Functional setting for a fixed-point controllability argument

Determine a function space in which the mapping that assigns to each \(\tilde y\) a null-controllable state \(y\) for the linear parabolic system \(y_t-\nabla\cdot[a(\nabla\tilde y)\nabla y]=\chi_\omega v\) is well defined, so that a fixed-point argument can be applied to obtain null controllability of the original quasi-linear system.

Background

As an alternative to global inversion, the authors consider seeking a fixed point of the map $\tilde y\mapsto y$, where $y$, together with a control $v$, solves a null controllability problem whose diffusion coefficient is frozen at $\nabla\tilde y$. For this strategy to work, the map must be well defined on a suitable function space. The paper explicitly states that identifying such a space is unresolved.