An invariance kernel representation of ISDS Lyapunov functions

We apply set valued analysis techniques in order to characterize the input-to-state dynamical stability (ISDS) property, a variant of the well known input-to-state stability (ISS) property. Using a suitable augmented differential inclusion we are able to characterize the epigraphs of minimal ISDS Lyapunov functions as invariance kernels. This characterization gives new insight into local ISDS properties and provides a basis for a numerical approximation of ISDS and ISS Lyapunov functions via set oriented numerical methods.