State-constrained optimal control of phase-field equations with obstacle
© Zheng et al.; licensee Springer. 2013
Received: 23 June 2013
Accepted: 25 September 2013
Published: 8 November 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.
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.
(H4) and there exists a constant such that forany .
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.
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.
Now we may combine the estimates (2.8), (2.10), (2.12), (2.13) and (2.18) to concludethe results. This completes the proof. □
whereisindependent of ε, n.
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. □
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.
Finally, (2.28) and (2.32)-(2.34) imply that is the optimal pair forproblem ().This concludes the proof of Lemma 2.3. □
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 ().
whereisthe sub-differential of.
which implies (3.2). This completes the proof. □
Using Lemma 2.2, we may pass to the limit in (3.2) and derive(1.3)1.
which together with (3.19) and Lemma 2.2 implies (1.3)2 (the secondinequality of (1.3)).
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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