A kinematic smoothing method for tightening convex relaxations of ordinary differential equations

This article presents a new approach for constructing convex enclosures of reachable sets of parametric ordinary differential equations (ODEs), for use in deterministic methods for global dynamic optimization. In our new approach, we modify an established ODE relaxation framework by Scott and Barton (2013), using kinematic intuition to replace certain discontinuous transitions between discrete modes with tighter, smoother transitions, and ultimately producing tighter, smoother relaxations of the original ODE solution that are more amenable to integration by off-the-shelf numerical ODE solvers. We refer to our new relaxation approach as “kinematic smoothing”. Our new ODE relaxations are straightforward to construct automatically based on established tools, and we present several numerical examples based on a proof-of-concept implementation in Julia.