Existence of Caratheodory solutions in nonlinear systems with discontinuous switching feedback controllers

In this note, we consider the existence of a Caratheodory solution in a single-input-single-output nonlinear system with a discontinuous switching feedback controller. The main contribution is to show that if the nonlinear system can be transformed into a global normal form, then we can specify the value of the discontinuous switching feedback controller on the switching hypersurface so that the closed-loop system has a Caratheodory solution.