In a multistage program without complete recourse, a solution, which is feasible at a particular stage, may lead to an infeasible program for a subsequent stage. When solving the multistage program, infeasibility cuts are usually added to forbid such a solution. However, the number of infeasibility cuts needed to ensure relatively complete recourse can grow rapidly as the number of stages considered increase. We show how to reduce the computation of feasible sets ensuring relatively complete recourse to the computation of controlled invariant sets of an hybrid system. Recent work in the theory of control of cyber-physical systems shows how to compute ellipsoidal controlled invariant sets for these system by using semidefinite programming. The containment of the solution to an ellipsoid only requires second order cone programming and it certifies relatively complete recourse. This allows to precompute new constraints to add to the programs at each stage and scenario to remove the need for infeasibility cuts without significantly increasing the complexity of solving each program.