Analyzing continuous switching systems: theory and examples

This paper details work on ordinary differential equations that continuously switch among regimes of operation. In the first part, we develop some tools for analyzing such systems. We prove an extension of Bendixson's theorem to the case of Lipschitz continuous vector fields. We also prove a lemma dealing with the robustness of differential equations with respect to perturbations that preserve a linear part, which we call the linear robustness lemma (LRL). We then give some simple propositions that allow us to use this lemma in studying certain singular perturbation problems. In the second part, the attention focuses on example systems and their analysis. We use the tools from the first part and develop some general insights. The example systems arise from a realistic aircraft control problem. The extension of Bendixson's theorem and the LRL have applicability beyond the systems discussed in this paper.