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State-constrained optimal control of phase-field equations with obstacle
Boundary Value Problems volume 2013, Article number: 234 (2013)
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:
where Ω is a bounded domain in ()with a smooth boundary ∂ Ω, ,denote given parameters, is the subdifferential of the indicator function on the closed interval , Bw is a givenforcing term on Q, is the outward normal derivative on ∂ Ω and , 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 in this paper, and denoting the range of the order parameteru is assumed to be a compact interval , and mean,respectively, that the physical situation at is of pure solid and pure liquid, while means that thephysical situation at is mushy.
A great deal of research has been done on the phase-field transition system, for whichwe refer to the book by Temam  and thereferences therein. Without the term , 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 (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  and . Recently, the Caginalp phase-field system with coupleddynamic boundary conditions, including the singular potentials, was presented in and .
Throughout this paper, the Hilbert space is equipped with the usualinner product and the norm.Define a closed subspace ofH by . We putwith ,where . If we identifyand with their dual space, then we have and ,where is the duality space of .
Throughout the paper, we suppose that the following assumptions hold.
Let U be a real Hilbert space, be a linear continuousoperator. Assuming that Z is a Banach space with the dualstrictly convex, let be aclosed convex subset with finite co-dimensionality.
(H1) is in the class of.
(H2) is measurable in t and for every , thereexists independent of t such that and
(H3) is lower semicontinuousand convex with the following growth property:
(H4) and there exists a constant such that forany .
We consider the following optimal control problem:
satisfying (1.1) is called a feasible pair, where .
The first question regarding problem (P) is if there is an admissible solution,i.e., if the set is nonempty. Taking into account  similarly,we may assume in the sequel that for , 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 and 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  and ,respectively. In , 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 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.Letbe optimal forproblem (P), then there exists a tetradwithand ameasuresuchthat
Moreover, ifis injective,then.
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 (),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 , we define a nondecreasing function on R by
where denotes the positive part of functions. Then , and
We fix a primitive of such that
Without loss of generality, we may assume ,therefore, the approximation of (1.1) is
Lemma 2.1 Suppose thatsatisfies (2.1)-(2.2),, weaklyinand, are the solutionsof (2.3) corresponding toand,respectively. Then there exists a subsequenceof, still denoted byitself, such that
Proof Replacing and w by and in (2.3),respectively, we obtain
Multiplying (2.5)1 and (2.5)2 by and ,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 ofthat
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 ,we derive
which together with ,(2.2), (2.8), Nirenberg’s inequality and Gronwall’s inequality implies that
Next, multiplying (2.5)1 by ,integrating over and invoking Young’s inequality, wederive
Thanks to (2.8), 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 ,then multiplying (2.5)2 by , 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 ()are independent of n. Inserting (2.15) into (2.14), we derive
Taking the supremum with respect to t in (2.16), choosingsufficiently small and applying Gronwall’s inequality, we end up with
which combined with (2.13) implies that . 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 thatsatisfies (2.1)-(2.2),letwithweaklyinas,be the solution of(2.3) corresponding to.Then, on some subsequenceof, there exists aquadsuch that
asandis a solution of(1.1) satisfying the following estimates:
whereisindependent 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, we get a.e. in . This completes theproof. □
Now, we assume that is optimal for problem(P). For each , let be the solution to
It follows from Lemma 2.2 that
Now, the approximating optimal control problems () are asfollows:
where , by
and is the solution of (2.3). Here, denotes the distance of to S,
is the approximations of g, where, is amollifier in ,is the projection of H on , which isthe finite dimensional space generated by , isan orthonormal basis in H, is the operator defined by ,.
First of all, we show the existence of optimal solutions for ().
Lemma 2.3 ()has at least one optimal solution.
Proof Let befixed. It is clear that . Let and be aminimizing sequence such that
which together with (H2), (H3) and (2.28) implies that isbounded in . Without loss ofgenerality, we may assume that in. Let and be the solutions of (2.3)corresponding to and, respectively. It follows fromLemma 2.1 that on some subsequence of , 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
Finally, (2.28) and (2.32)-(2.34) imply that is the optimal pair forproblem ().This concludes the proof of Lemma 2.3. □
Lemma 2.4 Letbeoptimal for problem ()andbe the solution of(2.3) corresponding to.Then, on some subsequence,
Proof Since is asolution to (),we have
which together with (2.27) implies that
which combined with (2.37) implies that
which implies that (2.39), that isbounded in . Without loss generality,we may assume that weakly in, which together withLemma 2.2 implies that there exists a sequence of suchthat
On the other hand, (2.28) and (2.39) imply that
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. □
3 The optimality condition for () and(P)
In the following we derive the optimality condition for problem (P) by showingthe relation between approximation problem () andproblem (P). We start this section with the necessary conditions forto be optimal for ().
Lemma 3.1 Suppose that (H1), (H2), (H3)and (H4) hold. Letbeoptimal for problem ()andbe the solution of(2.3) corresponding to.Then there exists a tetradsuchthat
whereisthe sub-differential of.
Proof Let , and be the solution of (2.3)corresponding to .Then it is clear that
Now, owing to the fact that isoptimal for problem (), wehave (for all ),hence
where is the solution to
Next, employing the same arguments as in the proof of , we conclude that
where denotes the gradient ofto the second variable at and denotes the gradient ofh at .Here, and is the sub-differential of , whichimplies (3.3). Thanks to S being convex, closed and beingstrictly convex, we may also infer that
and 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 ofand 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 and such that
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 , we can conclude that
where for all most. Thanks to (H1) andLemma 2.2, we also infer
Now we claim that
Indeed, let and , then we derive
With the help of (3.13), (3.16), (3.18)-(3.23), we can pass to the limit in (3.1) toderive that andsatisfies (1.2). On the other hand, observing that , we derive
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 . To this end,we suppose that .It follows from (3.17), (3.19) and (3.24) that there exist and suchthat
Since isa closed convex subset with finite co-dimensionality, so is , which together with (3.25)and (3.26) implies that .
Assuming that is injective and, thanks to (1.2), wederive ,which yields and . This is a contradictionwith . Thus, if is injective, then. We completethe proof. □
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The authors declare that they have no competing interests.
JZ carried out the optimal control problem for the phase-field transition system withstate constraint and obstacle and drafted the manuscript. HL and JL participated in thedesign of the study and examined the results carefully. All authors read and approvedthe final manuscript.