Planar infinite-horizon optimal control problems with weighted average cost and averaged constraints, applied to cheeger sets

Department of Applied Mathematics, University of Washington, Seattle, WA 98195, United States.
ibright@uw.edu

Received: 2 November 2013
Revised: 10 November 2014

Abstract

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.