Asymptotic behaviour of the solution of the projection Riccati differential equation

The solution of the Riccati differential equation (RDE) is shown to be asymptotically close to the solution of the projection Riccati differential equation (PRDE). The asymptotic behaviour of the latter is analyzed in an explicit formula. The almost-periodic asymptote of the solution of the PRDE is computed by an algorithm based upon the concepts of an aperiodic/almost-periodic generator (APG) decomposition of a linear map and unit row-staircase form of a polynomial matrix. The analysis ultimately provides a convergence criterion. In particular, it is shown that the solution of the PRDE always converges in the aperiodic case.