# Turnpike in optimal control of PDEs, ResNets, and beyond

arXiv: https://arxiv.org/abs/2202.04097 · HTML: https://mlsanity.com/paper/2202.04097/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 104 references that are in the map.
- First posted on arXiv 8 Feb 2022.

## Builds on (most cited first, 30 of 33)

- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [A Tour of Reinforcement Learning: The View from Continuous Control](https://mlsanity.com/paper/1806.09460/) (2018)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Deep learning as optimal control problems: models and numerical methods](https://mlsanity.com/paper/1904.05657/) (2019)
- [Selection dynamics for deep neural networks](https://mlsanity.com/paper/1905.09076/) (2019)
- [The mean field Schr"odinger problem: ergodic behavior, entropy estimates and functional inequalities](https://mlsanity.com/paper/1905.02393/) (2019)
- [The turnpike property in finite-dimensional nonlinear optimal control](https://mlsanity.com/paper/1402.3263/) (2014)
- [Interpolation and approximation via Momentum ResNets and Neural ODEs](https://mlsanity.com/paper/2110.08761/) (2021)
- [On Turnpike and Dissipativity Properties of Continuous-Time Optimal Control Problems](https://mlsanity.com/paper/1509.07315/) (2015)
- [Feedback, Dynamics, and Optimal Control in Climate Economics](https://mlsanity.com/paper/1812.08066/) (2018)
- [Turnpike in Lipschitz-nonlinear optimal control](https://mlsanity.com/paper/2011.11091/) (2020)
- [Ensemble controllability by Lie algebraic methods](https://mlsanity.com/paper/1603.07133/) (2016)
- [Integral and measure-turnpike properties for infinite-dimensional optimal control systems](https://mlsanity.com/paper/1705.02762/) (2017)
- [On the Nonuniqueness and Instability of Solutions of Tracking-Type Optimal Control Problems](https://mlsanity.com/paper/2007.08250/) (2020)
- [Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs](https://mlsanity.com/paper/2008.13001/) (2020)
- [H"older equivalence of the value function for control-affine systems](https://mlsanity.com/paper/1304.6649/) (2013)
- [Complexity of control-affine motion planning](https://mlsanity.com/paper/1309.2571/) (2013)
- [Optimal shape and location of sensors for parabolic equations with random initial data](https://mlsanity.com/paper/1403.6445/) (2014)
- [Greedy controllability of finite dimensional linear systems](https://mlsanity.com/paper/1601.02817/) (2016)
- [The Turnpike Property for Two-Dimensional Navier-Stokes Equations](https://mlsanity.com/paper/1601.04984/) (2016)
- [Optimal shape design for 2D heat equations in large time](https://mlsanity.com/paper/1705.02764/) (2017)
- [Long time behavior of the master equation in mean-field game theory](https://mlsanity.com/paper/1709.04215/) (2017)
- [Controllability under positivity constraints of semilinear heat equations](https://mlsanity.com/paper/1711.07678/) (2017)
- [Phase portrait control for 1D monostable and bistable reaction-diffusion equations](https://mlsanity.com/paper/1805.10786/) (2018)
- [Characterization by observability inequalities of controllability and stabilization properties](https://mlsanity.com/paper/1811.01543/) (2018)
- [Towards Velocity Turnpikes in Optimal Control of Mechanical Systems](https://mlsanity.com/paper/1907.01786/) (2019)
- [Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations](https://mlsanity.com/paper/1912.13066/) (2019)
- [A Dissipativity Characterization of Velocity Turnpikes in Optimal Control Problems for Mechanical Systems](https://mlsanity.com/paper/2002.04388/) (2020)

## Built on by (most cited first, 11 of 11)

- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [Data Selection via Optimal Control for Language Models](https://mlsanity.com/paper/2410.07064/) (2024)
- [Clustering in pure-attention hardmax transformers and its role in sentiment analysis](https://mlsanity.com/paper/2407.01602/) (2024)
- [Cluster-based classification with neural ODEs via control](https://mlsanity.com/paper/2312.13807/) (2023)
- [From NeurODEs to AutoencODEs: a mean-field control framework for width-varying Neural Networks](https://mlsanity.com/paper/2307.02279/) (2023)
- [Hamiltonian Mechanics of Feature Learning: Bottleneck Structure in Leaky ResNets](https://mlsanity.com/paper/2405.17573/) (2024)
- [Training-Induced Escape from Token Clustering in a Mean-Field Formulation of Transformers](https://mlsanity.com/paper/2605.07772/) (2026)
- [Machine Learning and Control: Foundations, Advances, and Perspectives](https://mlsanity.com/paper/2510.03303/) (2025)
- [Convergence, design and training of continuous-time dropout as a random batch method](https://mlsanity.com/paper/2510.13134/) (2025)
- [Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategies](https://mlsanity.com/paper/2608.10738/) (2026)
- [Deep learning from a control vantage point](https://mlsanity.com/paper/2610.14385/) (2026)

## Abstract (arXiv)

The turnpike property in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates the insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states, are close (often exponentially), during most of the time, except near the initial and final time, to the optimal control and corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade --the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal--, and present several novel applications, including, among many others, the characterization of Hamilton-Jacobi-Bellman asymptotics, and stability estimates in deep learning via residual neural networks.