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.