  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.
