Piecewise Linear Systems with Closed Sliding Poly-Trajectories

Abstract

In this paper we study piecewise linear (PWL) vector fields $$ F(x,y)=\left\{\begin{array}{l} F^+(x,y)\quad $if$ \quad x\geq0, \\ F^-(x,y)\quad $if$ \quad x\leq0,\end{array}\right. $$ where $\mathrm{x}=(x,y)\in \mathbb{R}^2$, $F^{+}(\mathrm{x})=A^{+}\mathrm{x}+b^{+}$ and $F^{-}(\mathrm{x})=A^{-}\mathrm{x}+b^{-}$, $A^{+}=(a_{ij}^{+})$ and $A^{-}=(a_{ij}^{-})$ are $(2\times2)$ constant matrices, $b^{+}=(b_{1}^{+},b_{2}^{+})\in\mathbb{R}^2$ and $b^{-}=(b_{1}^{-},b_{2}^{-})\in\mathbb{R}^2$ are constant vectors in $\mathbb{R}^2$. We suppose that the equilibrium points are saddle or focus in each half-plane. We establish a correspondence between the PWL vector fields and vectors formed by some of the following parameters: sets on $\Sigma$ (crossing, sliding or escaping), kind of equilibrium (real or virtual), intersection of manifolds with $\Sigma$, stability and orientation of the focus. Such vectors are called configurations. We reduce the number of configurations by an equivalent relation. Besides, we analyze for which configurations the corresponding PWL vector fields can have or not closed sliding poly-trajectories.