Perturbation analysis for Riccati equation

A perturbation analysis is applied to the Riccati equation of the linear quadratic regulator problem. In the limit where the weighting on the control input is zero, it is shown that the asymptotic expansion of the Riccati equation is equivalent to the limiting Riccati equation obtained by using the Goh transformation in the singular optimal control problem. Although the Goh transformation leads to an elegant general form, the perturbation approach is more direct and insightful. Furthermore, the asymptotic behavior of the Riccati equation near the limit is captured by using the perturbation approach