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{\it Differential Inclusions and the Control of Forest Fires}

Alberto Bressan\\
Penn State University \\
{\tt bressanatmath.psu.edu} 
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\noindent{\bf 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. 

{\bf References}: 
\begin{enumerate}
\item Alberto Bressan, Differential inclusions and the control of forest
  fires, {\it J. Differential Equations} (special volume in honor of
  A.~Cellina and J.~Yorke), {\bf 243} (2007), 179-207. 

\item Alberto Bressan and Camillo De Lellis, Existence of optimal strategies
  for a fire confinement problem, in preparation.
\end{enumerate} 

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