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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 \u2013 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 \u2013 and present several novel applications, including, among many others, the characterization of Hamilton\u2013Jacobi\u2013Bellman asymptotics, and stability estimates in deep learning via residual neural networks.<\/jats:p>","DOI":"10.1017\/s0962492922000046","type":"journal-article","created":{"date-parts":[[2022,6,9]],"date-time":"2022-06-09T11:15:48Z","timestamp":1654773348000},"page":"135-263","source":"Crossref","is-referenced-by-count":33,"title":["Turnpike in optimal control of PDEs, ResNets, and beyond"],"prefix":"10.1017","volume":"31","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7890-3352","authenticated-orcid":false,"given":"Borjan","family":"Geshkovski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enrique","family":"Zuazua","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2022,6,9]]},"reference":[{"key":"S0962492922000046_r5","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2015076"},{"key":"S0962492922000046_r176","volume-title":"Optimal Control of Partial Differential Equations: Theory, Methods, and Applications","volume":"112","author":"Tr\u00f6ltzsch","year":"2010"},{"key":"S0962492922000046_r105","doi-asserted-by":"publisher","DOI":"10.1137\/120891174"},{"key":"S0962492922000046_r89","first-page":"245","article-title":"Une remarque sur la stabilisation de certains syst\u00e8mes du deuxi\u00e8me ordre en temps","volume":"46","author":"Haraux","year":"1989","journal-title":"Port. 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