A Comprehensive Framework to Determine Lyapunov Functions for a Set of Continuous Time Stability Problems

Analyzing the stability of nonlinear systems, with or without external inputs, is still a very challenging task. An efficient way consists in using the Lyapunov theory which can be deployed by searching for a Lyapunov function satisfying constraints depending on the considered stability problem. By showing that a majority of stability problems can be deduced from the Input-to-State Stability (ISS) one, a generic framework is proposed for tackling a catalog of stability analysis problems for continuous time systems. More precisely, it allows the user to determine Lyapunov functions, whose structure can be arbitrarily chosen, and that maximizes the size of the corresponding estimation of the Domain Of Attraction (DOA). This generic work is illustrated by ISS example showing the efficiency of the approach.