A reduction of topological infinite-horizon optimization to periodic optimization in a class of compact 2-manifolds

We consider a general class of infinite-horizon optimization problems, where the phase space is a two dimensional manifold satisfying the Jordan curve theorem, and the set of feasible curves satisfies general conditions similar to, but weaker than the solution set of a control system. We verify the existence of an optimal periodic solution for this class of problems. Applying this result we obtain two new results: we generalize a previous result for discounted infinite-horizon control problems, and provide a Poincaré–Bendixson type result for continuous generalized semi-flows.