Optimal control of convection–diffusion process with time-varying spatial domain: Czochralski crystal growth

This paper considers the optimal control of convection–diffusion systems modeled by parabolic partial differential equations (PDEs) with time-dependent spatial domains for application to the crystal temperature regulation problem in the Czochralski (CZ) crystal growth process. The parabolic PDE model describing the temperature dynamics in the crystal region arising from the first principles continuum mechanics is defined on the time-varying spatial domain. The dynamics of the domain boundary evolution, which is determined by the mechanical subsystem pulling the crystal from the melt, are described by an ordinary differential equation for rigid body mechanics and unidirectionally coupled to the convection–diffusion process described by the PDE system. The representation of the PDE as an evolution system on an appropriate infinite-dimensional space is developed and the analytic expression and properties of the associated two-parameter semigroup generated by the nonautonomous operator are provided. The LQR control synthesis in terms of the two-parameter semigroup is considered. The optimal control problem setup for the PDE coupled with the finite-dimensional subsystem is presented and numerical results demonstrate the regulation of the two-dimensional crystal temperature distribution in the time-varying spatial domain.