Stabilization and optimization of switched linear systems

In this paper, we establish the equivalence among switched convergency, asymptotic stabilizability, and exponential stabilizability for force-free switched linear systems, and discuss the implication to the infinite-time horizon optimal switching problem. We show that, for a general cost function under mild assumptions, the finiteness of the optimal cost is equivalent to the asymptotic stabilizability of the switched linear system. Finally, we prove the equality between the optimal costs for the switched system and for the relaxed differential inclusion.