An improved algorithm for the non-iterative solution of the discrete-time finite-horizon LQ control problem with fixed final state

The non-iterative solution through Moore-Penrose inverse which applies to discrete-time finite-horizon LQ optimal control problems with fixed final state is subject to a constraint on the maximal length of the control time interval. This is a consequence of the limitation on the computational power available for processing the generalized inverse of properly constructed matrices. In this work, a computational framework where the dimensionality restriction is completely removed is presented. The core of the proposed algorithm consists in a procedure where the time interval taken into account doubles at each step. This routine guarantees a fast convergence to the solution. Moreover, the solution of the corresponding infinite-horizon problem is retrievable with arbitrary accuracy by setting the final state to zero and welding a sufficient number of arcs. The procedure returns an arbitrarily accurate solution of the infinite-horizon problem, with no additional complications, also when the to-be-controlled system is non-left-invertible.