Sliding mode on tangential sets of Filippov systems

TL;DR

This paper studies the behavior of piecewise smooth vector fields on manifolds where both fields are tangent, introducing a unique sliding mode called the tangential sliding vector field, and relates it to singular perturbation dynamics.

Contribution

It characterizes the conditions for the tangential sliding vector field and links it to reduced dynamics of a regularized singular perturbation problem.

Findings

Provides necessary and sufficient conditions for the tangential sliding vector field.

Shows conjugation of the sliding vector field to reduced singular perturbation dynamics.

Includes examples such as a model for intermittent HIV treatment.

Abstract

We consider piecewise smooth vector fields $Z=(Z_+, Z_-)$ defined in $\mathbb{R}^n$ where both vector fields are tangent to the switching manifold $\Sigma$ along a submanifold $M\subset \Sigma$. We shall see that, under suitable assumptions, Filippov convention gives rise to a unique sliding mode on $M$, governed by what we call the {\it 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.