We establish a Poincaré–Bendixson type result for a weighted averaged infinite horizon problem in the plane, with and without averaged constraints. For the unconstrained problem, we establish the existence of a periodic optimal solution, and for the constrained problem, we establish the existence of an optimal solution that alternates cyclicly between a finite number of periodic curves, depending on the number of constraints. Applications of these results are presented to the shape optimization problems of the Cheeger set and the generalized Cheeger set, and also to a singular limit of the one-dimensional Cahn–Hilliard equation.

DOI: 10.1051/cocv/2014060

Keywords: Infinite-horizon optimization, periodic optimization, averaged constraint, planar cheeger set, singular limit, occupational measures, Poincaré–Bendixson

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@article{COCV_2015__21_4_1108_0, author = {Bright, Ido}, title = {Planar infinite-horizon optimal control problems with weighted average cost and averaged constraints, applied to cheeger sets}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1108--1119}, publisher = {EDP-Sciences}, volume = {21}, number = {4}, year = {2015}, doi = {10.1051/cocv/2014060}, mrnumber = {3395757}, zbl = {1341.49001}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2014060/} }

TY - JOUR AU - Bright, Ido TI - Planar infinite-horizon optimal control problems with weighted average cost and averaged constraints, applied to cheeger sets JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 1108 EP - 1119 VL - 21 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014060/ DO - 10.1051/cocv/2014060 LA - en ID - COCV_2015__21_4_1108_0 ER -

%0 Journal Article %A Bright, Ido %T Planar infinite-horizon optimal control problems with weighted average cost and averaged constraints, applied to cheeger sets %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 1108-1119 %V 21 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014060/ %R 10.1051/cocv/2014060 %G en %F COCV_2015__21_4_1108_0

Bright, Ido. Planar infinite-horizon optimal control problems with weighted average cost and averaged constraints, applied to cheeger sets. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 4, pp. 1108-1119. doi : 10.1051/cocv/2014060. http://www.numdam.org/articles/10.1051/cocv/2014060/

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