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...

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Bibliographic Details
Published inNonlinearity Vol. 35; no. 4; pp. 1652 - 1701
Main Authors Esteve-Yagüe, Carlos, Geshkovski, Borjan, Pighin, Dario, Zuazua, Enrique
Format Journal Article
LanguageEnglish
Published IOP Publishing 07.04.2022
Subjects
deep learning
heat equation
neural ODEs
optimal control
ResNets
turnpike property
wave equation
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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.
Bibliography:NON-104973.R2
London Mathematical Society
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/ac4e61