Page 22 - Textos de Matemática Vol. 36

12 Chapter 2. Global aspects Here we show how the solutions of algebraic Riccati equations are related to
certain n-dimensional M-invariant subspaces.
Definition 2.1.1. (i) Throughout this lecture we denote by J the 2n × 2n
matrix
0 −In J= In 0 .
Obviously J∗ = J−1 = −J and J2 = −I2n.
(ii) The linear system L of differential equations corresponding to HRDE is
˙ A(t) −S(t) In Y = −Q(t) −A∗(t) Y, Y(t0)= W ,
(2.1)
(2.2)
and has the Hamiltonian (see (iii)) system matrix
A −S H(t) = −Q −A∗
(iii) A matrix H ∈ C2n×2n is called Hamiltonian if JH = (JH)∗,
(t).
or, equivalently,
This means that every Hamiltonian matrix has the form (2.2) with Q = Q∗
JH + H∗J = 0. A matrix S ∈ C2n×2n is called symplectic if
and S = S∗.
S∗JS = J. (2.5)
(iii) By LARE and LHARE we denote the sets of all solutions of ARE and HARE, respectively; further we denote by HHARE the set of all Hermitian solutions of HARE.
(iv) Every n-dimensional subspace
G(X) := Im X ⊂ C
is called a graph subspace of X.
Theorem 2.1.2. W ↔ G(W ) defines a bijective map between LARE and the set of all n-dimensional, M-invariant graph subspaces of Cn+m. Moreover, if M fulfills ARE,
In In
M W = W (M11 +M12W). (2.6)
In n+m
0
(2.3) (2.4)