Sliding Mode on Tangential Sets of Filippov Systems

We consider piecewise smooth vector fields Z = ( Z + , Z - ) defined in R n where both vector fields are tangent to the switching manifold Σ along a submanifold M ⊂ Σ . We shall see that, under suitable assumptions, Filippov convention gives rise to a unique sliding mode on M , governed by what we call the tangential sliding vector field . Here, we will provide the necessary and sufficient conditions for characterizing such a vector field. Additionally, we prove that the tangential sliding vector field is conjugated to the reduced dynamics of a singular perturbation problem arising from the Sotomayor–Teixeira regularization of Z around M . Finally, we analyze several examples where tangential sliding vector fields can be observed, including a model for intermittent treatment of HIV.