  In this paper, we consider a control synthesis problem for a class of
polynomial dynamical systems subject to bounded disturbances and with input
constraints. More precisely, we aim at synthesizing at the same time a
controller and an invariant set for the controlled system under all admissible
disturbances. We propose a computational method to solve this problem. Given a
candidate polyhedral invariant, we show that controller synthesis can be
formulated as an optimization problem involving polynomial cost functions over
bounded polytopes for which effective linear programming relaxations can be
obtained. Then, we propose an iterative approach to compute the controller and
the polyhedral invariant at once. Each iteration of the approach mainly
consists in solving two linear programs (one for the controller and one for the
invariant) and is thus computationally tractable. Finally, we show with several
examples the usefulness of our method in applications.
