D. Properties of Continuous-Time Riccati Equations
D.4 Convergence of Solutions of the CTARE to the Stabilizing Solution of the CTARE
Finally, we show that, under reasonable conditions, the solution of the CTDRE will converge to the unique stabilizing solution of the CTARE. In the sequel, we will be particularly interested in the stabilizing solution to the CTARE.
We observe that the eigenvalues of can be grouped so that contains only eigenvalues that lie in the left-half plane. We then have that
given that and are matrices with eigenvalues strictly inside the LHP.
Remark D.1 Actually, provided that is detectable, then it suffices to have in Lemma D.5