]}^{+}denotes the positive part of functions. Then {\beta }^{\epsilon }\in C^1,{({\beta }^{\epsilon })}^{\prime }\in W^{1,\mathrm{\infty }}(R) and

We fix a primitive {\hat{\beta }}^{\epsilon }of {\beta }^{\epsilon }such that

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

Lemma 2.1 Suppose that{\beta }^{\epsilon }(\cdot )satisfies (2.1)-(2.2),w_n\in L^2(0,T;U), w_n\to \tilde{w}weaklyinL^2(0,T;U)and(\tilde{u},\tilde{v}), (u_n,v_n)are the solutionsof (2.3) corresponding to\tilde{w}andw_n,respectively. Then there exists a subsequenceof(u_n,v_n), still denoted byitself, such that

Proof Replacing (u,v) and w by (u_n,v_n) and w_n in (2.3),respectively, we obtain

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

Therefore, we conclude with the help of Young’s inequality and the properties of{\beta }^{\epsilon }that

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

Now, testing (2.5)₁ by -\mathrm{△}{u}_n,we derive

which together with {\lambda }^{″}(t)\le \kappa,(2.2), (2.8), Nirenberg’s inequality and Gronwall’s inequality implies that

Next, multiplying (2.5)₁ by {\beta }^{\epsilon },integrating over [0,T] and invoking Young’s inequality, wederive

Thanks to (2.8), {\lambda }^{″}(t)\le \kappa andGronwall’s inequality, we derive

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

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

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

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

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

which combined with (2.13) implies that {\lambda }^{\prime }(u_n)u_{n,t}\in L^2(0,T;H). Therefore, employing thestandard parabolic theory to (2.5)₂ leads to

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{\beta }^{\epsilon }(\cdot )satisfies (2.1)-(2.2),letw_{\epsilon }\in L^2(0,T;U)withw_{\epsilon }\to w^{\ast }weaklyinL^2(0,T;U)as\epsilon \to 0,(u_{\epsilon },v_{\epsilon })be the solution of(2.3) corresponding tow_{\epsilon }.Then, on some subsequence(u_{{\epsilon }_n},v_{{\epsilon }_n})of(u_{\epsilon },v_{\epsilon }), there exists aquad(u,v,\eta )\in Y\times Y\times L^2(0,T;H)such that

while

as{\epsilon }_n\to 0and(u,v,\eta )is a solution of(1.1) satisfying the following estimates:

whereC>0isindependent of ε, n.

Proof Rewrite (2.3) as follows:

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

Now, we assume that (u^{\ast },v^{\ast },w^{\ast }) is optimal for problem(P). For each \epsilon >0, let(u_{\epsilon }^{\ast },v_{\epsilon }^{\ast },w_{\epsilon }^{\ast }) be the solution to

It follows from Lemma 2.2 that

Now, the approximating optimal control problems (P^{\epsilon }) are asfollows:

where L_{\epsilon }:L^2(0,T;U)\to R, by

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

is the approximations of g[23], wheren=[\frac{1}{\epsilon }], {\rho }_n is amollifier in R^n,P_n:H\to X_nis the projection of H on X_n, which isthe finite dimensional space generated by {\{e_i\}}_{i=1}^n,{\{e_i\}}_{i=1}^{\mathrm{\infty }} isan orthonormal basis in H, {\mathrm{\Lambda }}_n:R^n\to X_nis the operator defined by {\mathrm{\Lambda }}_n(\tau )={\sum }_{i=1}^n{\tau }_i{e}_i,\tau =({\tau }_1,{\tau }_2,\dots ,{\tau }_n).

First of all, we show the existence of optimal solutions for (P^{\epsilon }).

Lemma 2.3 (P^{\epsilon })has at least one optimal solution.

