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Global optimality conditions for a dynamic blockingproblem

Published online by Cambridge University Press:  23 December 2010

Alberto Bressan  and
Tao Wang
Alberto Bressan
Affiliation:
Department of Mathematics, Penn State University, University Park, Pa. 16802, USA. bressan@math.psu.edu; wangt@math.psu.edu
Tao Wang
Affiliation:
Department of Mathematics, Penn State University, University Park, Pa. 16802, USA. bressan@math.psu.edu; wangt@math.psu.edu

Abstract

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The paper is concerned with a class of optimal blocking problems in the plane. Weconsider a time dependent set R(t) ⊂ ℝ2,described as the reachable set for a differential inclusion. To restrict its growth, abarrier Γ can be constructed, in real time. This is a one-dimensionalrectifiable set which blocks the trajectories of the differential inclusion. In this paperwe introduce a definition of “regular strategy”, based on a careful classification ofblocking arcs. Moreover, we derive local and global necessary conditions for an optimalstrategy, which minimizes the total value of the burned region plus the cost ofconstructing the barrier. We show that a Lagrange multiplier, corresponding to theconstraint on the construction speed, can be interpreted as the “instantaneous value oftime”. This value, which we compute by two separate formulas, remains constant when freearcs are constructed and is monotone decreasing otherwise.

Keywords

Dynamic blocking problemoptimality conditionsdifferential inclusion with obstacles

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 124 - 156
Copyright
© EDP Sciences, SMAI, 2010

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