Families of solutions of matrix Riccati equations

The J. C. Willems-Coppel-Shayman geometric characterization of solutions of the algebraic Riccati equation (ARE) is extended to asymmetric Riccati dierential equations with time varying coefficients. The coefficients do not need to satisfy any definiteness, periodicity, or system-theoretic condition. More precisely, given any two solutions X1(t) and X2(t) of such equation on a given interval [t0; t1], we show how to construct a family of solutions of the same equation of the form X(t) = (I −(t))X1(t) + (t)X2(t), where is a suitable matrix-valued function. Even when specialized to the case of X1 and X2 equilibrium solutions of a symmetric equation with constant coefficients, our results considerably extend the classical ones, as no further assumption is made on the pair X1, X2 and on the coefficient matrices.