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We characterize the maximum controlled invariant (MCI) set for discrete- aswell as continuous-time nonlinear dynamical systems as the solution of aninfinite-dimensional linear programming problem. For systems with polynomialdynamics and compact semialgebraic state and control constraints, we describe ahierarchy of finite-dimensional linear matrix inequality (LMI) relaxationswhose optimal values converge to the volume of the MCI set; dual to these LMIrelaxations are sum-of-squares (SOS) problems providing a converging sequenceof outer approximations to the MCI set. The approach is simple and readilyapplicable in the sense that the approximations are the outcome of a singlesemidefinite program with no additional input apart from the problemdescription. A number of numerical examples illustrate the approach.

Convex computation of the maximum controlled invariant set for polynomial control systems