The algebraic riccati equation and its role in indefinite inner product spaces

Abstract

In this essay algebraic Riccati equations will be discussed. It turns out that Hermitian solutions of algebraic Riccati equations which originate from systems and control theory may be studied in terms of invariant Lagrangian subspaces of matrices which are selfadjoint in an indefinite inner product. The essay will describe briefly certain problems in systems and control theory where the algebraic Riccati equation plays a role. The focus in the main part of the essay will be on those aspects of the theory of matrices in indefinite inner product spaces that were motivated and largely influenced by the connection with the study of Hermitian solutions of algebraic Riccati equations. This includes the description of uniqueness and stability of invariant Lagrangian subspaces and of invariant maximal semidefinite subspaces of matrices that are selfadjoint in the indefinite inner product, which leads to the concept of the sign condition. Also, it is described how the inertia of solutions of a special type of algebraic Riccati equation may be described completely in terms of the invariant Lagrangian subspaces connected with the solutions.