Advances in computational Lyapunov analysis using sum-of-squares programming

The stability of an equilibrium point of a nonlinear dynamical system is typically determined using Lyapunov theory. This requires the construction of an energy-like function, termed a Lyapunov function, which satisfies certain positivity conditions. Unlike linear dynamical systems, there is no algorithmic method for constructing Lyapunov functions for general nonlinear systems. However, if the systems of interest evolve according to polynomial vector fields and the Lyapunov functions are constrained to be sum-of-squares polynomials then stability verification can be cast as a semidefinite (convex) optimization programme. In this paper we describe recent advances in sum-of-squares programming that facilitate advanced stability analysis and control design.