Turnpike in Lipschitz—nonlinear optimal control

We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying system. Our strategy combines the construction of quasi-turnpike controls via controllability, and a bootstrap argument, and does not rely...

Full description

Saved in:
Bibliographic Details
Published in:Nonlinearity 2022-04, Vol.35 (4), p.1652-1701
Main Authors: Esteve-Yagüe, Carlos, Geshkovski, Borjan, Pighin, Dario, Zuazua, Enrique
Format: Article
Language:English
Subjects:
deep learning
heat equation
neural ODEs
optimal control
ResNets
turnpike property
wave equation
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying system. Our strategy combines the construction of quasi-turnpike controls via controllability, and a bootstrap argument, and does not rely on analyzing the optimality system or linearization techniques. This in turn allows us to address several optimal control problems for finite-dimensional, control-affine systems with globally Lipschitz (possibly nonsmooth) nonlinearities, without any smallness conditions on the initial data or the running target. These results are motivated by applications in machine learning through deep residual neural networks, which may be fit within our setting. We show that our methodology is applicable to controlled PDEs as well, such as the semilinear wave and heat equation with a globally Lipschitz nonlinearity, once again without any smallness assumptions.
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/ac4e61