Computing the domain of attraction for polynomial systems via BMI optimization method

Concerning a time-invariant, autonomous and polynomial system, we propose a new approach to estimate the domain of attraction (DOA) around an asymptotically stable equilibrium. Special emphasis is laid on elaborating the connections between modern results of real algebraic geometry and Lyapunov´s stability theory, namely between the positive definite polynomials and the direct method of Lyapunov. The estimation problem can thereby be reduced to solving a sequence of low non-convexity-rank bilinear matrix inequalities (BMI) optimization problems. The BMI problem enables the calculation of the inner approximation to the relevant region of the DOA. We illustrate the presented approach with an example