A set valued characterization of ISDS Lyapunov functions

We use set valued analysis techniques in order to characterize Lyapunov functions for the input–to–state dynamical stability (ISDS) property, a quantitatively sharper but qualitatively equivalent variant of the well known input-to–state stability (ISS) property. We show that the epigraphs of minimal ISDS Lyapunov functions are invariance kernels of a suitable augmented differential inclusion. This identity provides theoretical insight into local ISDS properties and yields a basis for a numerical approximation of ISDS and ISS Lyapunov functions via set oriented numerical methods.