Due to its unique performance of high efficiency, fast heating speed and low power consumption, induction heating is widely and commonly used in many applications. In this paper, we study an optimal control problem arising from a metal melting process by using a induction heating method. Metal melting phenomena can be modeled by phase field equations. The aim of optimization is to approximate a desired temperature evolution and melting process. The controlled system is obtained by coupling Maxwell's equations, heat equation and phase field equation. The control variable of the system is the external electric field on the local boundary. The existence and uniqueness of the solution of the controlled system are showed by using Galerkin's method and Leray-Schauder's fixed point theorem. By proving that the control-to-state operator $P$ is weakly sequentially continuous and Fréchet differentiable, we establish an existence result of optimal control and derive the first-order necessary optimality conditions. This work improves the limitation of the previous control system which only contains heat equation and phase field equation.

This work is concerned with a Kirchhoff-Schrödinger-Poisson (KSP) system posed in a bounded domain of ${\mathbb{R}}^3$. The model features a singular nonlinearity $\alpha v^{-\tau }$ with $0<\tau <1$, together with a coupling term of the form $\phi |v{|}^{q-2}v$, where $2<q<3$. The singular term destroys differentiability of the energy functional while the nonlocal potential ${\phi }_v$ causes compactness issues. Using nonsmooth critical point theory, we establish a key estimate linking the weak slope with the derivative of the regular part, prove the Palais-Smale (PS) condition, and characterize critical points as weak solutions. By means of Ekeland's variational principle and the mountain pass theorem, we establish the existence of a constant $\mathrm{\Gamma }>0$ with the property that the system admits two distinct positive solutions whenever $\alpha \in (0,\mathrm{\Gamma })$.