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In this paper we combine two existing approaches for approximatingattractors. One of them approximates the attractors arbitrarily well bysublevel sets related to solutions of infinite dimensional linear programmingproblems. A downside there is that these sets are not necessarily positivelyinvariant. On the contrary, the second method provides supersets of theattractor which are positively invariant. Their method on the other hand hasthe disadvantage that the underlying optimization problem is notcomputationally tractable without the use of heuristics - and incorporatingthem comes at the price of losing guaranteed convergence. In this paper wemarry both approaches by combining their techniques and we get converging outerapproximations of the attractor consisting of positively invariant sets basedon convex optimization via sum-of-squares techniques. The method is easy to useand illustrated by numerical examples.

Converging approximations of attractors via almost Lyapunov functions and semidefinite programming