We develop a framework and numerical method for controlling the full space-time tube of a geometrically driven flow. We consider an optimal control problem for the mean curvature flow of a curve or surface with a volume constraint, where the control parameter acts as a forcing term in the motion law. The control of the trajectory of the flow is achieved by minimizing an appropriate tracking-type cost functional. The gradient of the cost functional is obtained via a formal sensitivity analysis of the space-time tube generated by the mean curvature flow. We show that the perturbation of the tube may be described by a transverse field satisfying a parabolic equation on the tube. We propose a numerical algorithm to approximate the optimal control and show several results in two and three dimensions demonstrating the efficiency of the approach.

> Optimal control of volume-preserving mean curvature flow

Controlling droplet shape via surface tension has numerous technological applications, such as droplet lenses and lab-on-a-chip. This motivates a partial differential equation-constrained shape optimization approach for controlling the shape of droplets on flat substrates by controlling the surface tension of the substrate. We use shape differential calculus to derive an L^2 gradient flow approach to compute equilibrium shapes for sessile droplets on substrates. We then develop a gradient based optimization method to find the substrate surface tension coefficient yielding an equilibrium droplet shape with a desired footprint (i.e. the liquid-solid interface has a desired shape). Moreover, we prove a sensitivity result with respect to the substrate surface tensions for the free boundary problem associated with the footprint. Numerical results are also presented to showcase the method.

Check the videos made by Shawn Walker who did the numerics for this paper > An ellipse target > A square target > A clover target

> Droplet footprint control

A bilevel shape optimization problem with the exterior Bernoulli free boundary problem as lower-level problem and the control of the free boundary as the upper-level problem is considered. Using the shape of the inner boundary as the control, we aim at reaching a specific shape for the free boundary. A rigorous sensitivity analysis of the bilevel shape optimization in the infinite-dimensional setting is performed. The numerical realization using two different cost functionals presented in this paper demonstrate the efficiency of the approach.

> A bilevel shape optimization problem for the exterior Bernoulli free boundary value problem