On the Decidability of the Reachability Problem for Planar Differential Inclusions

In this paper we develop an algorithm for solving the reachability problem of two-dimensional piece-wise rectangular differential inclusions. Our procedure is not based on the computation of the reach-set but rather on the computation of the limit of individual trajectories. A key idea is the use of one-dimensional affine PoincarE maps for which we can easily compute the fixpoints. As a first step, we show that between any two points linked by an arbitrary trajectory there always exists a trajectory without self-crossings. Thus, solving the reachability problem requires considering only those. We prove that, indeed, there are only finitely many "qualitative types" of those trajectories. The last step consists in giving a decision procedure for each of them. These procedures are essentially based on the analysis of the limits of extreme trajectories. We illustrate our algorithm on a simple model of a swimmer spinning around a whirlpool.