Differential Inclusions and the Control of Forest Fires

Alberto Bressan
Penn State University
bressanatmath.psu.edu

Abstract: The talk will describe a new class of variational problems, motivated by the control of forest fires. The area burned by the fire (or contaminated by a spreading agent) at time $t>0$ is modelled 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 walls. In the case of a forest fire, one may think of a thin strip of land which is either soaked with water poured from above (by airplane or helicopter), or cleared from all vegetation using a bulldozer.

The first part of the talk will examine under which conditions there exists a strategy that blocks the fire within a bounded domain.

Next, consider a function $\alpha(x)$ describing the unit value of the land at the location $x$, and a function $\beta(x)$ 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.

A general theorem on the existence of optimal strategies will be presented, together with various necessary conditions for optimality.

References:

  1. Alberto Bressan, Differential inclusions and the control of forest fires, J. Differential Equations (special volume in honor of A. Cellina and J. Yorke), 243 (2007), 179-207.

  2. Alberto Bressan and Camillo De Lellis, Existence of optimal strategies for a fire confinement problem, in preparation.