PhD in Applied Mathematics

## PROJECTS

We consider a controlled reaction-diffusion equation, motivated by a pest eradication problem. Our goal is to derive a simpler model, describing the controlled evolution of a contaminated set. In this direction, the first part of the paper studies the optimal control of 1-dimensional traveling wave profiles. Using Stokes' formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls. In the last section we introduce a family of optimization problems for a moving set. We show how these can be derived from the original parabolic problems, by taking a sharp interface limit.

This research is available as a paper here.

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Motivated by the control of invasive biological populations, we consider a class of optimization problems for moving sets

tâ†¦ â„¦ (t)⊂ IR2. Given an initial set â„¦0, the goal is to minimize the area of the contaminated set â„¦(t) over time, plus a cost related to the control effort. Here the control function is the inward normal speed along the boundary ∂â„¦(t). We prove the existence of optimal solutions, within a class of sets with finite perimeter. Necessary conditions for optimality are then derived, in the form of a Pontryagin maximum principle. Additional optimality conditions show that the sets â„¦(t) cannot have certain types of outward or inward corners. Finally, some explicit solutions are presented.

The detailed results are available here.

Image courtesy of Sven Torfinn.