Characterizations of input-to-state stability for hybrid systems

By Chaohong Cai, Andrew R. Teel. Systems & control letters (Print), 2009

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Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods (2010, methodology): “Investigation of ISS for hybrid systems can be found in [9].”
Input‐to‐state exponents and related ISS for delayed discrete‐time systems with application to impulsive effects (2018, unknown): “Similarly, we can derive that DDS 1 has IS-e0 (IS-e+) property if β = 0 (β > 0) with the following estimates: ||y(k)|| ⩽ cd̄d−1||φ||τ + ncd̄dk||w||[k0,k−1], k ∈ N, if β = 0; (16) ||y(k)|| ⩽ cd̄d−1||φ||τeβ(k−k0) + ncd̄d(eβ(k−k0) − 1) eβ − 1 ||w||[k0,k−1], k ∈ N, if β > 0.”
Input-to-state stability of impulsive and switching hybrid systems with time-delay (2011, background): “ISS notions for hybrid systems are investigated in Cai and Teel (2005) and Cai and Teel (2009), where the hybrid systems are defined on hybrid time domains.”