*Published Paper*

**Inserted:** 6 may 2011

**Journal:** Comm. Pure Appl. Math.

**Volume:** 62

**Number:** 6

**Pages:** 789-830

**Year:** 2009

**Notes:**

For the update version and eventual errata see the webpage http:/www.math.uzh.ch*delellis*

**Abstract:**

We consider a class of variational problems for differential inclusions, related to the control of forest fires. The area burned by the fire at time $t>0$ is modeled as the reachable set for a differential inclusion $\dot x\in F(x)$, starting from an initial set $R_0$. To block the fire, a barrier can be constructed progressively in time, at a given speed. In this paper we prove the existence of an optimal strategy, which minimizes the value of the area destroyed by the fire, plus the cost of constructing the barrier.