Extreme Points of Riccati Inequalities

Abstruct-Relations between solutions of the algebraic Riccati equation and the associated quadratic matrix inequalities are discussed and ex-plained. The purpose of this note is to clarify some of the relations that exist between solutions of the algebraic Riccati equation and the associated quadratic matrix inequalities. In particular, the main result of the note is to establish that there are extremepoints in the solutions set of quadratic matrix inequalities that are not solutions of the algebraic Riccati equa-tion. The history of this result is typical of many results involving Riccati equations in the engineering literature-total confusion. It has been part of the folklore for many years that the solutions of the algebraic Riccati equation are extreme points of the above-mentioned quadratic matrix inequalities. In Badawi’s thesis [l] a very elegant proof is given, however, in review it was discovered that there had in fact appeared a proof in the literature. In Faurre et nl. [3] there is indeed a proof and a footnote to th