On the Kamke–Müller conditions, monotonicity and continuity for bi-modal piecewise-smooth systems

We show that the Kamke–Müller conditions for bimodal piecewise-smooth systems are equivalent to simple conditions on the vector fields defining the system. As a consequence, we show that for a specific class of such systems, monotonicity is equivalent to continuity. Furthermore, we apply our results to derive a stability condition for piecewise positive linear systems.