Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two-dimensional (2-D) parabolic partial differential equation with time-varying domain is realized in this work. The transformation of boundary actuation to distributed control setting allows to represent the system's model in a standard evolutionary form. By exploring dynamical model evolution and generating data, a set of time-varying empirical eigenfunctions that capture the dominant dynamics of the distributed system is found. This basis is used in Galerkin's method to accurately represent the distributed system as a finite-dimensional plant in terms of a linear time-varying system. This reduced-order model enables synthesis of a linear optimal output tracking controller, as well as design of a state observer. Finally, numerical results are prepared for the optimal output tracking of a 2-D model of the temperature distribution in Czochralski crystal growth process which has nontrivial geometry. (c) 2014 American Institute of Chemical Engineers AIChE J, 61: 494-502, 2015