On the Decidability of the Reachability Problem for Planar Differential Inclusions

SPeeDI — A Verification Tool for Polygonal Hybrid Systems

“…The tool implements the algorithm published in [6] which is based on the analysis of a finite number of qualitative behaviors generated by a discrete dynamical system characterized by positive affine Poincaré maps. Since the number of such behaviors may be extremely big, the tool uses several powerful heuristics that exploit the topological properties of planar trajectories for considerably reducing the set of actually explored signatures.…”

“…The SPDI isẋ ∈ ∠ bP aP for x ∈ P , where ∠ b a denotes the angle on the plane between the vectors a and b. In [6] we have proved that (point-to-point, edge-to-edge and polygon-to-polygon) reachability is decidable and we have proposed a decision procedure that exploits the topological properties of the plane. Our procedure is not based on the computation of the reach-set but rather on the exploration of a finite number of types of qualitative behaviors obtained from the edge-signatures of trajectories (i.e., the sequences of their intersections with the edges of the polygons).…”

Relaxing Goodness Is Still Good

“…However, it was proved that edges which are entry-only in one region, and exit-only in the other result in matching induced directions: e ∈ E d (H) or e −1 ∈ E d (H), but not both [ASY01,MP93]. In an SPDI satisfying goodness, the only case where one can have both e and e −1 is when e is an entry-only (or exit-only) edge in both adjacent regions it belongs to.…”

“…The following analysis, taken from [ASY01], allows us to calculate the behaviour of cycles provided that the path along the cycle has a normal (not inverted) TAMF. Since, in good SPDIs, the TAMF between a pair of edges is normal, and the composition of two normal TAMFs is itself a normal TAMF, this approach is universally applicable as long as the goodness assumption holds.…”

“…For example, exact solutions can be difficult to obtain for a nonlinear differential equation, making a qualitative and approximative analysis necessary. An interesting class of hybrid systems for which the reachability question is known to be decidable, is the class of Polygonal Hybrid Systems (SPDIs) -a subclass of hybrid systems on the plane whose dynamics is defined by constant differential inclusions [ASY01,ASY07]. Informally, an SPDI consists of a partition of the plane into polygonal regions, each of which enforces different dynamics given by two vectors determining the possible directions a trajectory might take; a simple SPDI is depicted in Fig.…”