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

^{1}

@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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/cocv/2014060/

G. Alberti, Variational models for phase transitions, an approach via $\Gamma $-convergence. Calc. Var. Partial Differ. Equ. Edited by G. Buttazzo, A. Marino and M.K.V. Murthy. Springer, Berlin, Heidelberg (2000) 95–114. | MR | Zbl

Periodic optimization suffices for infinite horizon planar optimal control. SIAM J. Control Optim. 48 (2010) 4963–4986. | DOI | MR | Zbl

and ,Singularly perturbed control systems with one-dimensional fast dynamics. SIAM J. Control Optim. 41 (2002) 641–658. | DOI | MR | Zbl

and ,P. Billingsley, Convergence of probability Measures. Wiley, New York (1968). | MR | Zbl

A. Braides, A handbook of $\Gamma $-convergence. Vol. 3 of Stationary Partial Differential Equations. Edited by M. Chipot and P. Quittner. North-Holland (2006) 101–213. | Zbl

A reduction of topological infinite-horizon optimization to periodic optimization in a class of compact 2-manifolds. J. Math. Anal. Appl. 394 (2012) 84–101. | DOI | MR | Zbl

,Free energy of a nonuniform system. i. interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. | DOI | Zbl

and ,Structured phase transitions on a finite interval. Arch. Ration. Mech. Anal. 86 (1984) 317–351. | DOI | MR | Zbl

, and ,A lower bound for the smallest eigenvalue of the Laplacian. Probl. Anal. 625 (1970) 195–199. | MR | Zbl

,On the Poincaré–Bendixson theorem. Lect. Notes Nonlin. Anal. 3 (2002) 49–69. | Zbl

,The Poincaré–Bendixson Theorem: from Poincaré to the XXIst century. Cent. Eur. J. Math. 10 (2012) 2110–2128. | MR | Zbl

,Asymptotic properties of optimal solutions in planar discounted control problems. SIAM J. Control Optim. 27 (1989) 608. | DOI | MR | Zbl

and ,Linear programming approach to deterministic long run average problems of optimal control. SIAM J. Control Optim. 44 (2006) 2006–2037. | DOI | MR | Zbl

and ,Approximate solution of the HJB inequality related to the infinite horizon optimal control problem with discounting. Dynam. of Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 19 (2012) 65–92. | MR | Zbl

, and ,Generalized Cheeger sets related to landslides. Calc. Var. Partial Differ. Equ. 23 (2005) 227–249. | DOI | MR | Zbl

and ,Characterization of Cheeger sets for convex subsets of the plane. Pacific J. Math. 225 (2006) 103–118. | DOI | MR | Zbl

and ,One dimensional infinite-horizon variational problems arising in continuum mechanics. Arch. Ration. Mech. Anal. 106 (1989) 161–194. | DOI | MR | Zbl

and ,Un esempio di $\Gamma $-convergenza. Boll. Un. Mat. It. B 14 (1977) 285–299. | MR | Zbl

and ,Translation of J.D. van der Waals: The thermodynamik theory of capillarity under the hypothesis of a continuous variation of density. J. Stat. Phys. 20 (1979) 197–200. | DOI | MR | Zbl

,The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101 (1988) 209–260. | DOI | MR | Zbl

,J. Warga, Optimal control of differential and functional equations. Academic Press, New York (1972). | MR | Zbl

L.C. Young, Lectures on the calculus of variations and optimal control theory. Chelsea, New York (1980). | Zbl

*Cited by Sources: *