The Exponential Turnpike Property for Neural Differential Equations

We consider an optimal control problem where the tracking term in the objective function is related to a weighted $H^{1}$-norm. To be specific, we study a system that is governed by a neural ordinary differential equation that is related to machine learning. We show that a simultaneous exact controllability property of the system implies an exponential turnpike property for the state. Moreover, we show that also the simultaneous exponential stabilizability of the system yields an exponential turnpike property.