Stochastic control problems and a convergence result for horizontal mean curvature flow

Abstract: The evolution by horizontal mean curvature flow was broadly studied for its applications in neurogeometry and in image processing (e.g. Citti-Sarti model). It represents the contracting evolution of a hypersurface embedded in a particular geometrical setting, called sub-Riemannian geometry, in which only some curves (called horizontal curves) are admissible by definition. This may lead to the existence of some points of the hypersurface, called characteristic, in which is not possible to define the so-called horizontal normal. To avoid this problem, it is possible to use the notion of Riemannian approximation of a sub-Riemannian geometry applied to the horizontal mean curvature flow.

I will show the connection between the evolution of a generic hypersurface in this setting and the associated stochastic optimal control problem. Then, I will show some results of asymptotic optimal controls in the Heisenberg group and use them to show later a convergence result between the solutions of the approximated mean curvature flow and the horizontal ones. This is from some joint works with N. Dirr and F. Dragoni.

MathematicsPhysics

Audience: researchers in the topic

Nečas Seminar on Continuum Mechanics

Series comments: This seminar was founded on December 14, 1966.

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8. If not written otherwise, we will meet on Mondays at 15:40 in lecture hall K3 (2nd floor)