Albrecht Böttcher: The algebraic Riccati equation with Toeplitz matrices as coefficients

Author(s):

Albrecht Böttcher

Title:

The algebraic Riccati equation with Toeplitz matrices as coefficients

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 02, 2011

Mathematics Subject Classification:

47N70	[Applications in systems theory, circuits, and control theory]
15A24	[Matrix equations and identities]
15B05	[Toeplitz, Cauchy, and related matrices]
46L89	[Other "noncommutative" mathematics based on C*-algebra theory]
47B35	[Toeplitz operators, Hankel operators, Wiener-Hopf operators]
93C15	[Systems governed by ordinary differential equations]

Abstract:

It is shown that, under appropriate assumptions, the continuous algebraic Riccati equation with Toeplitz matrices as coefficients has Toeplitz-like solutions. Both infinite and sequences of finite Toeplitz matrices are considered, and also studied is the finite section method, which consists in approximating infinite systems by large finite truncations. The results are proved by translating the problem into C*-algebraic language and by using theorems on the Riccati equation in general C*-algebras. The paper may serve as another illustration of the usefulness of C*- algebra techniques in matrix theory.

Keywords:

algebraic Riccati equation, Toeplitz matrix, C*-algebra

Language:

English

Publication time:

01/2011