Motivated by the residual type neural networks (ResNet), this paper studies optimal control problems constrained by a non-smooth integral equation associated to a fractional differential equation. Such non-smooth equations, for instance, arise in the continuous representation of fractional deep neural networks (DNNs). Here the underlying non-differentiable function is the ReLU or max function. The control enters in a nonlinear and multiplicative manner and we additionally impose control constraints. Because of the presence of the non-differentiable mapping, the application of standard adjoint calculus is excluded. We derive strong stationary conditions by relying on the limited differentiability properties of the non-smooth map. While traditional approaches smoothen the non-differentiable function, no such smoothness is retained in our final strong stationarity system. Thus, this work also closes a gap which currently exists in continuous neural networks with ReLU type activation function.

受残差型神经网络（ResNet）的启发，本文研究了与分数阶微分方程相关的非光滑积分方程约束的最优控制问题。例如，这种非光滑方程出现在分数深度神经网络 (DNN) 的连续表示中。这里底层的不可微函数是 ReLU 或 max 函数。控制以非线性和乘法方式进入，我们另外施加控制约束。由于不可微映射的存在，排除了标准伴随微积分的应用。我们依靠非光滑映射的有限可微性特性得出强平稳条件。虽然传统方法平滑了不可微函数，我们最终的强平稳系统中没有保留这种平滑性。因此，这项工作也弥补了目前 ReLU 型激活函数连续神经网络中存在的空白。