Stabilising unstable steady states of falling liquid films using optimal feedback control

The dynamics of thin liquid films falling under gravity is an excellent example of a highly complex, nonlinear, interfacial flow. Such problems are typically too complex for standard control theoretical results to be applicable. We propose a method to stabilise unstable steady-state solutions to such systems using a finite number of sites at the lower boundary wall where fluid can either be injected or removed. Based on approaches proved to control the Kuramoto-Sivashinsky equation, we link the two problems we chain together a hierarchy of increasingly idealised approximations: an asymptotic expansion followed by linearising and then discretising. For the simplest approximation we can design optimal feedback controls using a linear quadratic regulator. We show that this can be successfully applied to direct numerical simulations of the full Navier-Stokes system over a wide range of parameters, even when observations are restricted to a finite number of points.