  We propose and analyse a new methodology based on linear-quadratic regulation
(LQR) for stabilising falling liquid films via blowing and suction at the base.
LQR methods enable rapidly responding feedback control by precomputing a gain
matrix, but are only suitable for systems of linear ordinary differential
equations (ODEs). By contrast, the Navier-Stokes equations that describe the
dynamics of a thin liquid film flowing down an inclined plane are too complex
to stabilise with standard control-theoretical techniques. To bridge this gap
we use reduced-order models - the Benney equation and a weighted-residual
integral boundary layer model - obtained via asymptotic analysis to derive a
multi-level control framework. This framework consists of an LQR feedback
control designed for a linearised and discretised system of ODEs approximating
the reduced-order system, which is then applied to the full Navier-Stokes
system. The control scheme is tested via direct numerical simulation (DNS), and
compared to analytical predictions of linear stability thresholds and minimum
required actuator numbers. Comparing the strategy between the two reduced-order
models we show that in both cases we can successfully stabilise towards a
uniform flat film across their respective ranges of valid parameters, with the
more accurate weighted-residual model outperforming the Benney-derived
controls. The weighted-residual controls are also found to work successfully
far beyond their anticipated range of applicability. The proposed methodology
increases the feasibility of transferring robust control techniques towards
real-world systems, and is also generalisable to other forms of actuation.
