The subject or PDE constrained optimization has recently received a strong collaborative impulse by scientists from mathematics. physics, engineering, etc. This subject is focused on optimization methods, which take into account structures that arise from Lagrangians, PDEs and their discretizations. Our paper starts from an important Kuiper geometrical open problem whose solvability requires techniques from PDE constrained optimization. does a surface have a constrained minimal total Gaussian curvature? In this context, we analyse the following problems: the distance Lagrangian, the Monge-Ampere-Tzitzeica PDE, approximations of Tzitzeica Lagrangians, Tzitzeica surfaces with minimum total curvature, affine deviated Gauss curvature, and controlled least squares approximations of Tzitzeica surfaces.