A Kleinman–Newton construction of the maximal solution of the infinite-dimensional control Riccati equation

Assuming only strong stabilizability, we construct the maximal solution of the algebraic Riccati equation as the strong limit of a Kleinman–Newton sequence of bounded nonnegative operators. As a corollary we obtain a comparison of the solutions of two algebraic Riccati equations associated with different cost functions. We show that the weaker strong stabilizability assumptions are satisfied by partial differential systems with collocated actuators and sensors, so the results have potential applications to numerical approximations of such systems. By means of a counterexample, we illustrate that even if one assumes exponential stabilizability, the Kleinman–Newton construction may provide a solution to the Riccati equation that is not strongly stabilizing.