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Global optimality conditions for a dynamic blocking problem

Imagine a fire developing in a windy zone. The burned area will grow developing in all directions and being influenced by the wind. A model for this situation is naturally given by a differential inclusion, in dimension two, and the burned area is the reachable set. Firefighters will start constructing walls (or barriers) to limit the fire expansion. However, one has to take into account the time needed for that. The optimization problem considered in this paper corresponds to the minimization of burned area and of costs for construction of wall, modeled by a one-dimensional rectifiable set. Necessary conditions are derived, which are obviously quite complicated to state, but explicit enough to solve some simple yet interesting problems. Moreover, they open the way to address the problem numerically.

Global optimality conditions for a dynamic blocking problem
Alberto Bressanand Tao Wang
ESAIM: COCV 18 (2012) 124-156
http://dx.doi.org/10.1051/cocv/2010053