Regulating discrete-time homogeneous systems under arbitrary switching

We consider discrete-time homogeneous control systems that undergo arbitrary switching. We propose an optimization-based, constructive method (which can be numerically realized) to generate a homogeneous control Lyapunov function and a homogeneous feedback law that stabilizes the origin for all possible switching scenarios. The established stability is robust with respect to small perturbations. We show that for linear systems the resulting Lyapunov function turns out to be convex. We also present a converse Lyapunov result where we state the equivalence of controllability to the origin and existence of a control Lyapunov function.