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.

中文翻译：

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最优状态反馈的深度神经网络表示的稳定性分析


最近的工作表明，确定性非线性自治系统的最佳状态反馈可以通过深度神经网络来近似。在本文中，我们考虑由最优反馈的网络表示控制的非线性系统的稳定性。首先，我们证明稳定性理论的主要方法很容易应用。然后，我们提出了一种基于微分代数技术的新方法来研究标称轨迹相对于初始条件扰动的鲁棒性。据我们所知，微分代数技术首次被证明可以对一般非线性系统、特别是神经控制系统的运动稳定性进行高阶分析。我们在四轴飞行器最优控制的二维情况下举例说明了所提出的方法，并针对不同的神经网络架构进行了演示。