A delta-sampling verification theorem for discrete-time, possibly discontinuous systems

This paper considers the problem of safety verification for discrete-time, possibly discontinuous dynamical systems. Typical solutions rely on finding invariant sets or Lyapunov functions and require solving optimization problems, which suffer from scalability and numerical solvers issues. Recently, a δ-sampling method for verifying invariance for Lipschitz continuous dynamics was proposed, which does not make use of optimization. In this work we present a δ-sampling verification theorem that extends the previous result to general discrete-time, possibly discontinuous dynamics. This opens up the application of δ-sampling verification to hybrid systems. Moreover, this paper proposes verification of stability on a set by jointly verifying (finite-step) Lyapunov type functions on an annulus with a (finite-step) Lyapunov function on the inner hole. We further indicate that δ-sampling can also be used to verify Lyapunov conditions on the annulus. Lastly, we employ finite-step invariant sets and finite-step Lyapunov functions, respectively, together with δ-sampling to achieve more practical safety verification methods.