On Filippov solutions of discontinuous DAEs of index 1

We study discontinuous differential-algebraic equations (DDAEs) with a co-dimension 1 discontinuity manifold Σ. Our main objectives are to give sufficient conditions that allow to extend the DAE along Σ and, when this is possible, to define sliding motion (the sliding DAE) on Σ, 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 Σ 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.