Turnpike in optimal control of PDEs, ResNets, and beyond
Geshkovski B, Zuazua Iriondo E (2022)
Publication Type: Journal article, Original article
Publication year: 2022
Journal
Book Volume: 31
Pages Range: 135-263
DOI: 10.1017/S0962492922000046
Abstract
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 insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states are close (often exponentially) most of the time, except near the initial and final times, to the optimal control and the 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.
Authors with CRIS profile
Enrique Zuazua Iriondo
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Additional Organisation(s)
Involved external institutions
United States (USA) (US)How to cite
APA:
Geshkovski, B., & Zuazua Iriondo, E. (2022). Turnpike in optimal control of PDEs, ResNets, and beyond. Acta Numerica, 31, 135-263. https://doi.org/10.1017/S0962492922000046
MLA:
Geshkovski, Borjan, and Enrique Zuazua Iriondo. "Turnpike in optimal control of PDEs, ResNets, and beyond." Acta Numerica 31 (2022): 135-263.
BibTeX: Download