Page 15 - Textos de Matemática Vol. 36

1.1. Riccati-transformation 5
The Lyapunov–Riccati transformation x → UTx decouples the system (1.1) completely and is sometimes called an LK-transformation (see [Ande69]).
In the case where A11, A12, A21, A22 are constant, we may choose a particu- lar constant solution K0 of (1.3), if it exists, (see Chapter 2), i.e., a solution of the algebraic Riccati equation ARE associated with (1.3). We then have K˙ 0 = 0, T˙ = 0, hence
A˜(K0,t) = TAT−1.
This yields the spectral factorization of the operator A. Since the spectrum σ(A)
of A coincides with that of A˜, we obtain from (1.5)
σ(A) = σ(A11 − A12K0) ∪ σ(K0A12 + A22), (1.9)
for any constant solution K0 of (1.3). In particular it is possible to choose n, m and the constant solution K0 of (1.3) such that the spectrum of A11 − A12K0 and of K0A12 + A22 consists of the desired parts of the spectrum of A. Spectral factorization problems became popular in engineering literature starting with the celebrated paper of Youla [Youl61] and with [Popo73], which showed the connection between linear quadratic control, spectral factorizations and stabi- lizing solutions to Hermitian Riccati equations. For instance, a factorization problem leading to the non-symmetric algebraic Riccati equation
K(A+ − b+cT+) + (AT− − c−bT−)K − Kb+bT−K − c−cT+ = 0,
was studied in [ClAn76]. A similar important role is played by Riccati equa- tions in more general factorization problems; Chapter 5 of the book by Bart, Gohberg and Kaashoek [BGK79] as well as the influential paper by Bart et. al. [BGKVD80] contains further information on the connection between factoriza- tion problems and algebraic Riccati equations. The work of Fuhrmann [Fuhr89] describes factorization theory from a polynomial point of view and [Fuhr95] is concerned with the characterization and parameterization of minimal spectral factors. J-unitary minimal factors of rational matrix–valued functions are also obtained by Dym [Dym01a], by exploiting the connections between the Her- mitian solutions of matrix Riccati equations and a class of finite dimensional reproducing kernel Krein spaces. Let us also mention that symmetric (Hermi- tian) and non-symmetric algebraic Riccati equations are in particular closely related to the factorization of matrix polynomials and to the associated Be- zoutians ; for details on this topic we refer to [LeRa97], [KaLe01b], [LeRo99a], [KLR01].
Investigating the existence of solutions to the infinite time–horizon quadratic optimal problem, Staffans [Staf98]–[Staf99] establishes a connection between co- prime factorizations and symmetric (Hermitian) Riccati equations for a certain class of linear infinite-dimensional systems, satisfying a regularity condition. Op- erator Riccati equations also play similarly an important role in connection with the block diagonalization of linear operators acting in Hilbert and Krein spaces