Approximation of viability kernels and capture basins for hybrid systems

This paper deals with hybrid dynamical systems with state constrains and target. We investigate the subset of initial positions from which there exists at least one run forever remaining in the constraint set - the hybrid viability kernel - or remaining in the constraint set until it reaches a given closed target in finite time - the hybrid capture basin. We present an algorithm for approximating those sets and, under some regularity assumptions, we construct viable hybrid feed-backs providing viable runs. One example illustrates this study. It deals with an academic dynamical system revealing the complexity of the structure of the hybrid viability kernel and showing hybrid solutions.