Control of variable structure systems in the plane

The qualitative behavior of trajectories of homogeneous bilinear systems in the plane are investigated. The controls are restricted to take values in a discrete set and, as such, a trajectory of the overall system is represented as a concatenation of trajectories of a set of constituent linear systems. The controllability properties of the overall system are related to the existence of opposition points, weak centers and local cyclic behavior. The analysis focuses on the geometry of trajectories of the constituent linear systems. The major result of the paper is a decomposition of the state space (R2) into distinct regions with well defined controllability properties.