  This paper presents numerical methods for computing regions of finite-time
invariance (funnels) around solutions of polynomial differential equations.
First, we present a method which exactly certifies sufficient conditions for
invariance despite relying on approximate trajectories from numerical
integration. Our second method relaxes the constraints of the first by sampling
in time. In applications, this can recover almost identical funnels but is much
faster to compute. In both cases, funnels are verified using Sum-of-Squares
programming to search over a family of time-varying polynomial Lyapunov
functions. Initial candidate Lyapunov functions are constructed using the
linearization about the trajectory, and associated time-varying Lyapunov and
Riccati differential equations. The methods are compared on stabilized
trajectories of a six-state model of a satellite.
