The nature of the attraction of the solution of the time-invariant matrix Riccati differential equation towards the stabilizing solution of the algebraic Riccati equation is studied. This is done on an explicit formula for the solution when the system is stabilizable and the hamiltonian matrix has no eigenvalues on the imaginary axis. Various aspects of this convergence are analysed by displaying explicit mechanisms of attraction, and connections are made with the literature. The analysis ultimately shows the exponential nature of the convergence of the solution of the Riccati differential equation and of the related finite horizon LQ-optimal state and control trajectories as the horizon recedes. Computable characteristics are given which can be used to estimate the quality of approximating the solution of a large finite-horizon LQ problem by the solution of an infinite-horizon LQ problem.