Abstract:
The area burned by the fire (or contaminated by a spreading chemical
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 which conditions guarantee the
existence of a strategy that completely blocks 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.
A general theorem on the existence of optimal strategies will be
presented, together with various necessary conditions for optimality.
REFERENCES:
[2] A.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.
[3] A. Bressan, M. Burago, A. Friend, and J.Jou,
Blocking Strategies for a Fire Control Problem,
{\it Analysis and Applications}, to appear.
[4] A. Bressan and C. De Lellis, Existence of optimal strategies
for a fire confinement problem. Preprint 2008.
