  These notes present preliminary results regarding two different
approximations of linear infinite-horizon optimal control problems arising in
model predictive control. Input and state trajectories are parametrized with
basis functions and a finite dimensional representation of the dynamics is
obtained via a Galerkin approach. It is shown that the two approximations
provide lower, respectively upper bounds on the optimal cost of the underlying
infinite dimensional optimal control problem. These bounds get tighter as the
number of basis functions is increased. In addition, conditions guaranteeing
convergence to the cost of the underlying problem are provided.
