This paper analyzes the stability of a direct current microgrid with a decentralized switched control using differential-algebraic equations and Lyapunov functions. The decentralized controllers regulate the voltage, achieve the power-sharing condition, and guaranty the non-Zeno condition. They also fulfill the droop-control condition for optimal power dispatch. The event of either connecting or disconnecting a converter is analyzed as a switched event. It is shown that the system is asymptotically stable under this class of switching events after either connecting or disconnecting a distributed generator.