# On Turnpike and Dissipativity Properties of Continuous-Time Optimal Control Problems

arXiv: https://arxiv.org/abs/1509.07315 · HTML: https://mlsanity.com/paper/1509.07315/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Sep 2015.

## Built on by (most cited first, 3 of 3)

- [Turnpike in optimal control of PDEs, ResNets, and beyond](https://mlsanity.com/paper/2202.04097/) (2022)
- [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)

This paper investigates the relations between three different properties, which are of importance in optimal control problems: dissipativity of the underlying dynamics with respect to a specific supply rate, optimal operation at steady state, and the turnpike property. We show in a continuous-time setting that if along optimal trajectories a strict dissipation inequality is satisfied, then this implies optimal operation at this steady state and the existence of a turnpike at the same steady state. Finally, we establish novel converse turnpike results, i.e., we show that the existence of a turnpike at a steady state implies optimal operation at this steady state and dissipativity with respect to this steady state. We draw upon a numerical example to illustrate our findings.