Turnpike in optimal control of PDEs, ResNets, and beyond

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.

From NeurODEs to AutoencODEs: A mean-field control framework for width-varying neural networks

“…Prior works, such as [20], have extended the control-theoretic analysis of NeurODEs to more general width-varying neural networks. Their model is based on an integro-differential equation that was first suggested in [34] in order to study the continuum limit of neural networks with respect to width and depth.…”

“…In such an equation the state variable has a dependency on both time and space since the changing dimension over time is viewed as an additional spatial variable. In [20,Section 6] the continuous space-time analogue of ResNets proposed in [34] has been considered and discretized in order to model variable width ResNets of various types, including convolutional neural networks. The authors assume a simple time-dependent grid, and use forward difference discretization for the time derivative and Newton-Cotes for discretizing the integral term, but refer to more sophisticated moving grids in order to possibly propose new types of architectures.…”

Optimal operation of dielectric elastomer wave energy converters under harmonic and stochastic excitation

“…The state-control pairs, on which the average stage costs are the lowest, define a turnpike. Optimal solutions approach this steady-state and dwell there for as long as boundary conditions and the time horizon given within the optimal control problem (OCP) allow for [10,27].…”

The turnpike property for mean-field optimal control problems

“…An early reference to the turnpike property is [36], and in [20,41,42], overviews on discrete-time and continuous-time turnpike properties are given. The turnpike phenomenon for systems governed by ordinary differential equations has been studied also in detail in [27,39,40]. Measure and integral turnpike properties have been studied in [38].…”