Some optimal control problems for a two-phase field model of solidification

Abstract

In this paper we deal with some optimal control problems for a solidification phase field model of metallic alloys. The model allows crystallizations of two kinds, each one described by its own phase field. Accordingly, the state is the triplet (τ, u, v), where τ is the temperature and u and v are phase field functions. The optimality conditions for the optimal control problems considered in this work are obtained by using the Dubovitskii-Milyutin formalism.

Local existence of solutions of a three phase‐field model for solidification

“…In this article the model is deduced from physical principles and supported by some numerical simulations. We remark that, from the mathematical point of view, the present model can be seen as a generalization of the solidification models treated by Hoffman and Jiang in [3] and is related to Boldrini et al in [20,21], where the interaction potentials were similar to the ones considered here, but the diffusion mechanism was much simpler than the present nonlinear ones.…”

Analysis of a two-phase field model for the solidification of an alloy

“…The present results will be used to study several optimal control problems in [7]. Furthermore, the techniques we use here can be regarded as preliminary for the study of a more complex three-phase field model for the solidification of an alloy, to be considered in forthcoming papers.…”

Optimal control for a conserved phase field system with a possibly singular potential

“…We would like to force the averaged temperature and phase variable to be closed to some fixed values ϑ Q and ϕ Q and their final values at time T to be closed to ϑ Ω and ϕ Ω , respectively. In order to do that we choose the following cost functional 3 2 Ω (ϑ(T ) − ϑ Ω ) 2 + κ 4 2 Ω (ϕ(T ) − ϕ Ω ) 2 (1.5) where (ϑ, ϕ) is the state corresponding to the control u, and the desired temperatures ϑ Q ∈ L 2 (Q), ϑ Ω ∈ L 2 (Ω), the target phases ϕ Q ∈ L 2 (Q), ϕ Ω ∈ L 2 (Ω), and the constants κ i ≥ 0, i = 1, . .…”