Parallels between control PDEs and systems of ODEs

The authors introduce a particular linear partial differential equation (PDE) (called a Kolmogorov equation) that relates to the PDE as the linear system relates to the ODE (ordinary differential equation). For this particular PDE they introduce an appropriate feedback that allows eigenvalue placement if the equation is hypoelliptic. The authors study the effect of this feedback on the spatial Fourier transform of the solution. They also mention the problem of transforming (by state coordinate changes and feedback) the linear PDE to a Kolmogorov equation as one would transform the nonlinear system to a controllable linear system