Georgia Institute of Technology College of Sciences

In this talk we examine the typical behavior of a trajectory of a piecewise smooth system in the neighborhood of a co-dimension 2 discontinuity manifold $\Sigma$. It is well known that (in the class of Filippov vector fields, and under commonly occurring conditions) one may anticipate sliding motion on $\Sigma$. However, this motion itself is not in general uniquely defined, and recent contributions in the literature have been trying to resolve this ambiguity either by justifying a particular selection of a Filippov vector field or by substituting the original discontinuous problem with a regularized one. However, in this talk, our concern is different: we look at what we should expect of a typical solution of the given discontinuous system in a neighborhood of $\Sigma$. Our ultimate goal is to detect properties that are satisfied by a sufficiently wide class of discontinuous systems and that (we believe) should be preserved by any technique employed to define a sliding solution on $\Sigma$.