We consider the control system (𝜕t - |D|)f = P+ 1ω u. This system is
not null-controllable, but we can ask which initial conditions can be
steered to 0.

It turns out that this control system is related to some holomorphic
functions spaces. In this talk, we will use this as an excuse to
introduce several complex analysis tools and use them to extract
information about the initial conditions that are null-controllable.
This involves the Hardy space and its reproducing kernel, the density
of polynomials in Bergman spaces, separation of singularities,
hyperfunctions, the Bloch-A¹ duality and boundary behavior of conformal
maps.

This is joint work with Andreas Hartmann