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This paper considers the problem of approximating the "maximal" region ofattraction (the set that contains all asymptotically stable sets) of any givenset of locally exponentially stable nonlinear Ordinary Differential Equations(ODEs) with a sufficiently smooth vector field. Given a locally exponentialstable ODE with a differentiable vector field, we show that there exists aglobally Lipschitz continuous converse Lyapunov function whose 1-sublevel setis equal to the maximal region of attraction of the ODE. We then propose asequence of d-degree Sum-of-Squares (SOS) programming problems that yields asequence of polynomials that converges to our proposed converse Lyapunovfunction uniformly from above in the L1 norm. We show that each member of thesequence of 1-sublevel sets of the polynomial solutions to our proposedsequence of SOS programming problems are certifiably contained inside themaximal region of attraction of the ODE, and moreover, we show that thissequence of sublevel sets converges to the maximal region of attraction of theODE with respect to the volume metric. We provide numerical examples ofestimations of the maximal region of attraction for the Van der Pol oscillatorand a three dimensional servomechanism.

Converse Lyapunov Functions and Converging Inner Approximations to Maximal Regions of Attraction of Nonlinear Systems