# Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results

arXiv: https://arxiv.org/abs/2011.13670 · HTML: https://mlsanity.com/paper/2011.13670/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Nov 2020.

## Built on by (most cited first, 2 of 2)

- [On the Turnpike to Design of Deep Neural Nets: Explicit Depth Bounds](https://mlsanity.com/paper/2101.03000/) (2021)
- [On Dissipativity of Cross-Entropy Loss in Training ResNets](https://mlsanity.com/paper/2405.19013/) (2024)

## Abstract (arXiv)

The turnpike property refers to the phenomenon that in many optimal control problems, the solutions for different initial conditions and varying horizons approach a neighborhood of a specific steady state, then stay in this neighborhood for the major part of the time horizon, until they may finally depart. While early observations of the phenomenon can be traced back to works of Ramsey and von Neumann on problems in economics in 1928 and 1938, the turnpike property received continuous interest in economics since the 1960s and recent interest in systems and control. The present chapter provides an introductory overview of discrete-time and continuous-time results in finite and infinite-dimensions. We comment on dissipativity-based approaches and infinite-horizon results, which enable the exploitation of turnpike properties for the numerical solution of problems with long and infinite horizons. After drawing upon numerical examples, the chapter concludes with an outlook on time-varying, discounted, and open problems.