Invariant manifolds and projective combinations of solutions of the Riccati differential equation

Abstract

In this paper, we show how families of solutions of the general Riccati differential equation (RDE) can be generated via projective combinations of a given number of reference solutions. Our approach is based upon the extension of the domain of the equation to the Grassmannian manifold and the application of the Radon Lemma. In this context, we briefly discuss the relevance of our results to the study of the invariant manifolds of the equation and compare them to existing results concerning representation formulas for solutions of (RDE). The results of the paper have been motivated by the recent characterization of solutions of the (RDE) given in M. Pavon, D. D'Alessandro, Families of solution of matrix Riccati differential equations, SIAM J. Control Optim., 35(1) (1997) 194–204, which extends the classical results on the algebraic Riccati equation due to Willems, Coppel and Shayman.