  We consider the application of the Douglas-Rachford (DR) algorithm to solve
linear-quadratic (LQ) control problems with box constraints on the state and
control variables. We split the constraints of the optimal control problem into
two sets: one involving the ODE with boundary conditions, which is affine, and
the other a box. We rewrite the LQ control problems as the minimization of the
sum of two convex functions. We find the proximal mappings of these functions
which we then employ for the projections in the DR iterations. We propose a
numerical algorithm for computing the projection onto the affine set. We
present a conjecture for finding the costates and the state constraint
multipliers of the optimal control problem, which can in turn be used in
verifying the optimality conditions. We carry out numerical experiments with
two constrained optimal control problems to illustrate the working and the
efficiency of the DR algorithm compared to the traditional approach of direct
discretization.
