Convergence of the Time-Invariant Riccati Differential Equation towards Its Strong Solution for Stabilizable Systems

We prove a necessary and sufficient condition for the solution of the time-invariant Riccati differential equation to converge towards the strong solution of the corresponding algebraic Riccati equation, when the system is stabilizable, without assuming that the Hamiltonian matrix has no eigenvalues on the imaginary axis (or equivalently that there are no critical unobservable modes). The condition is a generalization of an earlier one established by Callier and Willems. Our proof revises an earlier one by Faurre, Clerget, and Germain, leading to additional information of what can be assumed without loss of generality in our context. It is also shown that the convergence is not always exponential and that the presence of critical unobservable modes may slow down but does not prevent the convergence of the solution of the Riccati differential equation. The impact of the condition on linear-quadratic optimal control is briefly discussed.