UC Santa Cruz

Notions and tools for finite time stability of closed sets for hybrid dynamical systems modeled as hybrid inclusions are introduced. Finite time stability of a closed set is defined as the following two properties: Lyapunov stability, namely, the property that solutions that start close to the set stay close to it, and finite time convergence. In the latter property, the amount of time required to converge to the set of interest is captured by a settling-time function that depends on hybrid time, namely, the time elapsed during flows and the number of jumps (or events) of the hybrid system. Various sufficient conditions for such properties to hold for a given closed set are established. Conditions involving Lyapunov-like functions that strictly decrease during flows, that strictly decrease during jumps, and that strictly decrease along both regimes are proposed — these functions are only required to be locally Lipschitz. A link between (non-finite time) asymptotic stability and the proposed notion is also established. In addition, sufficient conditions for finite time attractivity involving properties at jumps of the system, the second derivative of a Lyapunov-like function, and a property of nested sets are provided. Throughout the paper, examples exercise the results. In addition, an application to finite time parameter estimation is provided and results are applied to it.