Families of Solutions of Matrix Riccati Differential Equations
Michele Pavon, Domenico D’Alessandro · SIAM Journal on Control and Optimization · 1997
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 definiteness, periodicity, or system-theoretic condition. More precisely, given any two solutions $X_1(t)$ and $X_2(t)$ of such equation on a given interval $[t_0,t_1]$, we show how to construct a family of solutions of the same equation of the form $X(t) = (I-\pi(t))X_1(t)+\pi(t)X_2(t)$, where $\pi$ is a suitable matrix-valued function. Even when specialized to the case of $X_1$ and $X_2$ 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 $X_1$, $X_2$ and on the coefficient matrices.