Optimal boundary control of parabolic PDE with time-varying spatial domain

This paper considers the optimal boundary control of reaction-diffusion process with time-varying spatial domain in the context of the Czochralski crystal growth process. A parabolic partial differential equation (PDE) model of the reaction-diffusion process which preserves the dynamical features attributed to the time-varying spatial domain is developed. The parabolic PDE is coupled to a second order ordinary differential equation (ODE) which describes the time-evolution of the spatial domain. The infinite-dimensional linear state space representation of the PDE system with control input at the boundary is reformulated into an abstract form and provides the framework for the optimal boundary control problem. The optimal control law is determined and numerical results of the closed-loop system are provided.