## Abstract

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 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 - π(t))X_{1}(t) + π(t)X_{2}(t), where π 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.

Original language | English (US) |
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Pages (from-to) | 194-204 |

Number of pages | 11 |

Journal | SIAM Journal on Control and Optimization |

Volume | 35 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1997 |

## Keywords

- Asymmetric Riccati differential equation
- Families of solutions
- Geometric characterization
- Invariant subspaces
- Projection-preserving differential equation

## ASJC Scopus subject areas

- Control and Optimization
- Applied Mathematics