Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation

Abstract

We consider piecewise smooth vector fields (PSVF) defined in open sets $M\subseteq R^n$ with switching manifold being a smooth surface $Σ$. The PSVF are given by pairs $X = (X_+, X_-)$, with $X = X_+$ in $Σ_+$ and $X = X_-$ in $Σ_-$ where $Σ_+$ and $Σ_-$ are the regions on $M$ separated by $Σ.$ A regularization of $X$ is a 1-parameter family of smooth vector fields $X^ε,ε>0,$ satisfying that $X^ε$ converges pointwise to $X$ on $M\setminusΣ$, when $ε\rightarrow 0$. Inspired by the Fenichel Theory , the sliding and sewing dynamics on the discontinuity locus $Σ$ can be defined as some sort of limit of the dynamics of a nearby smooth regularization $X^ε$. While the linear regularization requires that for every $ε>0$ the regularized field $X^ε$ is in the convex combination of $X_+ $ and $X_- $ the nonlinear regularization requires only that $X^ε$ is in a continuous combination of $X_+ $ and $X_- $. We prove that for both cases, the sliding dynamics on $Σ$ is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. \end{abstract}