Sliding Mode on Tangential Sets of Filippov Systems

We consider piecewise smooth vector fields \(Z=(Z_+, Z_-)\) $Z=(Z_{+},Z_{-})$ defined in \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ where both vector fields are tangent to the switching manifold \(\Sigma \) $\mathrm{\Sigma }$ along a submanifold \(M\subset \Sigma \) $M\subset \mathrm{\Sigma }$ . 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.