Convex enclosures for the reachable sets of nonlinear parametric ordinary differential equations

This article describes convex enclosures for the reachable sets of nonlinear parametric ordinary differential equations (ODEs). Such equations arise from the control parametrization of nonlinear control systems, and the computation of reachable sets for such systems is central to many control and verification problems. The method presented computes convex and concave relaxations of the ODE solutions with respect to the parameters, which are then used to compute a convex enclosure of the reachable set. This enclosure is expressed as an infinite intersection of halfspaces, so that a convex polyhedral enclosure of the reachable set can be computed by considering any finite m of these halfspaces. The method requires the solution of m dynamic optimization problems which are guaranteed to be convex, even when the reachable set is itself nonconvex.