Arizona Transfer and Articulation

 

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Dynamic Blocking Problems

Thursday, November 12, 2009

Speaker: Alberto Bressan

Title: Dynamic blocking problems

Abstract: The talk will be concerned with a new class of variational problems, motivated by the confinement of wild fires. The area burned by the fire (or contaminated by a spreading chemical agent) at time $ t0 $ is modeled as the reachable set for a differential inclusion $ \dot x\in F(x) $, starting from an initial set $ R_0 $. We assume that the spreading of the contamination can be controlled by constructing barriers. These are rectifiable sets $ \gamma(t) $ whose length is allowed to grow linearly in time.

A first problem that we shall address is the existence of a strategy that completely blocks the spreading of the fire, within a bounded domain.

Next, we consider functions $ \alpha(x) $ describing the unit value of the land at the location $ x $, and $ \beta(x) $ accounting for the cost of building a unit length of wall near $ x $. This leads to an optimization problem, where one seeks to minimize the total value of the burned region, plus the cost of building the barrier.

The existence of optimal strategies will be discussed, together with necessary conditions for optimality. Results of numerical simulations will be compared with some analytical solutions.