UNIQUENESS OF FILIPPOV SLIDING VECTOR FIELD ON THE INTERSECTION OF TWO SURFACES IN R^3 AND IMPLICATIONS FOR STABILITY OF PERIODIC ORBITS

In this paper, we consider the class of sliding Filippov vector fields in R^3 on the intersection of two smooth surfaces: S= 1 2, where S_i = {x : h_i(x) = 0}, and h_i : R^3 -> R, i = 1, 2, are smooth functions with linearly independent normals. Although, in general, there is no unique Filippov sliding vector field on , here we prove that –under natural conditions– all Filippov sliding vector fields determine the same solution trajectory on . In other words, the aforementioned ambiguity has no meaningful impact. We also examine the implication of this result in the important case of a periodic orbit a portion of which slides on S.