Alberto Bressan

Penn State University

`bressanatmath.psu.edu`

Penn State University

**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 is modelled as the
reachable set for a differential inclusion
, starting from an
initial set . 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 describing the unit value of the land at the location , and a function accounting for the cost of building a unit length of wall near . 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**:

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