A new proof of a theorem on optimal control of switched systems

Recently Sorin C. Bengea and Raymond A. DeCarlo [Sorin C. Bengea, Raymond A. DeCarlo, Optimal control of switching systems, Automatica J. IFAC 41 (2005) 11–27] have offered a key result that the set of trajectories of the two-switched system is dense in the set of trajectories of the embedded system. This result was proven by means of relaxed controls and the Chattering Lemma. In this paper we use the Lyapunov theorem to give a new simple proof.