Global optimality conditions for a dynamic blocking problem

Alberto Bressan; Tao Wang

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 1, page 124-156
  • ISSN: 1292-8119

Abstract

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

How to cite

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Bressan, Alberto, and Wang, Tao. "Global optimality conditions for a dynamic blocking problem." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 124-156. <http://eudml.org/doc/221891>.

@article{Bressan2012,
abstract = {The paper is concerned with a class of optimal blocking problems in the plane. We consider a time dependent set R(t) ⊂ ℝ2, described as the reachable set for a differential inclusion. To restrict its growth, a barrier Γ can be constructed, in real time. This is a one-dimensional rectifiable set which blocks the trajectories of the differential inclusion. In this paper we introduce a definition of “regular strategy”, based on a careful classification of blocking arcs. Moreover, we derive local and global necessary conditions for an optimal strategy, which minimizes the total value of the burned region plus the cost of constructing the barrier. We show that a Lagrange multiplier, corresponding to the constraint on the construction speed, can be interpreted as the “instantaneous value of time”. This value, which we compute by two separate formulas, remains constant when free arcs are constructed and is monotone decreasing otherwise. },
author = {Bressan, Alberto, Wang, Tao},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Dynamic blocking problem; optimality conditions; differential inclusion with obstacles; dynamic blocking problem},
language = {eng},
month = {2},
number = {1},
pages = {124-156},
publisher = {EDP Sciences},
title = {Global optimality conditions for a dynamic blocking problem},
url = {http://eudml.org/doc/221891},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Bressan, Alberto
AU - Wang, Tao
TI - Global optimality conditions for a dynamic blocking problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/2//
PB - EDP Sciences
VL - 18
IS - 1
SP - 124
EP - 156
AB - The paper is concerned with a class of optimal blocking problems in the plane. We consider a time dependent set R(t) ⊂ ℝ2, described as the reachable set for a differential inclusion. To restrict its growth, a barrier Γ can be constructed, in real time. This is a one-dimensional rectifiable set which blocks the trajectories of the differential inclusion. In this paper we introduce a definition of “regular strategy”, based on a careful classification of blocking arcs. Moreover, we derive local and global necessary conditions for an optimal strategy, which minimizes the total value of the burned region plus the cost of constructing the barrier. We show that a Lagrange multiplier, corresponding to the constraint on the construction speed, can be interpreted as the “instantaneous value of time”. This value, which we compute by two separate formulas, remains constant when free arcs are constructed and is monotone decreasing otherwise.
LA - eng
KW - Dynamic blocking problem; optimality conditions; differential inclusion with obstacles; dynamic blocking problem
UR - http://eudml.org/doc/221891
ER -

References

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