Results and open questions on the boundary control of moving sets

These notes provide a survey of recent results and open problems on the boundary control of moving sets. Motivated by the control of an invasive biological species, we consider a class of optimization problems for a moving set $t\mapsto \mathrm{\Omega }(t)$, where the goal is to minimize the area of the contaminated set $\mathrm{\Omega }(t)$ over time, plus a cost related to the control effort. Here the control function is the inward normal speed, assigned along the boundary $\mathrm{\partial }\mathrm{\Omega }(t)$. We also consider problems with geographical constraints, where the invasive population is restricted within an island. Existence and structure of eradication strategies, which entirely remove the invasive population in minimum time, is also discussed.