Optimally Controlled Moving Sets with Geographical Constraints

The paper is concerned with a family of geometric evolution problems, modeling the spatial control of an invasive population within a region V of the plane, bounded by geographical barriers. If no control is applied, the contaminated set Ω(t) ⊂ V expands with unit speed in all directions. By implementing a control, a region of area M can be cleared up per unit time. Given an initial set Ω(0) ⊆ V , three main problems are studied: (1) existence of an admissible strategy t → Ω(t) which eradicates the contamination in finite time, so that Ω(T) = ∅ for some T > 0. (2) Optimal strategies that achieve eradication in minimum time. (3) Strategies that minimize the average area of the contaminated set on a given time interval [0, T]. For these optimization problems, a sufficient condition for optimality is proved, together with several necessary conditions. Based on these conditions, optimal set-valued motions t → Ω(t) are explicitly constructed in a number of cases.