  Controlling the shape and position of moving and pinned droplets on a solid
surface is an important feature often found in microfluidic applications.
However, automating them, e.g., for high-throughput applications, does rarely
involve model-based optimal control strategies. In this work, we demonstrate
the optimal control of both the shape and position of a droplet sliding on an
inclined surface. This basic test case is a fundamental building block in
plenty of microfluidic designs. The static contact angle between the solid
surface, the surrounding gas, and the liquid droplet serves as the control
variable. By using several control patches, e.g., like done in electrowetting,
the contact angles are allowed to vary in space and time. In computer
experiments, we are able to calculate mathematically optimal contact angle
distributions using gradient-based optimization. The dynamics of the droplet
are described by the Cahn-Hilliard-Navier-Stokes equations. We anticipate our
demonstration to be the starting point for more sophisticated optimal design
and control concepts.
