<p><i>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</i> <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(t\mapsto \Omega (t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>↦</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation><i>, where the goal is to minimize the area of the contaminated set</i> <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Omega (t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> <i>over time plus a cost related to the control effort. Here, the control function is the inward normal speed assigned along the boundary</i> <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\partial \Omega (t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation><i>. 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, are also discussed.</i></p>