AMMCS-2011 Plenary Talk:

Dynamic Blocking Problems

by Alberto Bressan

Department of Mathematics,
Penn State University

The talk will describe a new class of optimization problems, motivated by the confinement of wild fires, or of the spreading of chemical contaminations.

In absence of control, the region burned by the fire is modeled as the reachable set for a differential inclusion. We assume that fire propagation can be controlled by constructing "barriers", in real time. These are represented by rectifiable sets in the plane, which cannot be crossed by trajectories of the differential inclusion. For this model, several results will be presented, concerning:

(1) The speed at which the barrier must be constructed, in order to eventually contain the fire.

(2) The existence of an optimal strategy, minimizing the total value of the burned region.

(3) Relations with Hamilton-Jacobi equations with obstacles.

(4) Necessary conditions for optimality, and the "instantaneous value of time".

(5) Examples of explicit solutions.

Some related questions and open problems will also be discussed.

Alberto Bressan completed his undergraduate studies at the University of Padova, Italy, and received a Ph.D. from the University of Colorado, Boulder, in 1982. He has held faculty positions at the University of Colorado and at the International School for Advanced Studies in Trieste, Italy. Presently he is Eberly Chair Professor of Mathematics at the Pennsylvania State University.

His scientific interests lie in the areas of differential inclusions, control theory, differential games, partial differential equations, and hyperbolic systems of conservation laws.

He gave a plenary lecture at the International Congress of Mathematicians, Beijing 2002. In 2006 he received the A. Feltrinelli prize for Mathematics, Mechanics, and Applications, from the Accademia Nazionale dei Lincei in Rome. He was awarded the S.I.A.M. "Analysis of Partial Differential Equations" prize in 2007 and the M. Bôcher prize by the American Mathematical Society, in 2008.