Converging Approximations of Attractors via Almost Lyapunov Functions and Semidefinite Programming

In this letter we combine two existing approaches for approximating global attractors. One of them approximates the global attractors arbitrarily well by sublevel sets related to solutions of infinite dimensional linear programming problems. A downside there is that these sets are not necessarily positively invariant. On the contrary, the second method provides supersets of the global attractor which are positively invariant. Their method on the other hand has the disadvantage that the underlying optimization problem is not computationally tractable without the use of heuristics – and incorporating them comes at the price of losing guaranteed convergence. In this letter we marry both approaches by combining their techniques and we get converging outer approximations of the global attractor consisting of positively invariant sets based on convex optimization via sum-of-squares techniques. The method is easy to use and illustrated by numerical examples.