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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.
@article{COCV_2012__18_1_124_0,
author = {Bressan, Alberto and Wang, Tao},
title = {Global optimality conditions for a dynamic blocking problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {124--156},
publisher = {EDP-Sciences},
volume = {18},
number = {1},
year = {2012},
doi = {10.1051/cocv/2010053},
mrnumber = {2887930},
zbl = {1258.49030},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010053/}
}
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 PY - 2012 SP - 124 EP - 156 VL - 18 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010053/ DO - 10.1051/cocv/2010053 LA - en ID - COCV_2012__18_1_124_0 ER -
%0 Journal Article %A Bressan, Alberto %A Wang, Tao %T Global optimality conditions for a dynamic blocking problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 124-156 %V 18 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010053/ %R 10.1051/cocv/2010053 %G en %F COCV_2012__18_1_124_0
Bressan, Alberto; Wang, Tao. Global optimality conditions for a dynamic blocking problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 124-156. doi: 10.1051/cocv/2010053
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