We consider flows of ordinary differential equations (ODEs) driven by path differentiable vector fields. Path differentiable functions constitute a proper subclass of Lipschitz functions which admit conservative gradients, a notion of generalized derivative compatible with basic calculus rules. Our main result states that such flows inherit the path differentiability property of the driving vector field. We show indeed that forward propagation of derivatives given by the sensitivity differential inclusions provide a conservative Jacobian for the flow. This allows to propose a nonsmooth version of the adjoint method, which can be applied to integral costs under an ODE constraint. This result constitutes a theoretical ground to the application of small step first order methods to solve a broad class of nonsmooth optimization problems with parametrized ODE constraints. This is illustrated with the convergence of small step first order methods based on the proposed nonsmooth adjoint.

翻译：我们考虑的是由路径不同矢量字段驱动的普通差异方程式(ODEs)的流动。路径不同功能构成利普西茨功能的适当小类,它接受保守的梯度,即符合基本微积分规则的通用衍生物概念。我们的主要结果显示,这种流动继承了驱动矢量字段的路径差异属性。我们确实表明,敏感度差异包含的衍生物的前向传播为流动提供了保守的雅各式。这允许提出非吸附式的辅助方法,该方法可适用于内合成本。这一结果构成理论基础,用于应用小步第一阶方法解决广泛的非移动优化问题,同时使用对等式矢量字段的限制。这与基于拟议的非移动连接的小型第一阶方法的趋同就是例证。