Under-Approximating Backward Reachable Sets by Polytopes

Under-approximations are useful for falsification of safety properties for nonlinear (hybrid) systems by finding counter-examples. Polytopic under-approximations enable analysis of these properties using reasoning in the theory of linear arithmetic. Given a nonlinear system, a target region of the simply connected compact type and a time duration, we in this paper propose a method using boundary analysis to compute an under-approximation of the backward reachable set. The under-approximation is represented as a polytope. The polytope can be computed by solving linear program problems. We test our method on several examples and compare them with existing methods. The results show that our method is highly promising in under-approximating reachable sets. Furthermore, we explore some directions to improve the scalability of our method.