A methodology for safety verification of continuous and hybrid systems using barrier certificates has been proposed recently. Conditions that must be satisfied by a barrier certificate can be formulated as a convex program, and the feasibility of the program implies system safety in the sense that there is no trajectory starting from a given set of initial states that reaches a given unsafe region. The dual of this problem, i.e., the reachability problem, concerns proving the existence of a trajectory starting from the initial set that reaches another given set. Using insights from the linear programming duality appearing in the discrete shortest path problem, we show in this paper that reachability of continuous systems can also be verified through convex programming. Several convex programs for verifying safety and reachability, as well as other temporal properties such as eventuality, avoidance, and their combinations, are formulated. Some examples are provided to illustrate the application of the proposed methods. Finally, we exploit the convexity of our methods to derive a converse theorem for safety verification using barrier certificates.
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