Decoupling Order Reduction via the Riccati Transformation

A program that dates back to Riccati [1724] and that seeks to replace a given differential equation with an equivalent collection of effectively uncoupled equations of lower orders through the use of suitable transformations has seen a notable advance in the last twenty years. The modern form of this Riccati transformation is useful partly because of the resulting order reduction, but more importantly it provides a tool for the decoupling of linear systems by block diagonalization which has proved useful as an analytical tool in many areas of applications. The present paper discusses applications of the transformation in optimal control theory, singular perturbation theory, and asymptotic approximation theory for differential equations. In the optimal control problem the decoupling permits the replacement of a two-point boundary value problem with an associated pair of simpler Cauchy problems. In the singular perturbation problem the Riccati transformation permits the decoupling of the slow and fast variables. Similarly, in the asymptotic integration of linear systems the transformation permits the decoupling of the increasing and decreasing solutions so as to provide Liouville–Green approximations. The results obtained are for the most part well known although novel results are obtained in some cases, and in every case the approach via the Riccati transformation is appealing due to its simplicity and directness.