Stabilization of switched systems via optimal control

We consider switched systems composed of LTI non Hurwitz dynamics and we deal with the problem of computing an appropriate switching law such that the controlled system is globally asymptotically stable. On the basis of previous results in this framework, we first present a method to design a feedback control law that minimizes a LQ performance index when an infinite number of switches is allowed and at least one dynamics is Hurwitz. Then, we show that this approach can be applied to stabilize switched systems whose modes are all unstable, by applying the proposed procedure to a ``dummy'' system, augmented with a stable dynamics. If the system with unstable modes is globally exponentially stabilizable, then our method can provide the feedback control law that minimizes the chosen quadratic performance index, and that guarantees the closed loop system to be globally asymptotically stable.