Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation

We consider piecewise smooth vector fields (PSVF) defined in open sets $M\subseteq R^n$ with switching manifold being a smooth surface $\mathrm{\Sigma }$. The PSVF are given by pairs $X=(X_{+},X_{-})$, with $X=X_{+}$ in ${\mathrm{\Sigma }}_{+}$ and $X = X_-$in$\Sigma_-$where$\Sigma _+$and$\Sigma _-$aretheregionson$M$separatedby$\Sigma.$Aregularizationof$X$ is a 1-parameter family of smooth vector fields $X^{ϵ},ϵ>0,$ satisfying that $X^{ϵ}$ converges pointwise to $X$ on $M\setminus \mathrm{\Sigma }$, when $ϵ\to 0$. Inspired by the Fenichel Theory , the sliding and sewing dynamics on the discontinuity locus $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$ is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. \end{abstract}