On some optimal control problems for moving sets

The talk is concerned with a family of geometric evolution problems, modeling the spatial control of an invasive population within a plane region bounded by geographical barriers. The "contaminated region" is a set moving in the plane, which we would like to shrink as much as possible. To control the evolution of this set, we assign the velocity in the inward normal direction at every boundary point. Three main problems are studied: existence of an admissible strategy which eradicates the contamination in finite time, optimal strategies that achieve eradication in minimum time, strategies that minimize the average area of the contaminated set on a given time interval. For these optimization problems, that we like to see as “time dependent isoperimetric problems”, a sufficient condition for optimality is proved, together with several necessary conditions. Based on these conditions, optimal set-valued motions are explicitly constructed in a number of cases.

Coauthors: Alberto Bressan and Vasile Staicu