Families of solutions of matrix riccati differential equations

The J. C. Willems-Coppel-Shayman geometric characterization of solutions of the algebraic Riccati equation (ARE) is extended to asymmetric Riccati differential equations with time-varying coefficients. The coefficients do not need to satisfy any definite-ness, 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 - π(t))X₁(t) + π(t)X₂(t), where π 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.