  We consider the control of a three-dimensional thin liquid film on a flat
substrate, inclined at a non-zero angle to the horizontal. Controls are applied
via same-fluid blowing and suction through the substrate surface. We consider
both overlying and hanging films, where the liquid lies above or below the
substrate, respectively. We study the weakly nonlinear evolution of the system,
which is governed by a forced Kuramoto--Sivashinsky equation in two space
dimensions. The uncontrolled problem exhibits three ranges of dynamics
depending on the incline of the substrate: stable flat film solution, bounded
chaotic dynamics, or unbounded exponential growth of unstable transverse modes.
We proceed with the assumption that we may actuate at every point on the
substrate. The main focus is the optimal control problem, which we first study
in the special case that the forcing may only vary in the spanwise direction.
The structure of the Kuramoto--Sivashinsky equation allows the explicit
construction of optimal controls in this case using the classical theory of
linear quadratic regulators. Such controls are employed to prevent the
exponential growth of transverse waves in the case of a hanging film, revealing
complex dynamics for the streamwise and mixed modes. We then consider the
optimal control problem in full generality, and prove the existence of an
optimal control. For numerical simulations, we employ an iterative gradient
descent algorithm. In the final section, we consider the effects of transverse
mode forcing on the chaotic dynamics present in the streamwise and mixed modes
for the case of a vertical film flow. Coupling through nonlinearity allows us
to reduce the average energy in solutions without directly forcing the linearly
unstable dominant modes.
