Optimal Control of Free Boundary Value Problems in Thermoelasticity

This work deals with the optimal control of a free boundary value problem in thermoelasticity where the geometry varies with respect to time. Typical applications are glass shaping for different purposes. We derive a mathematical model for one- and two-dimensional problems. The goal of the optimization is to achieve a desired shape of the domain by controlling the temperature inputs. In dimension one, both analysis and numerics are performed. The state-constrained problem is considered and its Moreau-Yosida regularization is analyzed. The numerical observations are presented which underline the findings from the theory. The deformation of a rectangular domain is studied in two-dimensional case and the goal is to reach a predefined profile of one of the free boundaries of the domain.