Solving the linear quadratic regulator problem in the policy space: the Policy Algebraic Riccati Equation

The linear quadratic regulator problem is solved here within the policy space alone. To this end, a quadratic algebraic equation is envisioned to comprise only the entries of the optimal feedback matrix as unknown variables. As a consequence, the resulting equation contains, in general, much fewer quadratic equations than the corresponding Algebraic Riccati Equation and it permits the characterization of all the symmetric solutions of the latter. Furthermore, it is shown that the derived policy equation is amenable for an iterative solution via Newton's method, which yields updated values identical to those obtained via the celebrated Kleinman's algorithm.