  We characterize the maximum controlled invariant (MCI) set for discrete- as
well as continuous-time nonlinear dynamical systems as the solution of an
infinite-dimensional linear programming problem. For systems with polynomial
dynamics and compact semialgebraic state and control constraints, we describe a
hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations
whose optimal values converge to the volume of the MCI set; dual to these LMI
relaxations are sum-of-squares (SOS) problems providing a converging sequence
of outer approximations to the MCI set. The approach is simple and readily
applicable in the sense that the approximations are the outcome of a single
semidefinite program with no additional input apart from the problem
description. A number of numerical examples illustrate the approach.
