# The turnpike property in nonlinear optimal control -- A geometric approach

## TL;DR

This paper introduces a geometric dynamical systems approach to analyze the turnpike property in nonlinear optimal control, providing new insights and simpler proofs for existing results.

## Contribution

It develops a geometric framework to study the turnpike property, extending understanding to more general conditions and removing some restrictions on initial and target states.

## Key findings

- Turnpike-like behavior appears in systems with hyperbolic equilibrium.
- Sufficient conditions for the turnpike property are established.
- Simpler proofs for existing turnpike results are provided.

## Abstract

This paper presents, using dynamical system theory, a framework for investigating the turnpike property in nonlinear optimal control. First, it is shown that a turnpike-like property appears in general dynamical systems with hyperbolic equilibrium and then, apply it to optimal control problems to obtain sufficient conditions for the turnpike occurs. The approach taken is geometric and gives insights for the behaviors of controlled trajectories, allowing us to find simpler proofs for existing results on the turnpike properties. Attempts to remove smallness restrictions for initial and target states are also discussed based on the geometry of (un)stable manifold and exponential stabilizability of control systems.