The linear time-invariant dynamical system \[ \begin{cases} \dot x(t) = Ax(t) + Bu(t),\quad x(0)=x_0,\\ y(t) ~=~ Cx(t), \end{cases} \tag{S} \] where \(A\), \(B\), \(C\) are matrices, is considered. In practice the square matrix \(A\) is \(n \times n\), and \(n\) is very large (of the order \(10^5\) or \(10^6\)). The authors are interested in the feedback control of the system (S), the corresponding cost functional being quadratic in an infinite horizon. Therefore, a new family of low-rank approximations of the solution of the algebraic Riccati equation is introduced. It is based on invariant subspaces of the Hamiltonian matrix. The stabilizing property of the feedback is obtained. In particular, the exact stabilizing solution of the Bernoulli equation is obtained. Numerical examples are presented.