Turnpike in Lipschitz-nonlinear optimal control

TL;DR

This paper proves a new version of the turnpike property for nonlinear optimal control problems with Lipschitz nonlinearities, applicable to finite-dimensional control-affine systems and certain PDEs, without smallness constraints.

Contribution

It introduces a novel proof technique for the turnpike property that avoids linearization, enabling analysis of broader nonlinear control problems including neural networks and PDEs.

Findings

Established turnpike property for Lipschitz nonlinear systems

Applicable to neural network training and PDE control problems

No smallness conditions required on initial data or targets

Abstract

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.