Optimal solutions for a class of set-valued evolution problems

The paper is concerned with a class of optimization problems for moving sets t↦Ω(t)⊂R2, motivated by the control of invasive biological populations. Assuming that the initial contaminated set Ω0 is convex, we prove that a strategy is optimal if an only if at each given time t∈[0,T] the control is active along the portion of the boundary ∂Ω(t) where the curvature is maximal. In particular, this implies that Ω(t) is convex for all t≥0. The proof relies on the analysis of a one-step constrained optimization problem, obtained by a time discretization.