The course is divided into two parts, each related to the control of expanding sets either with a barrier or by removing a fixed amount of area per unit time.
In the first part, we will address the following open problem (A. Bressan): assume that a set (the burning region or the fire region) is spreading in all directions with speed 1, and a barrier (a rectifiable region which the fire cannot cross) can be constructed with speed σ ≥ 1. Show that if σ ≤ 2, then the fire cannot be blocked; in other words, there is no barrier with finite length containing the burning region.
We will cover the following points:
1) Definition and setting of the problem.
2) Cost functional and existence of an optimal solution.
3) Necessary conditions and optimal solution in simple cases.
4) Spiral-like strategies and proof of the conjecture for this family of barriers.
In the second part, we will address the following problem: assume that a set (the contaminated set) is spreading with speed 1 in all directions, and we can sanitize an area A per unit time. Is it possible to control the contaminated set?
We will consider the following questions:
1) Formal derivation of this problem from a reaction-diffusion equation.
2) Existence of optimal solutions.
3) The case of convex initial data in the plane.
4) Necessary conditions for optimality and regularity of the optimal solution.
References will be provided during the course; the basic ones include:
- Alberto Bressan, "Results and Open Questions on Dynamic Blocking
Problems."
- Alberto Bressan, Maria Teresa Chiri, and Najmeh Salehi, "Optimal
Control of Moving Sets."