On the stability analysis of deep neural network representations of an optimal state-feedback

Recent works have shown that the optimal state feedback for deterministic, nonlinear autonomous systems can be approximated by deep neural networks. In this article, we consider the stability of nonlinear systems controlled by such a network representation of the optimal feedback. First, we show that principal methods from stability theory readily applies. We then propose a novel method based on differential algebra techniques to study the robustness of a nominal trajectory with respect to perturbations of the initial conditions. It is, to the best of our knowledge, the first time that differential algebraic techniques are shown to allow for the high-order analysis of motion stability for a nonlinear system in general and for a neurocontrolled system in particular. We exemplify the proposed method in the 2-D case of the optimal control of a quadcopter and demonstrate it for different neural network architectures.