Optimal control of volume-preserving mean curvature flow

Antoine Laurain (1) , Shawn W. Walker (2)

1 IME - Instituto de Matemática e Estatística
2 Department of Mathematics Louisiana State University

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.

Mots clés: mean curvature flow; optimal control; transverse field; PDEs on moving surfaces

Domaines: Mathématiques [math]

Antoine Laurain, Shawn W. Walker. Optimal control of volume-preserving mean curvature flow. 2020. ⟨hal-02934959⟩