Reachability of affine systems on polytopes in the plane

In this paper a problem related to reachability analysis of piecewise-affine hybrid systems is considered. We focus on one discrete mode of a hybrid system, and study the continuous dynamics in this mode, described by an affine autonomous system on a polytope. As soon as the continuous state leaves the polytope, a discrete event is triggered, transferring the hybrid system to a different mode. This discrete event depends on the facet through which the polytope is left. In this paper, the domains of attraction of these so-called exit facets are determined. This result describes the interplay between the continuous and discrete dynamics of a hybrid system, and may be useful in some approaches to reachability analysis for hybrid systems, proposed in the literature. The method presented in this paper is restricted to affine systems on two-dimensional polytopes. Since some results are based on argumentsfrom planar geometry, generalization of the approach to higher dimension srequires a rather extensive and detailed mathematical study that does not at all seem promising