This work presents a method to obtain inner and outer approximations of theregion of attraction of a given target set as well as an admissible controllergenerating the inner approximation. The method is applicable to constrainedpolynomial dynamical systems and extends to trigonometric and rational systems.The method consists of three steps: compute outer approximations, extract apolynomial controller while guaranteeing the satisfaction of the inputconstraints, compute inner approximations with respect to the closed-loopsystem with this controller. Each step of the method is a convex optimizationproblem, in fact a semidefinite program consisting of minimizing a linearfunction subject to linear matrix inequality (LMI) constraints. The innerapproximations are positively invariant provided that the target set isincluded in the inner approximation and/or is itself invariant. %The approachreadily extends to trigonometric dynamics and/or constraints.
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