Invariant ideals and its applications to the turnpike theory

TL;DR

This paper establishes the turnpike property for a non-convex discrete-time optimal control problem using the concept of invariant ideals, generalizing classical convergence notions to analyze solution behavior over infinite horizons.

Contribution

It introduces a novel approach by applying invariant ideals to prove the turnpike property in non-convex optimal control problems, extending existing theories.

Findings

Optimal solutions converge to a unique stationary point under ideal convergence.

The ideal convergence generalizes statistical and logarithmic density convergence.

The approach broadens the applicability of the turnpike theory in control problems.

Abstract

In this paper the turnpike property is established for a non-convex optimal control problem in discrete time. The functional is defined by the notion of the ideal convergence and can be considered as an analogue of the terminal functional defined over infinite time horizon. The turnpike property states that every optimal solution converges to some unique optimal stationary point in the sense of ideal convergence if the ideal is invariant under translations. This kind of convergence generalizes, for example, statistical convergence and convergence with respect to logarithmic density zero sets.