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The problem of determining domains of attraction is one of the most important when it comes to investigating nonlinear dynamical systems. In this contribution motivated by recently published theoretical results we suggest algorithms based on the mathematical theory of moments to solve the primal and the dual optimization problems related to determining domains of attraction of polynomial dynamical systems. We illustrate the algorithms with three examples.

Abstract. For bilinear control systems with quadratic cost, the so-called bilinear-quadratic problems, a feedback controller for the finite-time case is designed. An iteration procedure in close proximity to the Riccati approach is presented, and the proof of convergence is outlined. The potential of the new method is discussed, and the design procedure is illustrated for two examples.

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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.