Lyapunov functions for discontinuous difference inclusions

In this work we introduce the notion of finite-step Lyapunov functions for difference inclusions. First, we prove that the existence of a finite-step Lyapunov function is sufficient and necessary for the difference inclusion being KL-stable. Second, we establish four ways of constructing a Lyapunov function from a finite-step Lyapunov function. In contrast to Lyapunov functions obtained via converse Lyapunov theorems, the proposed Lyapunov functions do only depend on a finite sum resp. a maximum over finitely many terms. This crucial property allows to construct Lyapunov functions in principle, and to easily derive structure and regularity properties of the Lyapunov function obtained. Another advantage is that for all globally exponentially stable (GES) difference inclusions, norms are finite-step Lyapunov functions which yields a systematic approach for obtaining Lyapunov functions. Whereas all these finding are shown to hold also for discontinuous difference inclusion, we do also discuss robustness of the difference inclusion via upper semicontinuous Lyapunov functions.