The paper is concerned with a family of geometric evolution problems, modeling the spatial control of an invasive population within a region \(V\subset {\mathbb R} ^2\) bounded by geographical barriers. If no control is applied, the contaminated set \(\Omega (t)\subset V\) 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 \(\Omega (0)=\Omega _0\subseteq V\) , three main problems are studied: (1) existence of an admissible strategy \(t\mapsto \Omega (t)\) which eradicates the contamination in finite time, so that \(\Omega (T)=\emptyset \) 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 \(t\mapsto \Omega (t)\) are explicitly constructed in a number of cases.
