Families of Solutions of Algebraic Riccati Equations

TL;DR

This paper investigates the solution structure of homogeneous algebraic Riccati equations with mixed matrices, revealing multiple solution families and providing explicit formulas and conditions for their coverage.

Contribution

It introduces a comprehensive analysis of solution families for these Riccati equations, including explicit formulas and conditions for their completeness.

Findings

Multiple solution families exist for these Riccati equations.

Explicit formulas for solution families are derived.

Conditions are identified under which all solutions are covered.

Abstract

We consider Homogeneous Algebraic Riccati Equations in the general situation when the matrix of the dynamics can be "mixed". We show that in this case the equation may have infinitely many families of solutions. An analysis of these families is carried over and explicit formulas are derived. We also derive sufficient conditions under which the union of the families covers the whole set of solutions.