Optimal control of neural differential equations: The turnpike property

Consider a system that is governed by a neural differential equation.Such systems model deep neural networks with continuous time. For systems of this type, we study an optimal control problem where the objective function consists of a tracking term that is given by the sum of a squared weighted $H^1$-normand a control cost that is given by a squared $L^2$-norm. We prove that a simultaneous exact controllability property of the systemimplies a turnpike property for the optimal statethat depends on the weight in this differentiable objective function.We also show the finite-time turnpike propertyfor the non-differentiable objective function where the tracking term is not squared.Numerical results illustrate our findings.