Guaranteed overapproximations of unsafe sets for continuous and hybrid systems: Solving the hamilton-jacobi equation using viability techniques

Abstract

We show how reachable sets of constrained continuous and simple hybrid systems may be computed using the minimum time-toreach function. We present an algorithm for computing a discrete approximation to the minimum time-to-reach function, which we prove to be a converging underapproximation to the actual function. We use the discrete minimum time-to-reach function for simple hybrid systems to compute overapproximations of unsafe zones for aircraft in a sector of the Oakland Air Traffic Control Center, leading to the automatic generation of conflict-free aircraft maneuvers.