  This work presents an algorithmic scheme for solving the infinite-time
constrained linear quadratic regulation problem. We employ an accelerated
version of a popular proximal gradient scheme, commonly known as the
Forward-Backward Splitting (FBS), and prove its convergence to the optimal
solution in our infinite-dimensional setting. Each iteration of the algorithm
requires only finite memory, is computationally cheap, and makes no use of
terminal invariant sets; hence, the algorithm can be applied to systems of very
large dimensions. The acceleration brings in optimal convergence rates O(1/k^2)
for function values and O(1/k) for primal iterates and renders the proposed
method a practical alternative to model predictive control schemes for setpoint
tracking. In addition, for the case when the true system is subject to
disturbances or modelling errors, we propose an efficient warm-starting
procedure, which significantly reduces the number of iterations when the
algorithm is applied in closed-loop. Numerical examples demonstrate the
approach.
