Instability of dwell-time constrained switched nonlinear systems

Analyzing the stability of switched nonlinear systems under dwell-time constraints, this article investigates different scenarios where all the subsystems have a common globally asymptotically stable (GAS) equilibrium, but for the switched system, the equilibrium is not uniformly GAS for arbitrarily large values of dwell-time. We motivate our study with the help of examples showing that, if near the origin all the vector fields decay at a rate slower than the linear vector fields, then the trajectories are ultimately bounded for large enough dwell-time. On the other hand, if away from the origin, the vector fields do not grow as fast as the linear vector fields, then we can only guarantee local asymptotic stability for large enough dwell-times, with region of attraction depending on the dwell-time itself. We formalize our observations for homogeneous systems, and show that, even if the origin is not uniformly GAS with dwell-time switching for nonlinear systems, it still holds that the trajectories starting from a bounded set converge to a neighborhood of the origin if the dwell-time is large enough.