Proof Let \epsilon >0 befixed. It is clear that inf{L}_{\epsilon }(w)>-\mathrm{\infty }. Let d_{\epsilon }=inf\{L_{\epsilon }(w):w\in L^2(0,T;U)\} and w_n be aminimizing sequence such that

which together with (H₂), (H₃) and (2.28) implies thatw_n isbounded in L^2(0,T;U). Without loss ofgenerality, we may assume that w_n\to \tilde{w} inL^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,

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

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

and therefore

Finally, (2.28) and (2.32)-(2.34) imply that (\tilde{u},\tilde{v},\tilde{w}) is the optimal pair forproblem (P^{\epsilon }).This concludes the proof of Lemma 2.3. □

Lemma 2.4 Letw_{\epsilon }beoptimal for problem (P^{\epsilon })and(u_{\epsilon },v_{\epsilon })be the solution of(2.3) corresponding tow_{\epsilon }.Then, on some subsequence{\epsilon }_n,

Proof Since w_{\epsilon } is asolution to (P_{\epsilon }),we have

which together with (2.27) implies that

which combined with (2.37) implies that

which implies that (2.39), that w_{\epsilon } isbounded in L^2(0,T;U). Without loss generality,we may assume that w_{\epsilon }\to \tilde{w} weakly inL^2(0,T;U), which together withLemma 2.2 implies that there exists a sequence of {\epsilon }_n suchthat

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

and therefore

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

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

This completes the proof. □

In the following we derive the optimality condition for problem (P) by showingthe relation between approximation problem (P^{\epsilon }) andproblem (P). We start this section with the necessary conditions forw_{\epsilon }to be optimal for (P^{\epsilon }).

Lemma 3.1 Suppose that (H₁), (H₂), (H₃)and (H₄) hold. Letw_{\epsilon }beoptimal for problem (P^{\epsilon })and(u_{\epsilon },v_{\epsilon })be the solution of(2.3) corresponding tow_{\epsilon }.Then there exists a tetrad({\mu }_{\epsilon },p_{\epsilon },q_{\epsilon },{\zeta }_{\epsilon })\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^{\ast }suchthat

and

where\partial d_Sisthe sub-differential ofd_S.

Proof Let w\in L^2(0,T;U), w_{\epsilon }^{\chi }=w_{\epsilon }+\chi wand (u_{\epsilon }^{\chi },v_{\epsilon }^{\chi }) be the solution of (2.3)corresponding to w_{\epsilon }^{\chi }.Then it is clear that

Now, owing to the fact that w_{\epsilon } isoptimal for problem (P_{\epsilon }), wehave \frac{L_{\epsilon }(w_{\epsilon }^{\chi })-L_{\epsilon }(w_{\epsilon })}{\chi }\ge 0(for all \chi >0),hence

where (y_{\epsilon },{\overline{y}}_{\epsilon }) is the solution to

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

and

where \mathrm{\nabla }{g}_{\epsilon }(t,u_{\epsilon }) denotes the gradient ofg_{\epsilon }to the second variable at u_{\epsilon } and\mathrm{\nabla }h(w_{\epsilon }) denotes the gradient ofh at w_{\epsilon }.Here, {\zeta }_{\epsilon }\in \partial d_S(F(u_{\epsilon })) and \partial d_Sis the sub-differential of d_S, whichimplies (3.3). Thanks to S being convex, closed and Z^{\ast } beingstrictly convex, we may also infer that

Let

and (p_{\epsilon },q_{\epsilon }) be the solution of (3.1).It follows from (3.1), (3.5) and (3.6) that

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

The proof of Theorem 1.1 By using the properties of{\alpha }^{\epsilon }and {\beta }^{\epsilon }and Lemma 2.2, we have that, on a sequence of ε still denoted byε,

It follows from (3.10) and (3.11) that

Therefore, there exist generalized subsequences of {\mu }_{\epsilon }and {\zeta }_{\epsilon }such that

and

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

where \rho (t)\in \partial g(t,u^{\ast }) for all mostt\in (0,T). Thanks to (H₁) andLemma 2.2, we also infer