  This work presents a method to obtain inner and outer approximations of the
region of attraction of a given target set as well as an admissible controller
generating the inner approximation. The method is applicable to constrained
polynomial dynamical systems and extends to trigonometric and rational systems.
The method consists of three steps: compute outer approximations, extract a
polynomial controller while guaranteeing the satisfaction of the input
constraints, compute inner approximations with respect to the closed-loop
system with this controller. Each step of the method is a convex optimization
problem, in fact a semidefinite program consisting of minimizing a linear
function subject to linear matrix inequality (LMI) constraints. The inner
approximations are positively invariant provided that the target set is
included in the inner approximation and/or is itself invariant. %The approach
readily extends to trigonometric dynamics and/or constraints.
