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.

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We consider the continuous-time, neural ordinary differential equation (neural ODE) perspective of deep supervised learning, and study the impact of the final time horizon T in training. We focus on a cost consisting of an integral of the empirical risk over the time interval, and L 1 -parameter regularization. Under homogeneity assumptions on the dynamics (typical for ReLU activations), we prove that any global minimizer is sparse, in the sense that there exists a positive stopping time T * beyond which the optimal parameters vanish. Moreover, under appropriate interpolation assumptions on the neural ODE, we provide quantitative estimates of the stopping time T * , and of the training error of the trajectories at the stopping time. The latter stipulates a quantitative approximation property of neural ODE flows with sparse parameters. In practical terms, a shorter time-horizon in the training problem can be interpreted as considering a shallower residual neural network (ResNet), and since the optimal parameters are concentrated over a shorter time horizon, such a consideration may lower the computational cost of training without discarding relevant information.

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