Stratified discontinuous differential equations and sufficient conditions for robustness (Q255847)

The paper deals with the state-constrained discontinuous (stratified) ODE NEWLINE\[NEWLINE\begin{aligned}\dot x&=g_i(x) \text{ for }\,\,\text{a.e.}\,\,\,x\in{\mathcal M}_i,\\ x(0)&=x_0,\end{aligned}NEWLINE\]NEWLINE where \({\mathcal M}_i,\,\,i\in {\mathcal I}\), are the strata of a stratifiable closed set \({\mathcal K}\subseteq \mathbb{R}^N\). For a subset of indeces \({\mathcal I}_0\subseteq {\mathcal I}\) for which \(\{{\mathcal M}_i\}_{i\in {\mathcal I}_0}\) is dense in \({\mathcal K}\), the author introduces the stratified vector field, namely a family of tangent vector fields \(G=\{(g_i,{\mathcal M}_i)\}_{i\in {\mathcal I}_0}\) such that for any \(i\in {\mathcal I}_0\), \(g_i(x)\in {\mathcal T}_{{\mathcal M}_i(x)}\) for all \(x\in {\mathcal M}_i\), where \({\mathcal T}_{{\mathcal M}_i}(x)\) is the tangent space to \({\mathcal M}_i\) at \(x\). He investigates the existence of solutions and the robustness of trajectories with respect to external perturbations on the velocity for the above equation, by using the modulus outward-pointing. Some regularity properties over the stratification are also studied.