Discrete-time homogeneous Lyapunov functions for homogeneous difference inclusions

In this paper we consider homogeneity (of discrete-time systems) with respect to generalized dilations, which define a broader class of operators than dilations. The notion of generalized dilations allows us to deal with the stability of attractors that are more general than a single point, which may be unbounded sets. We study homogeneous difference inclusions where every solution passed through a homogeneous measure function is bounded from above by a class-KL estimate in terms of time and the initial state passed through the measure function. We show that for such inclusions, under some generic assumptions, there exists a continuous Lyapunov function that is homogeneous of arbitrary degree and smooth everywhere possibly except at the attractor.