On the stability of boundary equilibria in Filippov systems

Abstract

<p style='text-indent:20px;'>The leading-order approximation to a Filippov system <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> about a generic boundary equilibrium <inline-formula><tex-math id="M2">\begin{document}$ x^* $\end{document}</tex-math></inline-formula> is a system <inline-formula><tex-math id="M3">\begin{document}$ F $\end{document}</tex-math></inline-formula> that is affine one side of the boundary and constant on the other side. We prove <inline-formula><tex-math id="M4">\begin{document}$ x^* $\end{document}</tex-math></inline-formula> is exponentially stable for <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> if and only if it is exponentially stable for <inline-formula><tex-math id="M6">\begin{document}$ F $\end{document}</tex-math></inline-formula> when the constant component of <inline-formula><tex-math id="M7">\begin{document}$ F $\end{document}</tex-math></inline-formula> is not tangent to the boundary. We then show exponential stability and asymptotic stability are in fact equivalent for <inline-formula><tex-math id="M8">\begin{document}$ F $\end{document}</tex-math></inline-formula>. We also show exponential stability is preserved under small perturbations to the pieces of <inline-formula><tex-math id="M9">\begin{document}$ F $\end{document}</tex-math></inline-formula>. Such results are well known for homogeneous systems. To prove the results here additional techniques are required because the two components of <inline-formula><tex-math id="M10">\begin{document}$ F $\end{document}</tex-math></inline-formula> have different degrees of homogeneity. The primary function of the results is to reduce the problem of the stability of <inline-formula><tex-math id="M11">\begin{document}$ x^* $\end{document}</tex-math></inline-formula> from the general Filippov system <inline-formula><tex-math id="M12">\begin{document}$ f $\end{document}</tex-math></inline-formula> to the simpler system <inline-formula><tex-math id="M13">\begin{document}$ F $\end{document}</tex-math></inline-formula>. Yet in general this problem remains difficult. We provide a four-dimensional example of <inline-formula><tex-math id="M14">\begin{document}$ F $\end{document}</tex-math></inline-formula> for which orbits appear to converge to <inline-formula><tex-math id="M15">\begin{document}$ x^* $\end{document}</tex-math></inline-formula> in a chaotic fashion. By utilising the presence of both homogeneity and sliding motion the dynamics of <inline-formula><tex-math id="M16">\begin{document}$ F $\end{document}</tex-math></inline-formula> can in this case be reduced to the combination of a one-dimensional return map and a scalar function.</p>