The authors analyse a rational Krylov subspace projection method to solve the continuous time algebraic Lyapunov equation for for large scale control applications. The method finds an approximate solution on a significantly smaller space by imposing a Galerkin condition. The method’s convergence is studied and it is compared to the alternating direction implicit (ADI) method theoretically and numerically. The experiments show that this method is superior when the optimal parameters for ADI cannot be found.