Invariant Funnels around Trajectories using Sum-of-Squares Programming

Abstract

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.