We study discontinuous differential-algebraic equations (DDAEs) with a co-dimension 1 discontinuity manifold $\Sigma$. Our main objectives are to give sufficient conditions that allow to extend the DAE along $\Sigma$ and, when this is possible, to define sliding motion (the sliding DAE) on $\Sigma ,$ extending Filippov construction to this DAE case. Our approach is to consider discontinuous ODEs associated to the DDAE and apply Filippov theory to the discontinuous ODEs, defining sliding/crossing solutions of the DDAE to be those inherited by the sliding/crossing solutions of the associated discontinuous ODEs. We will see that, in general, the sliding DAE on $\Sigma$ is not defined unambiguously. When possible, we will consider in greater details two different methods based on Filippov’s methodology to arrive at the sliding DAE. We will call these the direct approach and the Singular Perturbation Approach and we will explore advantages and disadvantages of each of them. We illustrate our development with numerical examples.

我们研究具有一维不连续流形的不连续微分代数方程（DDAE） $\Sigma$。我们的主要目标是提供充分的条件，以允许将DAE扩展到$\Sigma$ 并在可能的情况下在以下位置定义滑动运动（滑动DAE）： $\Sigma ，$将Filippov结构扩展到此DAE案例。我们的方法是考虑与DDAE相关的不连续ODE，并将Filippov理论应用于不连续ODE，将DDAE的滑动/交叉解决方案定义为相关联的不连续ODE的滑动/交叉解决方案所继承的解决方案。通常，我们会看到滑动DAE$\Sigma$没有明确定义。在可能的情况下，我们将更详细地考虑基于Filippov的方法的两种不同方法，以得出滑动DAE。我们将这些称为直接方法和奇异摄动方法，并探讨每种方法的优缺点。我们通过数值示例来说明我们的发展。