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Optimal control of volume-preserving mean curvature flow

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-02934959
Contributor : Antoine Laurain <>
Submitted on : Wednesday, September 9, 2020 - 9:09:25 PM
Last modification on : Wednesday, October 14, 2020 - 4:10:59 AM
Long-term archiving on: : Thursday, December 3, 2020 - 1:15:01 AM

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

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