Guaranteed estimates of the domain of attraction for a class of hybrid systems

This paper addresses the estimation of the domain of attraction for a class of hybrid nonlinear systems where the state space is partitioned into several regions. Each region is described by polynomial inequalities, and one of these regions is the complement of the union of all the others in order to ensure complete cover of the state space. The system dynamics is defined on each region independently from the others by polynomial functions. The problem of computing the largest sublevel set of a Lyapunov function included in the domain of attraction is considered. An approach is proposed for addressing this problem based on linear matrix inequalities (LMIs), which provides a lower bound of the sought estimate by establishing negativity of the Lyapunov function derivative on each region. Moreover, a sufficient and necessary condition is provided for establishing optimality of the found lower bound. The results are illustrated by some numerical examples.