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Optimal control of a phase field model for solidification. (English) Zbl 0724.49003
A free boundary value problem is considered modelling solidification of a liquid. Besides temperature a function of space and time is introduced which defines the present state of the system. As a result two coupled parabolic partial differential equations describe the dynamic process of solidification. For this system existence, uniqueness and stability is proved making use of energy type estimates and Leray-Schauder’s fixed point principle. The solution exists in a weak sense but under mild assumptions. Furthermore the distributed optimal control problem for this system is considered. It is proved that an optimal control exists and satisfies necessary optimality conditions of Pontryagin type. For small time intervals this solution is unique. The proofs are based on sharp a priori estimates in appropriate Sobolev spaces.
Reviewer: K.-H.Hoffmann

MSC:
49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
35K20 Initial-boundary value problems for second-order parabolic equations
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References:
[1] DOI: 10.1007/BF00254827 · Zbl 0608.35080 · doi:10.1007/BF00254827
[2] DOI: 10.1093/imamat/38.3.195 · Zbl 0645.35101 · doi:10.1093/imamat/38.3.195
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[4] Zheng S., Global existence and stability of solutions of the phase field equation
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