Optimal boundary control of a diffusion-convection-reaction PDE model with time-dependent spatial domain: Czochralski crystal growth process

摘要

In this paper the optimal boundary control problem for diffusion-convection-reaction processes modeled by partial differential equations (PDEs) defined on time-dependent spatial domains is considered. The model of the transport system with time-varying domain arises in the context of high energy consuming Czochralski crystal growth process in which the crystal temperature regulation must successfully account for the change in the crystal spatial domain due to the crystal growth process realized by the pulling crystal out of melt. Starting from the first principles of continuum mechanics and transport theorem the time-varying parabolic PDE describing temperature evolution is derived and represented as a nonautonomous parabolic evolution system on an appropriately defined function space which is exactly transformed in the infinite-dimensional boundary control problem for which a boundary linear quadratic regulator is proposed. Properties of the solution of the time-varying parabolic PDEs given by the two-parameter evolutionary system are utilized in the synthesis of the optimal boundary regulator, and the control law is applied to the model given by a two-dimensional partial differential equation in the cylindrical coordinates representing the Czochralski crystal growth process with one-dimensional growth direction. Finally, numerical results demonstrate optimal stabilization of the two-dimensional temperature distribution in the crystal.