Motion Planning for Viscous Fingering

This letter considers the problem of controlling the shape of the moving interface between two fluids contained in a Hele-Shaw cell during its filling. This system is subject to viscous fingering. The interface shape dynamics is described using a modal decomposition where each mode is governed by a non-smooth ordinary differential equation with a random initial condition that is unknown. Once the interface is visibly non circular, a motion planning technique is employed to control the growth of the modes with the goal of approaching a shape of interest. This letter describes the dynamics at stake, studies its reachability property and proposes a constructive method to solve the motion planning problem. Some numerical experiments illustrate the presented methodology.