PDE backstepping control of one-dimensional heat equation with time-varying domain

In this work a PDE backstepping-based control law for one-dimensional unstable heat equation with time-varying spatial domain is developed. The underlying parabolic partial differential equation (PDE) with time-varying domain is a model emerging from process control applications such as crystal growth. The use of backstepping control methodology yields the inherent feature of a time-varying PDE describing the kernel of the associated Volterra integral. The well-posedness of PDE kernel is proven and a numerical method to compute the solution of PDE kernel augmented with the error analysis to establish the accuracy of the proposed numerical method is demonstrated. Finally, the explicit form of the full state-feedback control law is given and appropriate simulation is provided for the application of temperature regulation in the Czochralski crystal growth process.