Nonrecursive solution method for the linear-quadratic optimal control problem with a singular transition matrix

In the optimal linear regulator problem the control vector is usually determined by solving the algebraic matrix Riccati equation using successive substitutions. This, however, can be rather inefficient from a computational point of view. A nonrecursive method which requires that the transition matrix is nonsingular has been proposed by Vaughan (1970). In the present paper we present a nonrecursive solution to the matrix Riccati equation for the case that the transition matrix may be singular. We show that this procedure leads to the same numerical results as the standard iteration of the matrix Riccati equation.