Exponentially contractive invariant sets for discrete-time switching linear systems

By considering arbitrary Holder p-norms in the state space, the current paper studies the existence of exponentially contractive sets that are invariant with respect to the trajectories of discrete-time switching linear systems. We first provide necessary and sufficient conditions for testing if state-space sets (characterized by certain attributes such as shape, scaling factor and decreasing rate) are invariant with respect to the trajectories of a given switching system. Next, we synthesize state feedbacks so as the sets characterized by given attributes are invariant with respect to the trajectories of the closed-loop switching system. For the usual Holder p -norms, corresponding to p∈{1,2,∞}, we also discuss the numerical tractability of the theoretical results. A numerical example is included for practical illustration.