Fields Colloquium

Motivated by the control of invasive biological populations, we study evolving planar regions whose boundary velocity is controlled under a constraint on the total instantaneous effort. The central question is how geometry determines the best use of limited control resources. For convex initial configurations, we characterize optimal evolutions through a maximum-curvature principle: control is concentrated where boundary curvature is largest, and convexity is preserved. We then examine the sensitivity of the associated value functions to the initial configuration, presenting continuity results and counterexamples. For populations confined to a bounded domain, letting the uncontrolled expansion speed tend to zero connects minimum-time eradication to optimal slicing: sweeping out the domain by nested sets with prescribed area while minimizing the integrated length of their relative boundaries. We discuss existence, necessary optimality conditions, and continuity of the minimum slicing cost. This progression from controlled dynamics to geometric optimization leads to open questions involving relative isoperimetric profiles and the conjecture that the disk maximizes the minimum slicing cost among domains of fixed area.

Joint works with Stefano Bianchini, Alberto Bressan, and Elsa Marchini.